COMPUTATIONAL DEVELOPMENTS FOR AB INITIO MANY-BODY THEORY

By

Justin Gage Lietz

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics - Doctor of Philosophy

Computational Math, Science and Engineering - Dual Major

2019

COMPUTATIONAL DEVELOPMENTS FOR AB INITIO MANY-BODY

ABSTRACT

THEORY

By

Justin Gage Lietz

Quantum many-body physics is the body of knowledge which studies systems of many

interacting particles and the mathematical framework for calculating properties of these

systems. Methods in many-body physics which use a first principles approach to solving

the many-body Schr¨odinger equation are referred to as ab initio methods, and provide ap-

proximate solutions which are systematically improvable. Coupled cluster theory is an ab

initio quantum many-body method which has been shown to provide accurate calculations of

ground state energies for a wide range of systems in quantum chemistry and nuclear physics.

Calculations of physical properties using ab initio many-body methods can be computa-

tionally expensive, requiring the development of efficient data structures, algorithms and

techniques in high-performance computing to achieve numerical accuracy.

Many physical systems of interest are difficult or impossible to measure experimentally,

and so are reliant on predictive and accurate calculations from many-body theory. Neutron

stars in particular are difficult to collect observational data for, but simulations of infinite

nuclear matter can provide key insights to the internal structure of these astronomical ob-

jects. The main focus of this thesis is the development of a large and versatile coupled cluster

program which implements a sparse tensor storage scheme and efficient tensor contraction

algorithms. A distributed memory data structure for these large, sparse tensors is used so

that the code can run in a high-performance computing setting, and can thus handle the

computational challenges of infinite nuclear matter calculations using large basis sets. By

validating these data structures and algorithms in the context of coupled cluster theory and

infinite nuclear matter, they can be applied to a wide range of many-body methods and

physical systems.

Dedicated to my loving parents.

iv

ACKNOWLEDGMENTS

This dissertation is the culmination of six years of graduate school, all of which has been

under the guidance of my PhD advisor, Morten Hjorth-Jensen.

I am incredibly lucky to

have been on this journey with Morten, as he is not only an excellent teacher, scientist, and

mentor, but he is also a wonderful person who is overflowing with kindness. It is hard to

imagine what this dissertation would look like if I had never met Morten, as his influence is

in every one of its pages. My thoughts surrounding physics, research, and philosophy have

been profoundly shaped by Morten and that extends much beyond just this work. Thank

you so much Morten, while this dissertation marks the end of my time as your student, I

know it also marks the beginning of a lifelong friendship.

Next, I would like to thank Scott Bogner and Heiko Hergert who also have had a huge

impact on this dissertation by creating an intellectually stimulating and friendly environment

for the nuclear many-body group. It has been a pleasure to share this journey with you and

with the other grad students in the group: Sam Novario, Nathan Parzuchowski, Titus Morris

and Fei Yuan. The times I got to travel to summer schools with all of you and explore new

cities are some of my favorite memories of grad school.

Outside of those who directly contributed to this dissertion, I also want to thank those

who have made a positive impact on my life while I was working on it. First I want to thank

my girlfriend, Rachel, for all of your love and support. You helped make my time in grad

school the best years of my life, and I can’t wait for our next adventure together. I am also

grateful for the time shared with my other grad student friends, and in particular the great

times I got to spend with Dennis, Terri, Michael, and Brent in the last few months. It was

tough to push through at the end of grad school, but getting to spend time with you all

v

helped immensely.

Next I would like to thank my brother, Jordan who has been my friend and role model

for my entire life and continues to be to this day, and my sister-in-law Sarah who can bring

a smile to anyone in my family. You guys are the best, and I can’t wait for many more visits

to see you, Annabelle, and Ollie. Lastly, and certainly most importantly, I have to give

thanks to my parents, Kim and Steve, to whom this dissertation is dedicated to. They have

provided unconditional love and selfless support so that I could pursue what I love doing,

and I am incredibly grateful for everything they have done for me.

Thank you so much mom and dad.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

KEY TO ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Nuclear Theory as a Window into the the Stars
. . . . . . . . . . . . . . . .
1.2 Ab initio methods in Nuclear Theory . . . . . . . . . . . . . . . . . . . . . .
1.3 Quantum Many-Body Methods
. . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Infinite Matter Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Computational Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Quantum Many-Body Physics . . . . . . . . . . . . . . . . . . . .
2.1 Bra-ket Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Many-fermion wave functions and spaces . . . . . . . . . . . . . . . . . . . .
2.3 Occupation Number Representation . . . . . . . . . . . . . . . . . . . . . . .
2.4 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . .
2.5 Number Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Anti-commutation relations
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Operators in Second Quantized Form . . . . . . . . . . . . . . . . . . . . . .
2.8 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Generalized Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Slater-Condon Rules
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 The Fermi Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Configuration Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.13 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14 Many-Body Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . .
2.15 In-Medium Similarity Renormalization Group . . . . . . . . . . . . . . . . .
2.16 The Magnus Formulation of IM-SRG . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Physical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Pairing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Single-Particle Basis for Infinite Fermionic Matter . . . . . . . . . . . . . . .
3.3 Two-Nucleon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Homogeneous Electron Gas
. . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Coupled Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Prologue to Coupled Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
4
5
6
8

9
11
14
21
23
26
28
29
33
38
38
44
50
53
58
65
72

84
85
89
93
97

99
99

vii

4.2 Coupled Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.3 Coupled Cluster Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.4 Diagrammatic Derivation of the Coupled Cluster Equations . . . . . . . . . . 117
4.5 Computational Scaling of Coupled Cluster Theory . . . . . . . . . . . . . . . 127

Infinite Neutron Matter

Chapter 5 Computational Methodology . . . . . . . . . . . . . . . . . . . . . 132
5.1 Code Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.1.1 Pairing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
. . . . . . . . . . . . . . . . . . . . . . . . . 135
5.1.2
5.2 Taming the Two-Body Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3 Performance Testing Matrix-Matrix Multiplication . . . . . . . . . . . . . . . 141
5.4 Tensor Contractions as Matrix Multiplication . . . . . . . . . . . . . . . . . 145
5.5 Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.6 Distributed Memory Parallelization . . . . . . . . . . . . . . . . . . . . . . . 156
5.7 Final Parallel Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Chapter 6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1 Neutron Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2 Homogeneous Electron Gas
. . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Chapter 7 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . 184

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

viii

LIST OF TABLES

Table 2.1:

Single-Particle Index Conventions

. . . . . . . . . . . . . . . . . . .

45

Table 3.1:

Single-particle states and their quantum numbers and their energies
from Eq. (3.3). The degeneracy for every quantum number p is equal
. . . . . . . . . . . . . .
to two due to the two possible spin values.

Table 3.2:

y + n2

Total number of particle filling N↑↓ for various n2

z values
for one spin-1/2 fermion species. Borrowing from nuclear shell-model
terminology, filled shells correspond to all single-particle states for
one n2
z value being occupied. For matter with both protons
and neutrons, the filling degree increased with a factor of 2. . . . . .

x + n2

x + n2

y + n2

86

92

Table 3.3:

Parameters used to define the Minnesota interaction model [53].

. .

97

Table 4.1:

Sign table for the four terms of eq. (4.37). . . . . . . . . . . . . . . . 115

Table 5.1:

Coupled cluster and MBPT2 results for the simple pairing model with
eight single-particle levels and four spin-1/2 fermions for different
values of the interaction strength g.

. . . . . . . . . . . . . . . . . . 134

Table 5.2:

Table 5.3:

CCD and MBPT2 results for infinite neutron matter with N = 66
neutrons and a maximum number of single-particle states constrained
by Nmax = 36 (36 plane wave energy shells).

. . . . . . . . . . . . . 136

All possible particle-hole sectors are listed in the left column. In the
right column are 6 particle hole sectors which contain all of the infor-
mation of the whole matrix, plus how the other 10 can be equivalently
expressed.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Table 5.4:

A straight forward scheme to organize the two-body basis in columns. 139

ix

LIST OF FIGURES

Figure 2.1:

SRG with direct integration and with the Magnus expansion.

. . . .

81

Figure 2.2: Magnus SRG with the exact unitary transformation and with a BCH
expansion truncated after a fixed number of terms . . . . . . . . . .

Figure 2.3: Magnus SRG with the exact unitary transformation and with a BCH
. . . . . . . . . .

expansion truncated after a fixed tolerance is met

Figure 3.1: Correlation energy for the pairing model with exact diagonalization,
MBPT2 and perturbation theory to third order MBPT3 for a range of
interaction values. A canonical Hartree-Fock basis has been employed
in all MBPT calculations. . . . . . . . . . . . . . . . . . . . . . . . .

82

83

88

Figure 5.1: Correlation energy for the pairing model with exact diagonalization,
CCD and perturbation theory to third (MBPT3) and fourth order
(MBPT4) for a range of interaction values.

. . . . . . . . . . . . . . 135

Figure 5.2: The pp-pp sector of a two-body interaction matrix for a simple neu-
tron matter system with 40 single-particle states above the Fermi
level.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 5.3:

Implementation of the same mathematics can have very different run
times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Figure 5.4: MBPT2 contribution to the correlation for pure neutron matter with
N = 14 neutrons and periodic boundary conditions. Up to approx-
imately 1600 single-particle states have been included in the sums
over intermediate states in Eqs. (5.7) and (5.8) . . . . . . . . . . . . 148

Figure 5.5: A cartoon of how the interaction matrix might be split into work
loads for different threads of execution for the naive storage and the
block storage schemes.

. . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.6: Performance of MBPT2 calculations with increasing number of MPI
ranks. The speed of the calculation is measured in s−1, the black
data are the inverse time required to finish the calculation on the
fastest rank, and the red data are the speeds of the slowest rank.

. . 157

x

Figure 5.7: The blocks are distributed to the ranks to try and keep the num-
ber of non-zero matrix elements equal among ranks. The histogram
shows that the time it takes to load these blocks is dominated by an
enormous amount of small blocks, which is ideal for load balancing.

159

Figure 5.8: The t-amplitudes are permuted as needed for the tensor contractions
which are not aligned as a matrix-matrix product. In the histogram,
the larger blocks have begun to take more of the total time relative
to the previous step. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 5.9: The largest tensor contractions are now performed, which are already
aligned as matrix-matrix products. The O(N 3) scaling of the matrix-
matrix product causes the larger matrices to contribute significantly
to the total processing time in this step. . . . . . . . . . . . . . . . . 161

Figure 5.10: The t-amplitudes are summed together and the correlation energy
is calculated. If the energy has not converged to the set tolerance,
another iteration of the CC equations are performed, using these new
t-amplitudes in step 2.

. . . . . . . . . . . . . . . . . . . . . . . . . 161

Figure 6.1: Two different energy per particle plots at low densities of neutron
matter with the Minnesota potential [53] computed in the CCD ap-
proximation with 54 neutrons and an Nmax = 100 truncation (100
plane-wave energy shells), corresponding to 10754 single particle states.165

Figure 6.2: Energy per particle of pure neutron matter computed in the CCD

approximation with the Minnesota interaction model [53]. . . . . . . 166

Figure 6.3: The relative error shows how much the CCD correlation energy is
changing between subsequent calculations at different model spaces
sizes ranging from Nmax = 10 to 70 for neutron matter with the Min-
nesota potential at density 0.2 fm−3.

. . . . . . . . . . . . . . . . . 167

Figure 6.4: Finite-size effects in different energies of pure neutron matter com-
puted with the Minnesota interaction model [53] as a function of the
number of particles for both periodic boundary conditions (PBC) and
twist-averaged boundary conditions (TABC5).

. . . . . . . . . . . . 169

Figure 6.5: Energy per particle for pure neutron matter with the Minnesota po-
tential [53]. Here the calculations have been performed with IM-
SRG(2), CCD, CIMC [77], and the ADC(3) Self-Consistent Green’s
Function scheme [77].

. . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 6.6: The CCD energy per particle for the homogeneous electron gas for a

range of Wigner-Seitz Radii with A = 14 electrons . . . . . . . . . . 172

xi

Figure 6.7: Contributions to the energy from purely CCD many-body correlations.173

Figure 6.8: Fractional contribution to the energy from the Hartree-Fock reference

state.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Figure 6.9: The relative error shows how much the CCD correlation energy is
changing between model spaces sizes ranging from Nmax = 10 to 60
for the electron gas at rs = 0.5. . . . . . . . . . . . . . . . . . . . . . 174

Figure 6.10: The relative error shows how much the CCD correlation energy is
changing between model spaces sizes ranging from Nmax = 10 to 60
for the electron gas at rs = 0.1. . . . . . . . . . . . . . . . . . . . . . 175

Figure 6.11: The relative error of the CCD correlation energy is changing between
model spaces sizes ranging from Nmax = 100 to 200 for the neutron
matter with the Minnesota potential at A = 54 and ρ = 0.08. . . . . 176

Figure 6.12: The time required for the large basis set Minnesota potential calcu-

lations.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Figure 6.13: Strong scaling of distributed memory code, dark green line shows

ideal case.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 6.14: Weak scaling of distributed memory code, the dark green line shows

the ideal case.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6.15: Number of tensor elements required for the 3-body force in the infinite

matter basis with and without block-diagonal compression.

. . . . . 181

Figure 6.16: Size of tensor in gigabytes required for the 3-body force in the infinite

matter basis with and without block-diagonal compression.

. . . . . 182

Figure 6.17: On node timing tests for the tensor contraction of three-body force

diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

xii

KEY TO ABBREVIATIONS

ˆ NSCL - National Superconducting Cyclotron Laboratory (East Lansing MI, USA)

ˆ CC - Coupled Cluster

ˆ CC(S)(D)(T) - Coupled Cluster with (Singles)(Doubles)(Triples)

ˆ IM-SRG - In-Medium Similarity Renormalization Group

ˆ EoS - Equation of State

ˆ d.o.f. - Degree of Freedom

ˆ QCD - Quantum Chromodynamics

ˆ LQCD - Lattice QCD

ˆ GPU - Graphics Processing Unit

ˆ χ-EFT - Chiral Effective Field Theory

ˆ FCI - Full Configuration Interation

ˆ CISD - Configuration Interaction Singles and Doubles

ˆ CIMC - Configuration Interaction Monte Carlo

ˆ MBPT - Many-Body Perturbation Theory

ˆ BCH - Baker - Campbell - Hausdorff

ˆ HEG - Homogeneous Electron Gas

ˆ BLAS - Basic Linear Algebra Subroutines

ˆ cuBLAS - CUDA BLAS

ˆ OMP - Open Multi-Processing

ˆ MPI - Message Passing Interface

ˆ ADC - Algebraic Diagrammatic Construction

xiii

Chapter 1

Introduction

1.1 Nuclear Theory as a Window into the the Stars

Stars are objects of extreme forces. Spheres of crushing gravitational forces being held up

by the violent nuclear reactions at their core. Until they aren’t. Eventually the nuclear fuel

at the core of every star depletes, giving way to gravitational contraction, and in some cases

cataclysmic collapse. Many bright burning stars will collapse onto their own cores, creating

one of the brightest events in the galaxy, a supernova. The extreme explosions of some dying

stars blast off a significant fraction of their total material, and leave behind a spent core.

If the resulting core is large enough, the gravitational compression will crush the remaining

matter to extreme densities. In the cases that are not quite large enough for a black hole to

form, a neutron star remains at the center of a once bright burning star. Neutron stars get

their name because the extreme densities caused by the gravitational collapse have pushed

beyond the limits of electron degeneracy, collapsing the bulk of protons and electrons into

neutrons. In a region around 1-3 solar masses, the resulting neutron degeneracy pressure

together with the very close range nuclear force are enough to push back against further

gravitational collapse, forming an incredible astronomical object that is composed of a very

unique state of matter. Neutron stars tend be around one solar mass, but are only about

10km in size, as the entire macroscopic object is around nuclear density. After this bright

supernova, the resulting neutron star core is left cold and dim. This makes direct observation

1

of these fascinating objects very difficult, and can only rarely be done. In the case of pulsars,

direct electromagnetic radiation can be detected, however it is only from a directed beam

out of the magnetic axis of the star. Otherwise, indirect measurements of nearby stars must

be done. Looking for the gravitational footprint on the orbits of stars, such as a bright star

in a binary system with a neutron star can determine with some accuracy the mass of the

binary neutron star partner. A direct astronomical measurement of the neutron star radius

might not be possible, as they are just too dim and too far away. Since telescopes are largely

unable to see neutron stars, it is up to theoretical physics to help paint a picture of these

extreme objects.

To figure out how neutron stars respond to gravitational compression, and how they

eventually equilibrate to some radius, the equation of state (EoS) of the neutron star must

be known. Currently, there are many proposed equations of state [1, 2], which lead to a large

spread of possible radii given a particular observed mass. This is due to the difficult nature

of calculating the nuclear EoS, which in principle requires knowing the exact composition of

a neutron star, and calculating the energetic state of this quantum system. A large part of

a neutron star is thought to be pure neutron matter, although some amount of proton and

lepton matter is likely present in a state of β-stable matter. Some theories posit that the

extreme densities towards the core of the neutron star could cause the formation of hyperons

in the nuclear matter, or that the matter could be pushed into a state of pure quarks [1].

Regardless of the composition, the resulting calculation is a difficult problem of quantum

many-body physics. A comprehensive treatment of this problem would involve calculating

the strong interactions of an absurd number of particles, and many attempts have been made

using a slew of different approximations. Regardless of the exact framework, it seems that

theoretical nuclear physics, the study of some of the smallest particles, could be our best

2

tool to determine properties of these massive celestial bodies.

1.2 Ab initio methods in Nuclear Theory

Nuclear theory as a field is currently in an exciting period of growth due to theoretical and

computational developments in ab initio methods over the last two decades. Ab initio, latin

for “from the beginning”, is a phrase used to describe work done from first principles, which

for nuclear theory means starting from the building blocks of the atomic nucleus: protons

and neutrons. While nucleons are composite particles made up of quarks and gluons, they are

well bound (on the order of GeV) compared to the interactions between them (MeV) so they

function well as the basic degrees of freedom (d.o.f). The “hard core” nature of the nucleon-

nucleon force has made the ab initio approach to calculating nuclear properties intractable in

the past, as this leads to the coupling of high and low momentum modes, creating difficulties

in calculating all but the smallest nuclei. Many strategies have been used to evade this

problem, perhaps most notable are the phenomonological models used to great effect with

shell model (SM) calculations. By developing nuclear interactions using nuclear data input

near regions of interest, high accuracy calculations of properties of nuclei have been made [3].

In this approach however, many nucleonic d.o.f.’s are “frozen out”, meaning that they are

ignored and some contact with the underlying physics is lost. Another approach has been

the development of similarity renormalization group (SRG) methods, which have lead to the

proper decoupling of these high and low momentum modes, leading to softer interactions that

are able to converge much faster, and make many more calculations possible [4, 5]. Potentials

generated from chiral effective field theory (χ-EFT) [6, 7], which connect the nuclear force

to the underlying symmetries of the QCD langrangian, can now be softened with SRG [4, 5]

3

leading to a class of potentials that have connections to the underlying physics, and are

tractable for calculations of nuclei. This progress has allowed properties of medium mass

nuclei to be calculated with links to fundamental forces at modest computational cost. This

has opened a whole new way of studying nuclei, as now methods that keep track of all of

the nucleonic d.o.f.’s can be used in realistic calculations.

1.3 Quantum Many-Body Methods

There are many ways of taking on realistic calculations of nuclear properties, and at

the core of many approaches is the non-relativistic many-body Schr¨odinger equation. Low

energy nuclear physicists can get away without using relativistic quantum mechanics because

the typical binding energies of nuclei is in the range of 1 MeV to 10 MeV per nucleon,

while each nucleon itself is bound together at around 1 GeV. Even with this non-relativistic

approximation, the task of calculating any many-body problem is daunting. The second

approximation that must be done is to pick a finite basis to perform the calculation in.

These basis sets are in principle infinite, but we must have a finite system for our finite

computers. Within this framework, the task of a complete energy spectrum calculation of

N particles in a basis with M single-particle states, would require diagonalizing an(cid:0)M
(cid:1) sized matrix where(cid:0)M
(cid:0)M

N !(M−N )! is a factorially growing number. This means that
for all but small systems in small basis sets, this factorial growth will quickly grow beyond

(cid:1) =

N

(cid:1) by

N

N

M !

current computational power. Full configuration interaction (FCI) is a method which uses

a minimal number of approximations, and computes a nearly exact energy of the system,

but at a massive computational cost [8, 9, 10, 11]. This unfavorable scaling has led to the

development of an entire industry of approximations to solving the full many-body problem.

4

Each method approaches the many-body problem in its own way, with its own series of

advantages and disadvantages. In particular, coupled cluster theory [12, 13, 14, 15, 16] has

been in use in many-body theory since the 60’s, starting with the work of Coester [17, 18]

and Kummel [19], and saw enormous success in quantum chemistry, and more recently

nuclear theory as well [20, 21]. Coupled cluster theory is centered around a way to organize

the many-body basis by grouping states in excitation “clusters” that lead to very favorable

truncations of less important terms. Truncations are almost always needed in practical

calculations, but by restoring the truncations term by term we can systematically improve

the solution and eventually restore the exact FCI answer. With this improved many-body

basis truncation, coupled cluster theory has a favorable polynomial scaling, and sacrifices

only a small amount of accuracy.

In quantum chemistry, CC theory has been used to

calculate molecular properties to chemical accuracy at a fraction of the computational cost

of a total FCI calculation. In nuclear physics, coupled cluster’s many-body truncation errors

are typically minimal when compared to errors in the approximation of the nuclear forces.

With this success, CC has shown to be one of the premier ab initio many-body methods in

nuclear physics.

1.4

Infinite Matter Calculations

This thesis focuses on ab initio many-body calculations relevant for neutron stars. The

ultimate goal is to learn more about neutron stars while still maintaining a link to the

underlying theory of the strong force, quantum chromodynamics, while at the same time

studying the tools and approximations needed to make this possible. Simulating an entire

star at the quantum level is an impossible task. However, by studying a small periodic chunk

5

of neutron star matter, we can extract properties like the equation of state while maintaining

these important links[1].

This idea of studying a periodic box of quantum matter is not a new idea, and the so

called “infinite matter” problem is an often revisited system for many-body theorists [22].

By simulating a finite number of particles in a periodic box, an approximation to an infinite

space filled with these particles can be made. Infinite matter is studied in quantum chemistry

[23] to simulate electrons moving in a neverending lattice of atoms, and in nuclear physics

it can be studied as a large block of neutron star matter [2].

Many-body physicists in general study the infinite matter problem as a sandbox to ex-

amine their theoretical machinery. The periodic box in which infinite matter is frequently

studied leads to a natural choice of basis, that of plane waves. The flat periodic boundaries

that are chosen lead to a quantization of momentum modes, which give a basis of fixed

momenta waves moving through the box. Despite being a theoretically convenient basis, the

plane waves can be computationally challenging, as realistic simulations of neutron matter

can need hundreds of particles and thousands of basis states to converge to a point that

resembles a true infinite slab of matter.

1.5 Computational Challenges

Despite the polynomial scaling, ab initio methods like coupled cluster theory can begin

to struggle with the computational load that thousands of basis states require. It is here

that a deep inspection of the many-body tools in use must be done. Approximations in

coupled cluster theory organize the many-body basis in terms that (typically) decrease in

importance for higher excitation levels. For infinite matter calculations, the first non-zero

6

terms are that of doubles excitations, that is, exciting pairs of particles together, in what is

called coupled cluster doubles (CCD). Even in this restrictive approximation, storage of the

two-body interaction matrix scales as M 4, where the number of single-particle basis states

M can quickly grow to 103, leading to matrix sizes of 1012 elements, which is already too

large for modern computers to deal with. While these matrices are very large, by exploiting

the sparsity of this many-body basis, these matrices can be tamed to sizes that can run on a

single large computer, or a small cluster. However, studies of modern nuclear potentials have

shown that three-body forces are necessary for accurate calculations, leading to a three-body

interaction matrix that scales as M 6. Furthermore, CCD alone is often too restrictive of an

approximation, and either partial or full inclusion of triples excitations (CCDT) is necessary

for modern state-of-the-art calculations of infinite nuclear matter. These complications mean

that tools from high-performance computing are necessary to meet the precision and accuracy

demands of ab initio many-body theory. First, to even store the three-body interaction

matrix which can quickly grow to hundreds of terabytes (TB) in size, distributed memory

algorithms must be used. Only by distributing these large interaction matrices across a

computational cluster can the calculation even be started. Next, the floating point operations

(FLOPs) required by a CCDT calculation with three-body forces will scale as M 9, meaning

that massively parallel algorithms are needed to get these calculations finished in a reasonable

amount of time. This massively parallel paradigm of supercomputing usually means writing

custom code for the architecture of the computer that will be used, and in the case of the

current largest computers, this means leveraging the enormous power of graphics processing

units (GPUs) to get the job done. Along with a growing demand for computational power is

a growing problem with reproducible science and portable codes. To maximize the scientific

effort of computational physicists, research that is maintainable, extensible and reproducible

7

needs code that is well tested, well documented, well designed and version controlled.

1.6 Thesis Overview

This thesis will review the basics of quantum mechanics and many-body theory as it

pertains to coupled cluster calculations in Chapter 2 to establish the language and notation

needed for the subsequent chapters. To establish the context of the field, we will go through

the relevant theory derivations, and Chapter 3 will describe the quantum systems which will

be tackled with these methods. A focus will be on coupled cluster theory, and Chapter 4 will

show how to derive CC theory using the development of diagrammatic techniques. Once this

foundation is laid, we dive into the main object of this thesis: a large and versatile computer

program which can calculate properties of a variety of quantum systems using a variety

of many-body methods. In Chapter 5 we describe the distributed memory data stuctures

and algorithms that implement these many-body methods efficiently in a high performance

computing setting. Lastly, the numerical results and performance testing of the program for

infinite matter calculations are discussed in Chapter 6.

8

Chapter 2

Quantum Many-Body Physics

This work will focus on ab initio calculations of many interacting particles where ab

initio, meaning “from the beginning”, refers to the fact that we want these calculations to

be as fundamental as possible. By starting from the basic building blocks, we can make

accurate predictions of properties over a wide range of systems. Choosing these degrees of

freedom is a game of compromise. Degrees of freedom that are too “macroscopic” will have

limited applicability, but degrees of freedom that are too “microscopic” yield a wide range of

applicability, but coupling the very small scales up to the large scales of the system size can

become computationally impossible. The target systems of nuclear physics frequently land in

a regime where using the constituent protons and neutrons as the degrees of freedom can be

too microscopic, as calculating properties of nuclei rapidly grows too complex. Historically,

phenomenological models, like the shell model, have had success by generating effective

interactions for a few valence nucleons on top of an inert closed shell core. Accurate properties

can be computed relatively quickly by “freezing out” the nucleon degrees of freedom in the

closed shell. However, the shell model interactions rely on known experimental data in the

region of interest to fit the matrix elements, and so any given shell model interaction can

only function well in this limited space. Extrapolating into regions where there is little

experimental data is very challenging, as the interaction was only tuned to the region of

interest.

In the other extreme, the nucleons that make up nuclei are themselves composite particles

9

of quarks and gluons. So a truly fundamental calculation would build up the properties with

all quark and gluon degrees of freedom active. Such calculations are done in the field of

lattice quantum chromodynamics (LQCD) [24, 25], but due to the extreme microscopic

nature of these degrees of freedom they quickly become overwhelmingly difficult, leaving

only the smallest nuclear systems accessible this way .

This work operates in between these regimes, where the quarks and gluons are frozen

out, but all of the protons and neutrons remain active in the calculation. As a trade off,

the interactions between nucleons are generated from chiral effective field theory (χ-EFT)

[6, 7], which bring the symmetries from the fundamental QCD Lagrangian. While these

calculations are expensive, truncations are made to the possible configurations of nucleons

to make them feasible, and techniques in high performance computing allow relatively large

systems to be calculated. In principle, interactions fit once to few nucleon data have much

larger ranges of applicability, and would accurately compute properties from small to medium

mass nuclei to infinite nuclear matter.

In reality, the current state of the art predective

models need additional data like the binding energy and radii of small to medium mass

nuclei [26, 27]. In this sense, the philosophy of ab initio quantum many-body physics is to

keep all nucleons active, and while truncations are made, there is a systematically improvable

scheme for both the interaction and the many-body correlations. To provide the foundation

to make these statements concrete, this chapter will briefly walk through single-particle

quantum mechanics and then survey a few quantum many-body techniques.

10

2.1 Bra-ket Notation

A concise formalism for describing quantum states is bra-ket notation, also called Dirac

notation. In bra-ket notation, a quantum state is represented as an abstract ket, |ψ(cid:105). This
notation distinguishes itself from wave mechanics where the state is written explicitly as

ψ(x), a function of R space, or matrix mechanics where the state is expanded in some

orthonormal basis, and referenced as a set of basis components (c0, c1, . . . ). The abstract

nature of the ket allows for a formalism where derivations and manipulations can be done in

an invariant way, and a choice of coordinates or a basis can be chosen at any point where it

is convenient.

There are many texts covering the formalism of bra-ket notation, so we’ll just look at a

few interesting pieces to have footing for later discussions. In bra-ket formalism, the quantum

state |ψ(cid:105) ∈ H is an element of state space, which is a abstract complex Hilbert space that is
infinite dimensional and separable (i.e. can have a countable orthonormal basis).

One property of Hilbert spaces that is central to quantum mechanics is that they are

closed under linear combinations. As a consequence, superpositions of states are themselves

states in the space

|ψ(cid:105) = c1 |ψ1(cid:105) + c2 |ψ2(cid:105) .

(2.1)

Additionally, Hilbert spaces come with an inner product IP : (H,H) → C which can be

written many different ways. However the last notation written below using the angled

brackets is where bra-ket notation gets its name

IP(|φ(cid:105) ,|ψ(cid:105)) = (cid:104)|φ(cid:105) ,|ψ(cid:105)(cid:105) ≡ (cid:104)φ|ψ(cid:105) = (cid:104)ψ|φ(cid:105)∗ .

(2.2)

11

This leads to the definition of the bra state (cid:104)φ|. For any state |φ(cid:105) ∈ H, we can associate a
linear functional (cid:104)φ| ≡ f|φ(cid:105) : H → C, where for |ψ(cid:105) ∈ H

f|φ(cid:105)(|ψ(cid:105)) = (cid:104)φ| (|ψ(cid:105)) = IP(|φ(cid:105) ,|ψ(cid:105)) = (cid:104)φ|ψ(cid:105) .

(2.3)

Only the bra-ket notation of the inner product will be used from now on, the function

argument style was just to draw attention to the fact that these bras are linear maps from

the Hilbert space to the complex numbers. The Hermitian conjugate (conjugate transpose)

is used to go from a ket state to the correspoding bra state,

|ψ(cid:105)† = (cid:104)ψ|

(cid:104)ψ|† = |ψ(cid:105) .

(2.4)

A distinct advantage to this notation is now the projection operator Pψ onto the state |ψ(cid:105)
is compactly written as

Pψ = |ψ(cid:105)(cid:104)ψ| .

(2.5)

A set of ket vectors is considered an orthonormal basis {|ψi(cid:105) ≡ |i(cid:105)}i∈N if they satisfy the
orthonormality relation

and the completeness relation

(cid:104)i|j(cid:105) = δij,∀i, j ∈ N,
(cid:88)
i∈N|i(cid:105)(cid:104)i| = 1H.

(2.6)

(2.7)

From here, we can represent the quantum state in any basis of our choosing, by applying

12

the completeness relation (2.7) above on any ket |ψ(cid:105)

∞(cid:88)

i=0

|ψ(cid:105) =

|i(cid:105)(cid:104)i|ψ(cid:105) .

(2.8)

Next, we introduce linear operators on H. An operator A maps a ket |ψ(cid:105) into a new ket
A|ψ(cid:105) = |Aψ(cid:105) ∈ H and (cid:104)ψ| A = (cid:104)A†ψ|, where A† is the adjoint of A. The adjoint is defined
by the operation of the operator under the inner product

IP(|φ(cid:105) , A|ψ(cid:105)) = IP(A† |φ(cid:105) ,|ψ(cid:105)).

(2.9)

In bra-ket notation notice that (cid:104)φ|ψ(cid:105) = (cid:104)ψ|φ(cid:105)∗, so if |ψ(cid:105) is acted on by A then projected
onto (cid:104)φ| we get

(cid:104)φ| (A|ψ(cid:105)) = (cid:104)φ|Aψ(cid:105) = (cid:104)A†ψ|φ(cid:105)∗ ,

(2.10)

which defines the bra-ket notation of the adjoint. There are some technicalities regarding

the domain of A in contrast to the domain of A† which can in general cause problems, which

are detailed in[28]. If this is not a concern, as is the case in all calculations in this work,

then bra-ket notation actually provides an equivalent interpretation that the operator A acts

on the bra state, which is then projected onto by the ket state. In the case of self-adjoint

operators where A = A†, there are no worries even in bra-ket notation since

(cid:104)φ|Aψ(cid:105) = (cid:104)Aφ|ψ(cid:105) = (cid:104)φ|A|ψ(cid:105) .

(2.11)

Self-adjoint operators are of particular interest, since they correspond to physical observables

to ground the theory in reality.

13

Since this is a work in computational physics, everything must be truncated to fit the

calculations on a computer. In this case, one must select a representation for the calculation

to be carried out in, and then the basis is truncated

∞(cid:88)

i=0

|ψ(cid:105) =

N(cid:88)

i=0

(cid:104)i|ψ(cid:105)|i(cid:105) ≈

(cid:104)i|ψ(cid:105)|i(cid:105) .

(2.12)

The accuracy of this approximation will be discussed in later chapters. The quantum state is

fully encoded in N coefficients of orthonormal basis states and operators are fully described

by their matrix elements (cid:104)i|A|j(cid:105). In this regime, observables are represented by Hermitian
matrices where, (cid:104)i|A|j(cid:105) = (cid:104)j|A|i(cid:105)∗, which have real eigenvalues.

Let us now consider a Hamiltonian as an operator on a Hilbert space in the following way:

The eigenkets of ˆH, denoted |i(cid:105), provide an orthonormal basis for the Hilbert space. The
spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted

{εi}, solving the equation:

ˆH |i(cid:105) = εi |i(cid:105) .

(2.13)

Since ˆH is a Hermitian operator, the energy is always a real number.

2.2 Many-fermion wave functions and spaces

To calculate energies of quantum many-body systems, we must find the eigenkets of the

many-body Hamiltonian

ˆH |Ψµ(cid:105) = Eµ |Ψµ(cid:105) ,

ˆH = ˆZ + ˆV + ...

(2.14)

14

where ˆZ, ˆV are the one-body and two-body pieces of the Hamiltonian, and in general these

terms go up to A-body interactions for a system of A particles. The many-body state is an

element of the A-body Hilbert space

|Ψ(cid:105) ∈ HA.

(2.15)

To express the many-body state in terms of single-particle quantum mechanics, let’s first try

the A-body Hilbert space as the tensor product of A single-particle Hilbert spaces

(cid:124)

HA = H ⊗ H ⊗ ··· ⊗ H

,

(2.16)

(cid:123)(cid:122)

A

(cid:125)

where there ith single particle state is |ψi(cid:105) ∈ H. This accurately represents a many-body
state where none of the particles are interacting with each other

|Ψ(cid:105) = |ψp1ψp2 . . . ψpA(cid:105)

def

= |ψp1(cid:105) ⊗ |ψp2(cid:105) ⊗ ··· ⊗ |ψpA(cid:105) ,

(2.17)

where the positions of the kets matters, as |ψa(cid:105)|ψb(cid:105) means that particle 1 is in state |ψa(cid:105)
and particle 2 is in state |ψb(cid:105), and is in general different from |ψb(cid:105)|ψa(cid:105). The many-body
state written as a many-body wavefunction:

(cid:104)x1, x2, . . . , xA|Ψ(cid:105) = Ψ(x1, x2, . . . , xA) = (cid:104)x1, x2, . . . , xA|ψp1ψp2 . . . ψpA(cid:105)

= ψp1(x1)ψp2(x2) . . . ψpA(xA),

(2.18)

(2.19)

where x1, x2, . . . xA represent the coordinates of the degrees of freedom (like position and

spin for example) of each particle. These are called “product states” as the wave functions

15

simply multiply together, and they form a complete A-body basis

Ψ(x1, ..., xA) =

dp1...pAψp1(x1)...ψpA(xA),

(2.20)

(cid:88)

p1,...,pA

dp1...pA = (cid:104)ψp1...ψpA|Ψ(cid:105) ,

(2.21)

where dp1...pA defines the overlap between the states.

In the case of identical particles,

interchanging any two particles in a state should leave any observable unchanged. The

particle permutation operator Pij takes two particles, i in state |ψa(cid:105) and j in state |ψb(cid:105) and
swaps them so that they occupy each other’s state. Two states are physically equivalent if

they only differ by a complex phase, so

P12 |ψa(cid:105)|ψb(cid:105) = ±|ψb(cid:105)|ψa(cid:105) ,

(2.22)

gives two classes of indentical particles. Particles that are symmetric under particle inter-

change are called bosons, and particles that are antisymmetric under particle interchange

are called fermions. We will primarily be working with systems of fermions, so we intro-

duce the antisymmetrization operator because the product wave functions do not guarantee

antisymmetry

(cid:88)

ˆQ∈SA

ˆA =

1
A!

(−1)R ˆQ,

(2.23)

where A is the number of particles, SA is the symmetric group, and Q is a permutation
operator in the symmetric group, with (−1)R the associated phase of the permutation. This
will be talked about further, but for each pair of particles that are interchanged, a minus

16

sign is incurred. So if there are an even number of swaps, the sign is +, and if there are an

odd number of swaps, the sign is −. The antisymmetrizer is hermitian

and idempotent

ˆA† = ˆA,

ˆA2 = ˆA.

(2.24)

(2.25)

The antisymmetrizer projects any many-body wave function into an antisymmetric subspace.

A fermionic state is already antisymmetric, so the antisymmetrizer will act as the identity

operator when acting on a fermion wave function

|Ψ(cid:105)fermionic = ˆA|Ψ(cid:105)fermionic .

(2.26)

We can write our many-fermion ket state as the antisymmetric projection of the product ket

state space

|Ψ(cid:105)fermionic ≡ |Ψ(cid:105) =

(cid:88)

p1,...,pA

dp1...pA

ˆA|ψp1...ψpA(cid:105) .

(2.27)

Multiplying and dividing by √A!, we can rewrite this expression as

|Ψ(cid:105) =

and we will define

(2.28)

(2.29)

(cid:88)

p1,...,pA

1
√A!

dp1...pA

√A! ˆA|ψp1...ψpA(cid:105) ,

Dp1...pA ≡

1
√A!

dp1...pA,

17

and

|Φp1...pA(cid:105) ≡

√A! ˆA|ψp1...ψpA(cid:105) .

(2.30)

These many-body kets are explicitly antisymmetric, and they will form the basis for our

many-fermion basis.

If this expression is reorganized slightly, one can see that it can be written as the de-

terminant of a matrix, where the all of the permutations of different particles in different

single-particle states are the matrix entries, that is

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

|Φp1...pA(cid:105) →

1
√A!

ψp1(x1)

...

. . . ψp1(xA)
. . .

...

ψpA(x1)

. . . ψpA(xA)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .

(2.31)

Since the determinant of a matrix is unchanged up to a sign under row/column permutation,

this representation encodes the fermionic nature of the many-body state. This is a Slater

determinant [29], and they form a complete, orthogonal and antisymmetric many-body basis

to work with. For a simple example, we can look at the two-fermion case. We will try to

reserve capital phi (Φ) as a variable representing a Slater determinant throughout this text

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ψp1(x1) ψp1(x2)

ψp2(x1) ψp2(x2)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:19)

Φp1p2(x1, x2) =

1
√2

(cid:18)

=

1
√2

ψp1(x1)ψp2(x2) − ψp1(x2)ψp2(x1)

.

(2.32)

(2.33)

We see that both the antisymmetry and the Pauli exclusion principle are baked into these

18

Slater determinants

|Φp1p2...pA(cid:105) = −|Φp2p1...pA(cid:105) ,

(2.34)

and if p1 = p2 the Slater determinant is zero. So again, any fermionic wave function can be

expanded in this Slater determinant basis

(cid:88)

p1...pA

|Ψ(cid:105) =

Dp1...pA |Φp1...pA(cid:105) .

(2.35)

Looking at the coefficients we see that

(cid:122)(cid:125)(cid:124)(cid:123)

antisymmetric

Hermitian

(cid:122)(cid:125)(cid:124)(cid:123)
ˆA Ψ(cid:105)

dp1...pA = (cid:104)ψp1...ψpA|

Ψ (cid:105) = (cid:104)ψp1...ψpA|

= (cid:104) ˆA(ψp1...ψpA)|Ψ(cid:105) =

1
√A! (cid:104)Φp1...pA|Ψ(cid:105) .

(2.36)

(2.37)

Thus a sign change in Φ will imply the same change in D. Also, since the Slater determinant

is zero if any two particles occupy the same state

(cid:88)

(cid:88)

|Ψ(cid:105) =
p1 = p2 =...= pA

Dp1...pA |Φp1...pA(cid:105) = A!

p1<p2<...<pA

Dp1...pA |Φp1...pA(cid:105) ,

(2.38)

where the last expression accounts for the permutation of ordered states.

If we define

cp1...pA = A!Dp1...pA, then our many-fermionic state can be written as

|Ψ(cid:105) =

cp1...pA |Φp1...pA(cid:105) .

p1<...<pA

(2.39)

(cid:88)

19

To keep track,

cp1...pA = A!

1
√A!

dp1...pA = √A! dp1...pA = (cid:104)Φp1...pA|Ψ(cid:105) .

(2.40)

In conclusion, any quantum state of A fermions can be written as a linear combination of

Slater determinants of single particle states. Let’s now prove that the Slater determinant

basis is orthonormal. Given an orthonormal single particle basis, let p1 < ... < pA and

q1 < ... < qA

(cid:104)Φp1...pA|Φq1...qA(cid:105) = √A!√A!(cid:104) ˆA(ψp1...ψpA)| ˆA(ψq1...ψqA)(cid:105)

= A!(cid:104)ψp1...ψpA| ˆA2(ψq1...ψqA)(cid:105)
= A!(cid:104)ψp1...ψpA| ˆA(ψq1...ψqA)(cid:105)

(2.41)

ψ∗p1(x1)...ψ∗pA

(xA)ψqQ1

(x1)...ψqQA

(xA)dx1...dxA,

(cid:90)

(−)Q

(cid:88)

ˆQ∈SA

=

A!
A!

which if any of the q’s are different from the p’s, then the integral is zero. Assume p1 =

q1

... pA = qA,

(cid:104)Φp1...pA|Φp1...pA(cid:105) =

(cid:90)

(−)Q

(cid:88)

ˆQ∈SA

ψ∗p1(x1)...ψ∗pA

(xA)ψpQ1

(x1)...ψpQA

(xA)dx1...dxA.

But since our single-particle basis is orthonormal:

(cid:90)

ψ∗pi

(x)ψpj (x)dx = δij,

(2.42)

(2.43)

any of the permutations will make the integral equal to zero. Only the trivial permutation

20

of Q = 1 is nonzero, and in that case

(cid:90)

(cid:104)Φp1...pA|Φp1...pA(cid:105) =

ψ∗p1(x1)...ψ∗pA

(xA)ψp1(x1)...ψpA(xA)dx1...dxA = 1.

(2.44)

Thus the Slater determinants are orthonormal. To recap, if we are given a complete, or-

thonormal single-particle space

H = Span(cid:8)ψp(x), p = 1, 2, ...(cid:9) ,

we can create a complete A-fermion Hilbert space

(cid:2)
= ˆA

Hfermion

A

(cid:122)

(cid:125)(cid:124)

A

(cid:123)

(cid:3).

H ⊗ ... ⊗ H

(2.45)

(2.46)

where the antisymmetrizer must be used to project onto the antisymmetric subspace of the

full A-particle Hilbert space.

2.3 Occupation Number Representation

We are now in the proper position to begin talking about second quantization [16, 30] and

occupation number representation. Given a set of ordered single particle states 1, 2, ..., p, ...

we can write our Slater determinant in a slightly different way

p1(cid:122)(cid:125)(cid:124)(cid:123)

1 0...0

pA(cid:122)(cid:125)(cid:124)(cid:123)

1 0....0(cid:105) .

(2.47)

|{p1, ..., pA}(cid:105) ≡ |Φp1,...,pA(cid:105)

isomorphic

←−−−−−−→ |0...0

21

Or more compactly

|n1n2...np...(cid:105) , np = 0, 1,

(2.48)

where each of the np terms corresponds to a one or zero from Eqn. (2.47). This now allows

us to work in the Fock space [31], where the particle number is not fixed. The Fock space

is the space spanned by all of such kets. In this text, we do most of our calculations with

a fixed particle number, but the ability to represent a two-particle two-hole excitation as

annihilating to particles below the Fermi surface and creating two particles above the Fermi

surface proves to be very convenient. To ensure that the particle number is fixed, we write

our A-fermion Hilbert space as

(cid:40)

Hfermion

A

= Span

|n1...np(cid:105) ,

(cid:88)

p

(cid:41)

np = A

,

(2.49)

while the full Fock space is the direct sum of all of such Hilbert spaces from A = 0, 1, 2, . . . .

∞(cid:77)

A=0

F fermion =

Hfermion

A

.

(2.50)

A four particle example of such a state |{p1, ..., pA}(cid:105) with p1 = 0, p2 = 1, p3 = 6, p4 = 9, is
in this representation given by

|{0 1 6 9}(cid:105) = |{ψ0ψ1ψ6ψ9}(cid:105) = |Φ0 1 6 9(cid:105) = |110000100100...(cid:105) ,

(2.51)

where here the trailing zeros can either be finite or infinite depending on how many states

are in the single-particle basis.

22

2.4 Creation and annihilation operators

To define the annihilation operator, here is an example of what it does to a ket state,

(cid:80)p−1

k=1

ˆXp |n1 . . . np . . .(cid:105) = (−1)

(nk)

np |...(1 − np)...(cid:105) .

(2.52)

We want to annihilate a particle in a particular state while keeping the proper phases from

fermion statistics. So the (1 − np) term is going to change a 1 to a 0 at spot p if there is
a particle occupying that state. If there was not a particle occupying that state, then the

np coefficient in front of the ket will be 0.

If we multiply a ket by zero, we take this to

mean that the state cannot exist, and we discard it. This is different from a ket where all

of the np’s are zero, as that is a valid physical state, with zero particles. The sum in the

expression keeps track of the phase by determining how many particles occupy lower lying

states than the state we are trying to annihilate a particle from. To understand what this

means more concretely, let’s introduce some new notation of identifying if a single particle

state is occupied or not in a ket. If a particle is in a state labeled p, with m states occupied

before it, we will write

where

|...p...(cid:105) = | ...(cid:124)(cid:123)(cid:122)(cid:125)m

p(cid:122)(cid:125)(cid:124)(cid:123)

1 ...(cid:105) , np = 1,

p−1(cid:88)

k=1

nk,

m =

and if a particle is not in state p with m states occupied before it, we will write it as

|...
p...(cid:105) = | ...(cid:124)(cid:123)(cid:122)(cid:125)m

0 ...(cid:105) , np = 0.

p(cid:122)(cid:125)(cid:124)(cid:123)

23

(2.53)

(2.54)

(2.55)

The annihilation operator acting on these kets yields

ˆXp |...p...(cid:105) = (−1)m1|...
p...(cid:105) ,

ˆXp |...
p...(cid:105) = (−1)m0 = 0,

(2.56)

(2.57)

which matches with the intuition that was described above. The annihilation operator got

rid of the particle occupying state p if there was a particle there, and set the whole state to

0 otherwise. Now before the creation operator is introduced, we first need to introduce the

true vacuum state.

|0000....0...(cid:105) = |0(cid:105) .

A single particle state can be written in second quantized form as

ψp = |p(cid:105) = |00...0 p(cid:124)(cid:123)(cid:122)(cid:125)

np=1

0...(cid:105) ,

so that annihilating that single particle returns us to the true vacuum

ˆXp |p(cid:105) = |00...0...(cid:105) = |0(cid:105) .

(2.58)

(2.59)

(2.60)

We then define the creation operator as the Hermitian adjoint of the annihilation operator

ˆXp |p(cid:105) = |0(cid:105) → (cid:104)p| ˆX†p = (cid:104)0| .

(2.61)

24

It then follows that

(cid:104)p| ˆX†p|0(cid:105) = (cid:104)0|0(cid:105) = 1 =⇒ ˆX†p |0(cid:105) = |p(cid:105) .

When the creation operator acts on a state that is not the true vacuum

ˆX†p |{p1...pA}(cid:105) = |{p p1...pA}(cid:105) ,

ˆX†p |{p p1...pA}(cid:105) = |{p p p1...pA}(cid:105) = 0,

(2.62)

(2.63)

(2.64)

where Eqn. (2.64) is zero since the determinant of a matrix with a repeated row is 0. This

is the manifestation of the Pauli exclusion principle in the second quantization formalism,

since you cannot have two fermions occupying the same state

ˆX†p |{p1p2...pA}(cid:105) ,

p1 < p < p2

= |{p p1p2...pA}(cid:105) = −|{p1 p p2...pA}(cid:105) ,

or the same statement in the more compact notation

m(cid:122)(cid:125)(cid:124)(cid:123)... 
p...(cid:105) = (−1)m |...p...(cid:105) ,

ˆX†p |

and

ˆX†p |...p...(cid:105) = 0.

25

(2.65)

(2.66)

(2.67)

Using these pieces together, we can encode these properties of the creation operator into the

following definition

(cid:80)p−1

k=1

ˆX†p |...np...(cid:105) ≡ (−1)

nk (1 − np)|...(1 − np)...(cid:105) .

(2.68)

2.5 Number Operator

The number operator is defined as

ˆNp ≡ ˆX†p ˆXp.

(2.69)

This operator conserves particle number (it has the same number of creation and annihilation

operators) and is Hermitian

ˆN†p = ( ˆX†p ˆXp)† = ˆX†p ˆX††p = ˆNp.

(2.70)

When ˆNp acts on a ket where p is occupied, we obtain

ˆNp |...p...(cid:105) = ˆX†p ˆXp | ...(cid:124)(cid:123)(cid:122)(cid:125)m

p...(cid:105)

= (−1)m ˆX†p |...
p...(cid:105) = (−1)m(−1)m |...p...(cid:105)
= (−1)2m |...p...(cid:105)|...p...(cid:105) .

(2.71)

(2.72)

(2.73)

26

We can see that the operator ˆNp does not change the ket state. When ˆNp acts on a ket

where p is unoccupied we get

Thus

ˆNp |...
p...(cid:105) = ˆX†p ˆXp |...
p...(cid:105) = 0.

ˆNp |...np...(cid:105) = np |...np...(cid:105) .

(2.74)

(2.75)

The state |p(cid:105) is an eigenstate of Np in number occupancy representation with eigenvalue np,
just as a state with definite position |x(cid:105) is an eigenstate of the position operator ˆx in position
space representation. In fact, these eigenkets define these representations. For most of this

text, we will prefer to use the number occupancy representation of our many-fermion states.

Now we define the total number operator

where(cid:80)

k nk = A since we are restricting ourselves to the A-fermion Hilbert space. Thus

ˆN |Ψ(cid:105) = A|Ψ(cid:105) .

(2.78)

27

ˆN =

ˆX†p ˆXp =

(cid:88)

p

ˆN |...np...(cid:105) =

(cid:122)
(cid:125)(cid:124)
(cid:18)(cid:88)

A

nk

k

ˆNp,

(cid:88)

p

(cid:123)
(cid:19)

|...np...(cid:105) ,

(2.76)

(2.77)

2.6 Anti-commutation relations

The most defining characteristic of the creation and annihilation operators are their anti-

commutation rules,

(1)

(2)

(3)

(cid:110)

ˆX†p, ˆXq

(cid:111)

= ˆX†p ˆXq + ˆXq ˆX†p = δpq 1,

(cid:110)

ˆXp, ˆXq

(cid:110)

ˆX†p, ˆX†q

(cid:111)

(cid:111)

= 0,

= 0.

(2.79)

(2.80)

(2.81)

As an example of a calculation that can be done with second quantization, let’s compute the

overlap of two Slater determinants. Starting with a pair of two-particle Slater determinants:

|Φpq(cid:105) = |{pq}(cid:105) , |Φrs(cid:105) = |{rs}(cid:105), what is s = (cid:104){pq}|{rs}(cid:105)?

s = (cid:104){pq}|{rs}(cid:105) = (cid:104)0| ˆXq ˆXp ˆX†r ˆX†s|0(cid:105)
= (cid:104)0| ˆXq(δpr − ˆX†r ˆXp) ˆX†s|0(cid:105)
= δpr (cid:104)0| ˆXq ˆX†s|0(cid:105) − (cid:104)0| ˆXq ˆX†r ˆXp ˆX†s|0(cid:105)
= δpr (cid:104)q|s(cid:105) − (cid:104)0| ˆXq ˆX†r (δps − ˆX†s ˆXp)|0(cid:105)
= δprδqs − δps (cid:104)0| ˆXq ˆX†r|0(cid:105) + (cid:104)0| ˆXq ˆX†r ˆX†s ˆXp|0(cid:105)
= δprδqs − δps (cid:104)q|r(cid:105) + (cid:104)0| ˆXq ˆX†r ˆX†s
= δprδqs − δpsδqr.


ˆXp|0(cid:105)

28

Here, the general strategy was to push one of the annihilation operators as far right as possible

until it hit the vacuum ket state. Once the annihilation operator acts on the vacuum ket,

the state is gone entirely, and the expression simplifies quite a bit to simple Kronecker delta

functions which we can more easily deal with. While this is a powerful tool, more efficient

tools will be developed later.

2.7 Operators in Second Quantized Form

Let’s again take a look at our Hamiltonian with at most three-body operators

ˆH = ˆZ + ˆV + ˆW ,

(2.82)

where we split it into a 1-body piece

A(cid:88)

i=1

A(cid:88)

ˆz(xi),

ˆv(xi, xj),

ˆZ =

ˆV =

a two-body piece

and a three-body piece

1≤i<j

A(cid:88)

1≤i<j<k

ˆW =

ˆw(xi, xj, xk).

(2.83)

(2.84)

(2.85)

This is a generic procedure, as any operator can be split into a 0-body piece, 1-body piece,

etc.

A(cid:88)

k=0

ˆok,

ˆO =

A(cid:88)

ˆok(xi1, . . . , xik ).

ˆok =
1≤i1<···<ik

(2.86)

29

The next goal is to develop how these operators act on our many-fermion kets ˆOk |{p1 . . . pN}(cid:105).
Before we get to generic operators, we need to further develop the second quantized formal-

ism. We are now writing our ket states as strings of operators, and we will use capital pi

(Π) to represent the product of many of these operators. The new representation is then

A(cid:89)

k=1

ˆX†pk |0(cid:105) ≡ ˆX†p1 . . . ˆX†pA |0(cid:105) .

(2.87)

Acting with an annihilation operator on such a string of operators yields

A(cid:89)

k=1

ˆXq

ˆX†pk |0(cid:105) =

A(cid:88)
(−1)i−1δqpi

i=1

A(cid:89)

k=1
k = i

ˆX†pk |0(cid:105) .

(2.88)

To help parse what this chain of symbols means let’s break it down. We want to act with

an annihilation operator of state q on a ket state, but now our ket state is represented by

a chain of creation operators. The result is a sum, where the kronecker delta eliminates

any term in the sum that does not correspond to the annihilated state q. Then for each

term where q = pi, there is an induced phase (−1)i−1 and a new ket with the corresponding
missing creation operator.

Now to operators in second quantization.

In first quantization, a one-body operator

acting on A particles can be written as

ˆO1 =

A(cid:88)

i=1

ˆo1(xi).

30

(2.89)

In second quantization, the one-body operator will look like

where

ˆO1 =

(cid:104)p|ˆo1|q(cid:105) =

(cid:88)
pq (cid:104)p|ˆo1|q(cid:105) ˆX†p ˆXq,
(cid:90)

ψ∗p(x)ˆo1(x)ψq(x)dx,

(2.90)

(2.91)

where the integral over dx is a symbolic notation, meaning integrate over all 3 spatial di-

mensions and sum over all interal (spin) degrees of freedom. The two-body operator in first

quantization looks like

(cid:88)

i<j

ˆO2 =

ˆo2(xi, xj).

(2.92)

And in second quantization the two-body operator looks like

(cid:88)
pqrs(cid:104)pq|ˆo2|rs(cid:105) ˆX†p ˆX†q ˆXs ˆXr,

ˆO2 =

1
2

(2.93)

where

(cid:90)

(cid:104)pq|ˆo2|rs(cid:105) =

ψ∗p(x1)ψ∗q (x2)ˆo2(x1, x2)ψr(x1)ψs(x2)dx1dx2.

(2.94)

Note the ordering of the annihilation operators. The fact that the indices in the ket are

reversed from the indices of the operators is crucial. Also note the sum, we have chosen

to sum over the entire range of all of the indices, but due to the symmetries of particle

interchange, we do not have to if we do not want to. For a general k-body operator (rank

31

k), we can write it in second quantization representation in the Goldstone form [32] as

(cid:18) 1

(cid:19) (cid:88)

k!

p1...pk
q1...qk

ˆOk =

(cid:104)p1 . . . pk|ˆok|q1 . . . qk(cid:105) ˆX†p1 . . . ˆX†pk

ˆXqk . . . ˆXq1,

(2.95)

where we notice that we are summing over two times the rank number of indices, and where

(cid:90)

(cid:90)

···

(cid:104)p1 . . . pk|ˆok|q1 . . . qk(cid:105) =

ψ∗p1(x1) . . . ψ∗pk

(xk)ˆok(x1, . . . , xk)ψq1(x1) . . . ψqk (xk)dx1 . . . dxk.

(2.96)

Alternatively we can write the operator in the Hugenholtz form [33]

(cid:18) 1

(cid:19)2 (cid:88)

k!

p1...pk
q1...qk

ˆOk =

(cid:104)p1 . . . pk|ˆok|q1 . . . qk(cid:105)A

ˆX†p1 . . . ˆX†pk

ˆXqk . . . ˆXq1,

(2.97)

where we are now using the anti-symmetrized matrix element

(cid:88)

ˆR∈Sk

(cid:104)p1 . . . pk|ˆok|q1 . . . qk(cid:105)A =

(−1)R (cid:104)p1 . . . pk|ˆok|qR1 . . . qRk(cid:105) .

(2.98)

As an example, let’s look at the two-body operator in Hugenholtz form:

(cid:88)
pqrs(cid:104)pq|ˆo2|rs(cid:105)A

ˆO2 =

1
2

ˆX†p ˆX†q ˆXs ˆXr,

(2.99)

where we have defined the two-body matrix elements as

(cid:104)pr|ˆo2|rs(cid:105)A = (cid:104)pq|ˆo2|rs(cid:105) − (cid:104)pq|ˆo2|sr(cid:105) .

(2.100)

32

It is usually the case that writing equations using the antisymmetrized matrix elements is

more compact, and so we will be using them frequently.

2.8 Wick’s Theorem

The purpose of Wick’s theorem [34] is to give us a more concise way of dealing with these

huge cumbersome strings of creation and annihilation operators. To deal with the matrix

elements of many-body operators, we run into expressions like

(cid:104)Φp1...pA| ˆOk|Φq1...qA(cid:105) =

(cid:18) 1

(cid:19)2 (cid:88)

(cid:104)r1 . . . rk|ˆok|s1 . . . sk(cid:105)A ×

k!

r1...rk
s1...sk

(cid:104)0| ˆXpA . . . ˆXp1

ˆX†r1 . . . ˆX†rk

ˆXsk . . . ˆXs1

ˆX†q1 . . . ˆX†qA |0(cid:105) ,

which leads to a very unwieldy chain of operators, especially since our only tool to calculate

what this means is by anti-commuting these operators around between each other. Wick’s

theorem will give us a much more manageable way to deal with such chains of operators.

First however, we must develop the tools with which Wick’s theorem is expressed. The first

thing we do is to become blind to whether an operator is a creation or annihilation operator,

and we simply label each of them as a capital ˆM . In this new notation, we would write a

generic vacuum expectation value of a chain of operators as

(cid:104)0| ˆM1 . . . ˆMm|0(cid:105) ,

ˆMi ≡ ˆXpi or ˆX†pi,

(2.101)

33

and an operator acting on a Slater determinant would become

ˆOk |Φp1...pA(cid:105) → ˆM1 . . . ˆMm |0(cid:105) .

(2.102)

To write down Wick’s theorem, we need normal ordered products of operators, contractions

of operators, and the normal product with contractions. First, we define normal product n

acting on a chain of operators as

n[ ˆM1 . . . ˆMm] = (−1)R ˆX†R1

. . . ˆX†Rj

ˆXRj+1 . . . ˆXRm,

(2.103)

where again, the (−1)R term is keeping track of if this is an even or odd permutation. The
permutation R can be written as

 ,

 1

R =

2

. . .

j

j + 1 . . . m

R1 R2

. . . Rj Rj+1

. . . Rm

where the top row indicates the original ordering of the indices on the left-hand side of

the equation, and the second row indicates the final ordering of the indices on the right

hand side of the equation. The take away here is that the normal ordering operation on

a chain of operators pushes all of the annihilation operators to the right. This is useful,

because as we said before, a common strategy in calculations is to push the annihilation

operators rightwards so that they annihilate the vacuum state, yielding zeros, which tidy up

the algebra. Here are some examples of the normal ordering in action.

Ex:

34

n[ ˆX†p ˆXq ˆX†r ] = − ˆX†p ˆX†r ˆXq

p q r

p r q

 = (p)(qr) =⇒ (−1)R = (+1)(−1) = −1

R =

The notation (p)(qr) denotes the permutation cycles of the permutation. A permutation

cycle is a subset of a permutation whose elements trade places with each other.

In the

example above, p does not move, so it is a one-cycle, and q and r trade places with each

other, so they are a two-cycle. Cycles containing an odd number of elements do not induce

a phase, while cycles with an even number induce a phase of (−1). So to read the equation
above, it is saying that the phase of permutation R = {p, q, r} → {p, r, q} is equal to the
phase of a one-cycle times the phase of a two-cycle which is (−1). This is usually the easiest
and fastest way to compute the phase of a permutation.

Contraction:

A contraction between two operators tells you how different from a normal ordering they

are. A contraction is defined as

ˆM1 ˆM2 = ˆM1 ˆM2 − n[ ˆM1 ˆM2].

(2.104)

Here are the four cases that can arise out of this definition:

(i) ˆM1 = ˆXp, ˆM2 = ˆXq,

ˆXp ˆXq = ˆXp ˆXq − n[ ˆXp ˆXq] = ˆXp ˆXq − ˆXp ˆXq = 0.

35

(ii) ˆM1 = ˆXp, ˆM2 = ˆX†q ,

ˆXp ˆX†q = ˆXp ˆX†q − n[ ˆXp ˆX†q ] = ˆXp ˆXq + ˆX†q ˆXp = δpq.

(iii) ˆM1 = ˆX†p, ˆM2 = ˆXq,

ˆX†p ˆXq = ˆX†p ˆXq − n[ ˆX†p ˆXq] = ˆX†p ˆXq − ˆX†p ˆXq = 0.

(iv) ˆM1 = ˆX†p, ˆM2 = ˆX†q ,

ˆX†p ˆX†q = ˆX†p ˆX†q − n[ ˆX†p ˆX†q ] = ˆX†p ˆX†q − ˆX†p ˆX†q = 0.

Therefore ˆM1 ˆM2 = 0 except in the case of ˆXp ˆX†p = 1.

Normal Product with Contractions:

Once two operators are contracted together, they become a simple real number, and thus

can be pulled out of the normal product. This can be written as

n[ ˆM1 . . . ˆMi1 . . . ˆMiλ . . . ˆMj1 . . . ˆMjλ . . . ˆMm]

 1

i1

R =

= (−1)R ˆMi1

ˆMj1 . . . ˆMiλ

ˆMjλn[ ˆMk1 . . . ˆMkµ]

2

. . .

(2λ − 1)

(2λ)

(2λ + 1)

. . . m

j1

. . .

iλ

jλ

k1

. . . kµ



(2.105)

where 2λ + µ = m. Here, all of the contracted pairs have been pulled out of the normal

product and the phase is kept track in this permutation. When doing these operations, note

that ˆM1 ˆM2 = ˆM2 ˆM1.

36

Ex: n[ ˆXp ˆXq ˆX†r ] = − ˆXp ˆX†r n[ ˆXq] = −δpr ˆXq

p q r

p r q

R =

 = (p)(qr) =⇒ (−1)R = (+1)(−1) = (−1) = −1.

Without proof, we will now state some useful properties of contractions and normal ordered

operators.

1. n[ ˆM1 . . . ˆMm]|0(cid:105) = 0, unless all of the ˆM ’s are creation operators.

2. (cid:104)0|n[ ˆM1 . . . ˆMm]|0(cid:105) = 0, unless m = 0.

3. n[ ˆM1 . . . ˆMi1 . . . ˆMj1 |0(cid:105) = 0, If at least one uncontracted operator ˆM is an annihilator.

4. (cid:104)0|n[ ˆM1 . . . ˆMi1 . . . ˆMj1 . . . ˆMiλ . . . ˆMjλ . . . ˆMm]|0(cid:105) = 0, unless all operators are con-

tracted.

We now finally write down the definition of Wick’s theorem on an arbitrary chain of creation

and annihilation operators.

ˆM1 . . . ˆMm = n[ ˆM1 . . . ˆMm] +

n[ ˆM1 . . . ˆMi1 . . . ˆMj1 . . . ˆMm]

n[ ˆM1 . . . ˆMi1 . . . ˆMj1 . . . ˆMi2 . . . ˆMj2 . . . ˆMm]

(cid:88)

1≤i1<j1≤m

(cid:88)

+
1≤i1<j1≤m
1≤i2<j2≤m
i1<i2,j1 = j2

+ sum of all possible 3 contractions

+ . . .

+ sum of all possible N contractions,

37

where N is the maximum number of contractions possible. For an even number of operators,

this means they are all contracted, but for an odd number of operators, all but one will be

contracted. In words, Wick’s theorem is the statement that an arbitrary chain of creation

an annihilation operators can be written entirely as the normal product of that chain of

operators and the normal product of all possible ways to contract that chain of operators.

2.9 Generalized Wick’s Theorem

Wick’s theorem is a very powerful tool now at our disposal. However, we would like to

be able to tackle objects like ˆM1n[ ˆM2 ˆM3] or n[ ˆM1 ˆM2]n[ ˆM3 ˆM4]. The generalized Wick’s

theorem accounts for the partially normal producted cases for us and is stated as

ˆM1 . . . ˆMkµ−1n[ ˆMkµ−1+1 . . . ˆMkµ] ˆMkµ+1 . . . ˆMm
(cid:88)
= n[ ˆM1 . . . ˆMkµ−1

ˆXpα+1 . . . ˆXpβ

ˆX†p1 . . . ˆX†pα

ˆMkµ+1 . . . ˆMm]

+
modified

n[ ˆM1 . . . ˆMkµ−1

ˆX†p1 . . . ˆX†pα

ˆXpα+1 . . . ˆXpβ

ˆMkµ+1 . . . ˆMm],

where this modified sum excludes contraction terms from the original n product. The purpose

of these rules is to speed up tedious pen and paper manipulations.

2.10 Slater-Condon Rules

The Slater-Condon rules [29, 9] give simplified expressions for manipulations of the Hamil-

tonian in second quantizaiton. To see how they arise let’s look at how the Hamiltonian acts

on these many-body kets. For this section, we will restrict our Hamiltonian to a one-body

38

piece and a two-body piece

ˆH |Ψ(cid:105) = E |Ψ(cid:105) ,

ˆH = ˆZ + ˆV ,

(2.106)

(2.107)

and we write the solution to the Schr¨odinger equation as a linear combination of our complete

and orthonormal antisymmetrized A body states

(cid:88)

|Ψ(cid:105) =

Cp1...pA |Φp1...pA(cid:105) .

(2.108)

In this basis, we can write the Schr¨odinger equation as a matrix eigenvalue problem, since

|Ψ(cid:105) can be written as the vector of Cp1...pA coefficients, and the Hamiltonian can be written
as a matrix with matrix elements

Hij = (cid:104)Φi| ˆH|Φj(cid:105) = (cid:104)Φi| ˆZ + ˆV |Φj(cid:105) .

(2.109)

39

The Slater rules are rules about the Hamiltonian matrix elements. First, let’s look at the

matrix elements of the one-body operator

(cid:104){p1 . . . pA}|Z|{q1 . . . qA}(cid:105)
=

(cid:88)
r,s (cid:104)0| ˆXpA . . . ˆXp1
(cid:88)
(cid:88)
r,s (cid:104)r|ˆz|s(cid:105)
(cid:88)
(cid:88)
r,s (cid:104)r|ˆz|s(cid:105)

F.C.

F.C.

=

=

ˆX†r ˆXs ˆX†q1 . . . ˆX†qA|0(cid:105)(cid:104)r|ˆz|s(cid:105)

(cid:104)0| ˆXpA . . . ˆXp1

ˆX†r ˆXs ˆX†q1 . . . ˆX†qA|0(cid:105)

(cid:104)0|n[ ˆXpA . . . ˆXp1]n[ ˆX†r ˆXs]n[ ˆX†q1 . . . ˆX†qA]|0(cid:105) .

Remember that only contractions of the form ˆX ˆX† are non-zero. Also remember that the

generalized Wick’s theorem means we get rid of contraction terms between normal products.

This means that ˆX†r must contract with an annihilation operator to its left, and that ˆXs

must contract with a creation operator to its right. The fully contracted sum must obey

these rules, and we get

(cid:88)
r,s (cid:104)r|ˆz|s(cid:105)
(cid:88)
r,s (cid:104)r|ˆz|s(cid:105)

=

F.C.

(cid:88)
(cid:104)0|n[ ˆXpA . . . ˆXp1]n[ ˆX†r ˆXs]n[ ˆX†q1 . . . ˆX†qA]|0(cid:105)
(cid:88)

F.C.

(cid:104)0| ˆXpA . . . ˆXpi . . . ˆXp1

ˆX†r ˆXs ˆX†q1 . . . ˆX†qj

. . . ˆX†qA|0(cid:105) ,

where the r and s terms are clearly contracted. Note that for ˆXpk , where k = i, it must be
contracted fully with ˆX†ql where l = j. This means that these pk indices must be permutations

of the ql indices. This gives rise to our first Slater rule.
One-body Slater rules: (cid:104){p1 . . . pA| ˆZ|{q1 . . . qA}(cid:105) = 0 when:

(1) {pk}A

k=1 and {ql}A

l=1 differ by one spin-oribtal

40

(2) {pk}A

k=1 and {ql}A

l=1 do not differ. (May differ by permutation)

These two non-zero cases lead to different expressions for the one-body matrix elements.

p1 = q1, p2 = q2, . . . pn = qn.

(cid:88)
(cid:104){p1 . . . pA}| ˆZ|{q1 . . . qA}(cid:105)
r,s (cid:104)r|ˆz|s(cid:105)(cid:104)0| ˆXpA . . . ˆXp2
=
(cid:88)
r,s (cid:104)r|ˆz|s(cid:105) δrp1δsq1

=

= (cid:104)p1|ˆz|q1(cid:105) ,

ˆXp1

ˆX†r ˆXs ˆX†q1

ˆX†q2 . . . ˆX†qA|0(cid:105)

where we see the expression for when one basis state differs, is simply the one-body matrix

element between these two states. Now we have the case where the two sets of basis states are

the same up to a permutation. Without loss of generality, we write this as, p1 = q1, . . . , pA =

qA,

(cid:88)
(cid:104){p1 . . . pA}| ˆZ|{p1 . . . pA}(cid:105)
r,s (cid:104)r|ˆz|s(cid:105)
=

(cid:88)

F.C.

(cid:104)0| ˆXpA . . . ˆXpi . . . ˆXp1

ˆX†r ˆXs ˆX†p1 . . . ˆX†pj

. . . ˆX†pA|0(cid:105) .

41

Notice here that if pi = pj, then there will be a mismatch somewhere with the other con-

tractions. For the next case, we have pi = pj,

(cid:122)
(cid:104)0| ˆXpA . . . ˆXpi . . . ˆXp1

(cid:123)
(cid:125)(cid:124)
ˆX†r ˆXs ˆX†p1 . . . ˆX†pi . . . ˆX†pA|0(cid:105)

Even number of crossings

=

A(cid:88)

(cid:104){p1 . . . pA}| ˆZ|{p1 . . . pA}(cid:105)
(cid:88)
r,s (cid:104)r|ˆz|s(cid:105)
(cid:88)
A(cid:88)
A(cid:88)

(cid:104)r|ˆz|s(cid:105) δrpiδspi

i=1

i=1

r,s

=

=

i=1

(cid:104)pi|ˆz|pi(cid:105) ,

where here we sum over all of the single-particle states in the many-body state. In other

words, the expectation value of the one-body operator of a many-body state is the sum of

single-particle expectation values of the states in the many-body state.

Two-Body Slater Rules:

We want to understand when (cid:104){p1 . . . pA}| ˆV |{q1 . . . qA}(cid:105) = 0. Writing this out further we
obtain

(cid:104){p1 . . . pA}| ˆV |{q1 . . . qA}(cid:105)
=

(cid:104)rs|ˆv|tu(cid:105)(cid:104)0| ˆXpA . . . ˆXp1

1
2

(cid:88)

rstu

ˆX†r ˆX†s ˆXu ˆXt ˆX†q1 . . . ˆX†qA|0(cid:105) .

Now we play a similar game as we did with the one-body operator. Due to the partial normal-

ordering of the rsut operators, the rs creation operators must contract to the left to some

pi, pj annihilation operators, and the ut annihilation operators must contract to the right

42

against some qk, ql creation operators. After those four have been contracted, the remaining
ˆXpm and ˆX†qn must be contracted, where m = i, j and n = k, l. Thus, these operators cannot

differ, and they must be permutations of each other at most. That is, {pm}m = i,j, {qn}n = k,l
must represent permutations. This means that we get (cid:104){p1 . . . pA}| ˆV |{q1 . . . qA}(cid:105) = 0 if {pm}
and {qn} differ by more than two single-particle basis states. There are three cases where
we do not get zero, which will not be proven here. These are (1) p1 = q1, p2 = q2, with all

other pm = qm identical,

(cid:104){p1p2p3 . . . pA}| ˆV |{q1q2p3 . . . pA}(cid:105) = (cid:104)p1p2|ˆv|q1q2(cid:105)A .

(2) p1 = q1, p2 = q2 . . . pA = qA,

(cid:104){p1p2 . . . pA}| ˆV |{q1p2 . . . pA}(cid:105) =

A(cid:88)

k=1

(cid:104)p1pk|ˆv|q1pk(cid:105)A ,

where here, starting the sum from k = 1 is equally valid, as this corresponds to adding in a

zero term in the form of the Pauli exclusion breaking term (cid:104)p1p1|ˆv|q1p1(cid:105)A = 0. Finally the
last case is (3) p1 = q1, p2 = q2 . . . pA = qA

(cid:104){p1 . . . pA}| ˆV |{p1 . . . pA}(cid:105) =

1
2

(cid:104)pkpl|ˆv|pkpl(cid:105)A .

A(cid:88)

k,l=1

It is with expressions like this that the power of second quantization really shines, since we

have turned matrix elements in the full many-body basis into simple sums of matrix elements

in the one or two-body basis. This is an enormous simplification, and helps with calculations

immensely.

43

2.11 The Fermi Vacuum

As a reminder, once we have settled on our single-particle basis, we can expand any

many-fermion ket in terms of our anti-symmetrized Slater determinants,

(cid:88)

q1<···<qA

|Ψ(cid:105) =

cq1...qA |{q1 . . . qA}(cid:105) ,

for some set of constants cq1...qA. This sum is organized pretty arbitrarily, with no stress put

on what the first item in this sum is. However, depending on the physics of the situation,

we can select out a many-body state we determine as important, and organize around how

close we are to this “reference state”. We will denote the reference state as |Φ(cid:105). For now,
let’s say that single-particle states labeled with p’s are occupied in |Φ(cid:105) and states labeled
with q’s are unoccupied,

|Φ(cid:105) = ˆX†p1 . . . ˆX†pN |0(cid:105) = |{p1 . . . pN}(cid:105) .

We can now write any other N -fermion ket relative to our reference state. For example, a

state that differs by one single-particle state from our reference state can be written as

|{p1 . . . pµ−1qµpµ+1 . . . pN}(cid:105) = ˆX†qµ
= ˆX†qµ

ˆXpµ |{p1 . . . pµ−1pµpµ+1 . . . pN}(cid:105)
ˆXpµ |Φ(cid:105) ,

44

where now we are only dealing with 2 operators. Or in the case that we want to examine a

state that differs by 2 single-particle states

|{p1 . . . pµ−1qµpµ+1 . . . pν−1qνpν+1 . . . pN}(cid:105)
= ˆX†qµ

ˆX†qν

ˆXpν |{p1 . . . pµ−1pµpµ+1 . . . pν−1pνpν+1 . . . pN}(cid:105)
ˆXpν |Φ(cid:105) ,

ˆXpµ

= ˆX†qµ

ˆXpµ

ˆX†qν

where we now only have 4 operators. This is a huge simplification from having to deal with

the full N creation operators that we would typically have! This pattern continues; every

single-particle state different from our reference state that differs costs us one creation and

one annihilation operator.

Notation:

We are now going to change our notation a bit. We assign single-particle labels to have a

specific meaning in Table 2.1. Note that this notation only makes sense with respect to a

Table 2.1: Single-Particle Index Conventions

i, j, k, . . . (i1, i2, . . . ) =⇒
s.p. state occupied in |Φ(cid:105)
s.p. state unoccupied in |Φ(cid:105)
a, b, c, . . . (a1, a2, . . . ) =⇒
p, q, r, . . . (p1, p2, . . . ) =⇒ s.p. state is a generic state |Φ(cid:105)

particular reference state. We will also define a compressed notation for states that differ

from the reference state as

ˆX†a ˆXi |Φ(cid:105) = |Φa
i (cid:105) ,

(2.110)

45

which indicates a one-particles one-hole excitation

ˆX†a ˆXi ˆX†b

ˆXj |Φ(cid:105) = |Φab

ij (cid:105) = ˆX†a ˆX†b

ˆXj ˆXi |Φ(cid:105) ,

(2.111)

which indicates a two-particle two-hole excitation, and

a1...an
i1...in (cid:105) = ˆX†a1
|Φ

ˆXi1 . . . ˆX†an

ˆXin |Φ(cid:105) =

n(cid:89)

µ=1

ˆX†aµ

ˆXiµ |Φ(cid:105) ,

(2.112)

which indicates an n-particle n-hole excitation. In each of these cases, an occupied index

is being annihilated, and replaced by an unoccupied index. Let’s look at an example with

N = 6 spin-1/2 fermion in a basis with 12 spin-orbitals. The Fermi level is defined as the

energy where the six particles are occupying the six lowest spin-orbitals.

9,10
7,8

11,12}a,b
5,6 }i,j

3,4
1,2

Fermi level

|Φ(cid:105)

|Φ7
5(cid:105)

In this graphic, we have a cartoon of two different many-body states. We have defined

the reference state |Φ(cid:105) = |{123456}(cid:105) to be filling all of the lowest spin-orbitals up to the
Fermi level . This is a very common choice for the reference state. To the right we have

5(cid:105) = ˆX†7
|Φ7
that is above the Fermi level.

ˆX5 |Φ(cid:105), which has had one particle under the Fermi level with another particle

With the definition of a Fermi level we define our ansatz for the ground state, represented

by a Slater determinant Φ0. Switching notation a bit, the annihilation operator will now be

represented as a, where this is a more common symbol in nuclear physics, and we drop the

46

hat and from context determine that it is an operator. We can rewrite the ansatz for the

ground state as

(cid:89)

i≤F

a†i|0(cid:105),

|Φ0(cid:105) =

(2.113)

where we have introduced the shorthand labels for states below the Fermi level F as i, j, . . . ≤
F . For single-particle states above the Fermi level we reserve the labels a, b, . . . > F , while

the labels p, q, . . . represent any possible single-particle state.

The focus of the work is on infinite systems, where the one-body part of the Hamiltonian

is given by the kinetic energy operator only. In second quantization it is defined as

(cid:88)
pq (cid:104)p|ˆt|q(cid:105)a†paq,

ˆH0 = ˆT =

(2.114)

where the matrix elements (cid:104)p|ˆt|q(cid:105) represent the expectation value of the kinetic energy op-
erator (see the discussion below as well). The two-body interaction reads

(cid:88)
pqrs(cid:104)pq|ˆv|rs(cid:105)ASa†pa†qasar,

ˆHI = ˆV =

1
4

(2.115)

where we have defined the anti-symmetrized matrix elements

(cid:104)pq|ˆv|rs(cid:105)AS = (cid:104)pq|ˆv|rs(cid:105) − (cid:104)pq|ˆv|sr(cid:105).

(2.116)

We can also define a three-body operator

ˆW =

1
36

(cid:104)pqr| ˆw|stu(cid:105)ASa†pa†qa†rauatas,

(2.117)

(cid:88)

pqrstu

47

with the anti-symmetrized matrix element

(cid:104)pqr| ˆw|stu(cid:105)AS = (cid:104)pqr| ˆw|stu(cid:105) + (cid:104)pqr| ˆw|tus(cid:105) + (cid:104)pqr| ˆw|ust(cid:105)
− (cid:104)pqr| ˆw|sut(cid:105) − (cid:104)pqr| ˆw|tsu(cid:105) − (cid:104)pqr| ˆw|uts(cid:105).

In this and the forthcoming chapters we will limit ourselves to two-body interactions at most.

Throughout this chapter and the subsequent three we will drop the subscript AS and use

only anti-symmetrized matrix elements.

Using the ansatz for the ground state |Φ0(cid:105) as new reference vacuum state, we need to

redefine the anticommutation relations to

(cid:110)

a†p, aq

(cid:111)

= δpq, p, q ≤ F,

(2.118)

= δpq, p, q > F.

(2.119)

and

It is easy to see that

(cid:110)

ap, a†q

(cid:111)

ai|Φ0(cid:105) = |Φi(cid:105) (cid:54)= 0,

a†a|Φ0(cid:105) = |Φa(cid:105) (cid:54)= 0,

and

a†i|Φ0(cid:105) = 0

aa|Φ0(cid:105) = 0.

With the new reference vacuum state the Hamiltonian can be rewritten as,

(2.120)

(2.121)

ˆH = ERef + ˆHN ,

(2.122)

48

with the reference energy defined as the expectation value of the Hamiltonian using the

reference state Φ0

ERef = (cid:104)Φ0| ˆH|Φ0(cid:105) =

(cid:88)

i≤F

(cid:104)i|ˆh0|i(cid:105) +

1
2

(cid:88)

ij≤F

(cid:104)ij|ˆv|ij(cid:105),

(2.123)

and the new normal-ordered Hamiltonian (all creation operators to the left of the annihilation

operators) is defined as

(cid:110)

(cid:88)
pq (cid:104)p|ˆh0|q(cid:105)

a†paq

(cid:111)

(cid:88)
pqrs(cid:104)pq|ˆv|rs(cid:105)

+

1
4

(cid:110)

a†pa†qasar

(cid:111)

+

(cid:88)

pq,i≤F

(cid:104)pi|ˆv|qi(cid:105)

(cid:110)

a†paq

(cid:111)

, (2.124)

ˆHN =

where the curly brackets represent normal-ordering with respect to the new reference vacuum

state. The normal-ordered Hamiltonian can be rewritten in terms of a new one-body operator

and a two-body operator as

with

where

ˆHN = ˆFN + ˆVN ,

(cid:88)
pq (cid:104)p| ˆf|q(cid:105)

(cid:110)

a†paq

(cid:111)

,

ˆFN =

(cid:104)p| ˆf|q(cid:105) = (cid:104)p|ˆh0|q(cid:105) +

(cid:88)

i≤F

(cid:104)pi|ˆv|qi(cid:105).

(2.125)

(2.126)

(2.127)

The last term on the right hand side represents a medium modification to the single-particle

Hamiltonian due to the two-body interaction. Finally, the two-body interaction is given by

(cid:110)

(cid:88)
pqrs(cid:104)pq|ˆv|rs(cid:105)

ˆVN =

1
4

(cid:111)

a†pa†qasar

.

49

2.12 Configuration Interaction

With this in place, let’s look again at the many-body Schr¨odinger equation.

ˆH |Ψ(cid:105) = E |Ψ(cid:105)

(2.128)

and we expand the solution to the Schr¨odinger equation in the basis of our complete and

orthonormal antisymmetrized Slater determinants

(cid:88)
(cid:88)

i

i∈all SD’s

(cid:88)

i

ˆH

|Ψ(cid:105) =

Ci |Φi(cid:105) ,

Ci |Φi(cid:105) = E

Ci |Φi(cid:105) .

(2.129)

(2.130)

As stated in the Slater rules section, we can project Eqn. (2.128) onto (cid:104)Φj|, yielding

(cid:88)
(cid:88)

i

(cid:88)

i

(cid:104)Φj| ˆH

Ci |Φi(cid:105) = (cid:104)Φj| E

HijCi = ECi,

Ci |Φi(cid:105) ,

(2.131)

(2.132)

i

which is exactly the index notation for ←→H (cid:126)C = E (cid:126)C, which is the classic statement of the

eigenvalue problem in linear algebra, where the Hamiltonian can be written as a matrix with

matrix elements

Hij = (cid:104)Φi| ˆH|Φj(cid:105) .

To begin solving this equation, first a single-particle basis truncation must be made. Since

the many-body basis is built up from a single-particle basis, this ensures there are a finite

50

number of Slater determinants, and thus a finite number of matrix elements. Once this

matrix is finite, the eigenvalues and eigenvectors can be found by diagonalizing this matrix,

giving the exact solution within this truncated space. Recomputing this solution for larger

and larger single-particle basis set cutoffs will create a series of solutions that can hopefully

generate a smooth curve and an infinite basis limit can be extrapolated to. Thus the many-

body problem is solved.

Well, not quite. Since the size of the Hamiltonian matrix is N × N where N is the
number of Slater determinants, and since the Slater determinants are generated by creating

all permutations in the symmetry group Sn for n single-particle states which grows factorially,
this matrix gets very large, very fast. So computing these matrix diagonalizations with even

the largest modern supercomputers quickly becomes impossible even for systems on the

order of 10 particles. This means that further approximations are necessary to compute

properties of larger quantum systems. Along with the truncation to the single-particle basis,

we can truncate the basis of Slater determinants. While it is valid to take the list of Slater

determinants and throw a large fraction away so that the Hamiltonian matrix is smaller, not

all truncations are equal. This is where the machinery developed in the Fermi level formalism

section comes into use. In most many-body problems, not all configurations of particles and

states are equally important. For example, the probability of finding all A particles in the

A highest lying states in your basis is usually vanishingly small. So the scheme developed

in terms of particle and hole configurations with respect to some reference state keep the

“important” states at the front of our attention. Then, if truncations need to be made, we

can hopefully truncate Slater determinants which do not have much overlap with the ground

state. This of course assumes that an “important” reference state can indeed be picked out,

and this procedure will be expanded on further in the Hartree-Fock section.

51

(cid:88)

i,a

(cid:88)

i,j,a,b

(cid:88)

C
i1,i2,...,iA
a1,a2,...aA

So the complete expansion of a many body state as in Eqn. (2.129) can be rewritten as

|ΨF CI(cid:105) = C0 |Φ0(cid:105) +

i |Φa
Ca

i (cid:105) +

ij |Φab
Cab

ij (cid:105) + ··· +

a1a2...aA
i1i2...iA |Φ

a1a2...aA
i1i2...iA (cid:105) ,

(2.133)

where F CI stands for full configuration interaction, meaning that the full spaces of inter-

acting configurations is being used in this expansion. This is a complete A-body basis, just

reorganizing the Slater determinants in terms of how many excitations away from the refer-

ence state they are. The first approximation is a finite cutoff for the infinite single-particle

basis. Once the single-particle basis has been truncated, the above expression gives the exact

answer in this subspace, as FCI includes all Slater determinants up to A-body excitations,

at which point the series naturally truncates. The next natural approximation, is to not

include every Slater determinant. For example, if only singles and doubles excitations are

included, then we get the configuration interaction singles doubles (CISD) approximation,

which looks like:

|ΨCISD(cid:105) = C0 |Φ0(cid:105) +

(cid:88)

i,a

Ca
i |Φa

i (cid:105) +

(cid:88)

i,j,a,b

Cab
ij |Φab
ij (cid:105) .

(2.134)

While this is a conceptually nice place to truncate the series, it turns to be a pretty poor

way to include correlations into the target many-body state for a given computational cost.

The various ways to optimize these truncated many-body correlations is a widely studied

topic in its own right. Unfortunately, there is no one many-body method to rule them all,

as each approximation has its own strengths and weaknesses. This spans from FCI which

includes every many-body correlation, down to Hartree-Fock which includes only a single

52

Slater determinant.

2.13 Hartree-Fock Theory

Hartree-Fock (HF) theory [35, 36], is an algorithm for finding an approximative expression

for the ground state of a given Hamiltonian. The basic ingredients contain a single-particle

basis {ψα} defined by the solution of the following eigenvalue problem

ˆhHFψα = εαψα,

(2.135)

with the Hartree-Fock Hamiltonian defined as

ˆhHF = ˆt + ˆuext + ˆuHF.

(2.136)

The term ˆuHF is a single-particle potential to be determined by the HF algorithm. The

HF algorithm means to select ˆuHF in order to have

(cid:104) ˆH(cid:105) = EHF = (cid:104)ΦHF

0

| ˆH|ΦHF

0

(cid:105),

(2.137)

as a local minimum with a Slater determinant ΦHF

0

being the ansatz for the ground state.

The variational principle ensures that EHF ≥ E0, with E0 representing the exact ground
state energy.

We will show that the Hartree-Fock Hamiltonian ˆhHF equals our definition of the operator

ˆf discussed in connection with the new definition of the normal-ordered Hamiltonian, that

53

is we have, for a specific matrix element

(cid:104)p|ˆhHF|q(cid:105) = (cid:104)p| ˆf|q(cid:105) = (cid:104)p|ˆt + ˆuext|q(cid:105) +

(cid:88)

i≤F

(cid:104)pi| ˆV |qi(cid:105),

(2.138)

meaning that

(cid:88)

i≤F

(cid:104)pi| ˆV |qi(cid:105).

(cid:104)p|ˆuHF|q(cid:105) =

(2.139)

The so-called Hartree-Fock potential ˆuHF adds an explicit medium dependence due to the

summation over all single-particle states below the Fermi level F . It brings also in an explicit

dependence on the two-body interaction (in nuclear physics we can also have complicated

three- or higher-body forces). The two-body interaction, with its contribution from the

other bystanding fermions, creates an effective mean field in which a given fermion moves,

in addition to the external potential ˆuext which confines the motion of the fermion. For

systems like nuclei or infinite nuclear matter, there is no external confining potential. Nuclei

and nuclear matter are examples of self-bound systems, where the binding arises due to the

intrinsic nature of the strong force. For nuclear systems thus, there would be no external

one-body potential in the Hartree-Fock Hamiltonian.

Another possibility is to expand the single-particle functions in a known basis and vary

the coefficients, that is, the new single-particle wave function is written as a linear expansion

in terms of a fixed chosen orthogonal basis (for example the well-known harmonic oscillator

functions or the hydrogen-like functions, etc). We define our new Hartree-Fock single-particle

basis by performing a unitary transformation on our previous basis (labelled with Greek

indices) as

ψHF
p =

(cid:88)

λ

54

Cpλφλ.

(2.140)

In this case we vary the coefficients Cpλ. If the basis has infinitely many terms, we need to

truncate the above sum. We assume that the basis φλ is orthogonal. A unitary transforma-

tion keeps the orthogonality, which is desired.

It is normal to choose a single-particle basis defined as the eigenfunctions of parts of the

full Hamiltonian. The typical situation consists of the solutions of the one-body part of the

Hamiltonian, that is we have

ˆh0φλ = 
λφλ.

(2.141)

For infinite nuclear matter, ˆh0 is given by the kinetic energy operator and the states are given

by plane wave functions. Due to the translational invariance of the two-body interaction,

the Hartree-Fock single-particle eigenstates are also given by the same functions. Thus, for

infinite matter it is only the single-particle energies that change when we solve the Hartree-

Fock equations.

The single-particle wave functions φλ(r), defined by the quantum numbers λ and r are

defined as the overlap

φλ(r) = (cid:104)r|λ(cid:105).

(2.142)

In our discussions we will use our definitions of single-particle states above and below the

Fermi (F ).

We use Greek letters to refer to our original single-particle basis. The expectation value

for the energy with the ansatz Φ0 for the ground state reads

(cid:88)

µ≤F

(cid:88)

µ,ν≤F

E[Φ0] =

(cid:104)µ|h|µ(cid:105) +

1
2

(cid:104)µν|ˆv|µν(cid:105).

(2.143)

Now we are interested in defining a new basis defined in terms of a chosen basis as defined

55

in Eq. (2.140). We define the energy functional as

E[ΦHF ] =

(cid:104)i|h|i(cid:105) +

1
2

(cid:88)

i≤F

(cid:88)

ij≤F

(cid:104)ij|ˆv|ij(cid:105),

(2.144)

where ΦHF is the new Slater determinant defined by the new basis of Eq. (2.140).

Using Eq. (2.140) we can rewrite Eq. (2.144) as

(cid:88)

(cid:88)

i≤F

αβ

(cid:88)

(cid:88)

ij≤F

αβγδ

E[Ψ] =

C∗iαCiβ(cid:104)α|h|β(cid:105) +

1
2

C∗iαC∗jβCiγCjδ(cid:104)αβ|ˆv|γδ(cid:105).

(2.145)

In order to find the variational minimum of the above functional, we introduce a set of

Lagrange multipliers, noting that since (cid:104)i|j(cid:105) = δi,j and (cid:104)α|β(cid:105) = δα,β, the coefficients Ciγ
obey the relation

(cid:104)i|j(cid:105) = δi,j =

C∗iαCiβ(cid:104)α|β(cid:105) =

C∗iαCiα,

(2.146)

(cid:88)

αβ

(cid:88)

α

(cid:88)

(cid:88)

i≤F

α

which allows us to define a functional to be minimized that reads

F [ΦHF ] = E[ΦHF ] −


i

C∗iαCiα.

(2.147)

Minimizing with respect to C∗iα (the equations for C∗iα and Ciα can be written as two

independent equations) we obtain

E[ΦHF ] −

 = 0,

C∗jαCjα

(cid:88)

j


j

(cid:88)

α

(2.148)

d
dC∗iα

which yields for every single-particle state i and index α (recalling that the coefficients Ciα

are matrix elements of a unitary matrix, or orthogonal for a real symmetric matrix) the

56

following Hartree-Fock equations

(cid:88)

β

Ciβ(cid:104)α|h|β(cid:105) +

(cid:88)

(cid:88)

j≤F

βγδ

C∗jβCjδCiγ(cid:104)αβ|ˆv|γδ(cid:105) = 
HF

i Ciα.

(2.149)

We can rewrite this equation as (changing dummy variables)

(cid:104)α|h|β(cid:105) +

(cid:88)

β

(cid:88)

(cid:88)

j≤F

γδ

 Ciβ = 
HF

C∗jγCjδ(cid:104)αγ|ˆv|βδ(cid:105)

i Ciα.

(2.150)

Note that the sums over Greek indices run over the number of basis set functions (in principle

an infinite number).

Defining

hHF
αβ = (cid:104)α|h|β(cid:105) +

(cid:88)

(cid:88)

j≤F

γδ

C∗jγCjδ(cid:104)αγ|ˆv|βδ(cid:105),

(2.151)

we can rewrite the new equations as

(cid:88)

β

hHF
αβ Ciβ = 
HF

i Ciα.

(2.152)

The latter is nothing but a standard eigenvalue problem. Our Hartree-Fock matrix is thus

ˆhHF
αβ = (cid:104)α|ˆh0|β(cid:105) +

C∗jγCjδ(cid:104)αγ|ˆv|βδ(cid:105).

(2.153)

(cid:88)

(cid:88)

j≤F

γδ

The Hartree-Fock equations are solved in an iterative way starting with a guess for the

coefficients Cjγ = δj,γ and solving the equations by diagonalization till the new single-particle

energies 
HF

i

do not change anymore by a user-defined small quantity.

Normally we assume that the single-particle basis |β(cid:105) forms an eigenbasis for the operator

57

ˆh0, meaning that the Hartree-Fock matrix becomes

(cid:88)

(cid:88)

j≤F

γδ

ˆhHF
αβ = 
αδα,β +

C∗jγCjδ(cid:104)αγ|ˆv|βδ(cid:105).

(2.154)

2.14 Many-Body Perturbation Theory

Hartree-Fock theory is incredibly powerful considering how simple the idea is: transform

the single-particle basis to optimize a single Slater determinant. This optimization problem

can be thought of as summing the interactions from the surrounding particles to create a

sort of external “mean field” that each particle feels. In some cases, this mean field approach

is sufficient to answer the physics questions being asked. However, the heart of many-body

theory is the correlation between particles that are purely many-body in nature, meaning

that they cannot be described in the independent particle picture. Historically, many-body

perturbation theory (MBPT) [37, 38, 33, 16] has been a first attempt to build these many-

body correlations on top of a reference state. Like perturbation theory in other branches of

physics, or in single particle quantum mechanics, it is assumed that the problem at hand

is a small “perturbation” away from a reference problem. In the many-body case, starting

from a good reference state is essential, as a bad starting point will require many additional

corrections, or in some cases adding additional corrections will not work at all. To set the

stage for deriving MBPT, we assume here that we are only interested in the non-degenerate

ground state of a given system and expand the exact wave function in terms of a series of

Slater determinants

∞(cid:88)

m=1

Cm|Φm(cid:105),

|Ψ0(cid:105) = |Φ0(cid:105) +

58

(2.155)

where we have assumed that the true ground state is dominated by the solution of the

unperturbed problem, that is

ˆH0|Φ0(cid:105) = W0|Φ0(cid:105),

and that the full Hamiltonian is given by this term plus a small interaction term

ˆH = ˆH0 + ˆHI .

(2.156)

(2.157)

The state |Ψ0(cid:105) is not normalized and we employ again intermediate normalization via
(cid:104)Φ0|Ψ0(cid:105) = 1.

The Schr¨odinger equation is given by

ˆH|Ψ0(cid:105) = E|Ψ0(cid:105),

and multiplying the latter from the left with (cid:104)Φ0| gives

(cid:104)Φ0| ˆH|Ψ0(cid:105) = E(cid:104)Φ0|Ψ0(cid:105) = E,

and subtracting from this equation

(2.158)

(2.159)

(cid:104)Ψ0| ˆH0|Φ0(cid:105) = W0(cid:104)Ψ0|Φ0(cid:105) = W0,

(2.160)

and using the fact that the operators ˆH and ˆH0 are hermitian results in

∆E = E − W0 = (cid:104)Φ0| ˆHI|Ψ0(cid:105),

(2.161)

59

which is an exact result. The total energy can be separated into two terms

E = ERef + ∆E,

(2.162)

where ∆E is the correlation energy, and the reference energy is given by

ERef = (cid:104)Φ0| ˆH|Φ0(cid:105).

(2.163)

Equation (2.161) forms the starting point for all perturbative derivations. However, as

it stands it represents nothing but a mere formal rewriting of Schr¨odinger’s equation and is

not of much practical use. The exact wave function |Ψ0(cid:105) is unknown. In order to obtain a
perturbative expansion, we need to expand the exact wave function in terms of the interaction

ˆHI .

Here we have assumed that our model space defined by the operator ˆP is one-dimensional,

meaning that

and

ˆP = |Φ0(cid:105)(cid:104)Φ0|,
∞(cid:88)

ˆQ =

|Φm(cid:105)(cid:104)Φm|.

m=1

We can thus rewrite the exact wave function as

|Ψ0(cid:105) = ( ˆP + ˆQ)|Ψ0(cid:105) = |Φ0(cid:105) + ˆQ|Ψ0(cid:105).

(2.164)

(2.165)

(2.166)

Going back to the Schr¨odinger equation, we can rewrite it, adding and a subtracting a term

60

ω|Ψ0(cid:105) as

(cid:17)

(cid:16)

ω − ˆH0

|Ψ0(cid:105) =

(cid:16)

ω − E + ˆHI

(cid:17)

|Ψ0(cid:105),

(2.167)

where ω is an energy variable to be specified later.

We assume also that the resolvent of

ω − ˆH0

defines the unperturbed Green’s function as

(cid:17)

exits, that is it has an inverse which

(cid:16)

(cid:17)−1

(cid:16)

ω − ˆH0

|Ψ0(cid:105) =

1

ω − ˆH0

(cid:16)

=

(cid:17).

1

ω − ˆH0

(cid:17)

|Ψ0(cid:105),

ω − E + ˆHI

(cid:16)

We can rewrite Schr¨odinger’s equation as

(2.168)

(2.169)

(2.170)

and multiplying from the left with ˆQ results in

(cid:16)
ω − E + ˆHI

(cid:17)

|Ψ0(cid:105),

ˆQ|Ψ0(cid:105) =

ˆQ

ω − ˆH0

which is possible since we have defined the operator ˆQ in terms of the eigenfunctions of ˆH0.

Since these operators commute we have

(cid:16)

ˆQ

(cid:17) ˆQ = ˆQ

(cid:16)

(cid:17) =

(cid:17).

ˆQ(cid:16)
ω − ˆH0

1

ω − ˆH0

1

ω − ˆH0

(2.171)

With these definitions we can in turn define the wave function as

(cid:16)

(cid:17)

ω − E + ˆHI

|Ψ0(cid:105).

(2.172)

|Ψ0(cid:105) = |Φ0(cid:105) +

ˆQ

ω − ˆH0

61

So far, this is just a reorganization of the Schr¨odinger equation. It is a non-linear equation

in two unknown quantities, the energy E and the exact wave function |Ψ0(cid:105). We can however
start with a guess for |Ψ0(cid:105) on the right hand side of the last equation.

The most common choice is to start with the function which is expected to exhibit the

largest overlap with the wave function we are searching after, namely |Φ0(cid:105). This can again
be inserted in the solution for |Ψ0(cid:105) in an iterative fashion and if we continue along these
lines we end up with

(cid:40) ˆQ

∞(cid:88)

(cid:16)

(cid:17)(cid:41)i

|Ψ0(cid:105) =

i=0

ω − ˆH0

ω − E + ˆHI

|Φ0(cid:105),

(2.173)

for the wave function and

∆E =

∞(cid:88)

i=0

(cid:104)Φ0| ˆHI

(cid:40) ˆQ

(cid:16)

ω − ˆH0

(cid:17)(cid:41)i

ω − E + ˆHI

|Φ0(cid:105),

(2.174)

which is now a perturbative expansion of the exact energy in terms of the interaction ˆHI

and the unperturbed wave function |Ψ0(cid:105).

In our equations for |Ψ0(cid:105) and ∆E in terms of the unperturbed solutions |Φi(cid:105) we have
still an undetermined parameter ω and a dependency on the exact energy E. Not much has

been gained thus from a practical computational point of view.

In Brillouin-Wigner perturbation theory [16, 39] it is customary to set ω = E. This

62

results in the following perturbative expansion for the energy ∆E

∞(cid:88)
(cid:32)
(cid:104)Φ0| ˆHI

i=0

(cid:40) ˆQ

(cid:16)

ω − ˆH0
ˆQ

∆E =

(cid:17)(cid:41)i

ω − E + ˆHI

|Φ0(cid:105) =

(cid:104)Φ0|

ˆHI + ˆHI

ˆHI + ˆHI

ˆQ

E − ˆH0

ˆHI

ˆQ

E − ˆH0

E − ˆH0

(cid:33)

ˆHI + . . .

|Φ0(cid:105).

(2.175)

(2.176)

This expression depends however on the exact energy E and is again not very convenient

from a practical point of view. It can obviously be solved iteratively, by starting with a guess

for E and then solve till some kind of self-consistency criterion has been reached.

Defining e = E − ˆH0 and recalling that ˆH0 commutes with ˆQ by construction and that ˆQ
is an idempotent operator ˆQ2 = ˆQ, we can rewrite the denominator in the above expansion

for ∆E as

ˆQ

1

ˆe − ˆQ ˆHI ˆQ

= ˆQ

(cid:20)1

ˆe

+

1
ˆe

ˆQ ˆHI ˆQ

1
ˆe

+

1
ˆe

ˆQ ˆHI ˆQ

1
ˆe

ˆQ ˆHI ˆQ

1
ˆe

+ . . .

(cid:21)

ˆQ.

Inserted in the expression for ∆E, we obtain

∆E = (cid:104)Φ0| ˆHI + ˆHI ˆQ

1

E − ˆH0 − ˆQ ˆHI ˆQ

ˆQ ˆHI|Φ0(cid:105).

(2.177)

(2.178)

In Rayleigh-Schr¨odinger (RS) perturbation theory [40, 41, 16] we set ω = W0 and obtain the

following expression for the energy difference

∞(cid:88)

i=0

∆E =

(cid:40)

(cid:32)
(cid:104)Φ0| ˆHI

ˆQ

W0 − ˆH0
ˆQ

(cid:17)(cid:41)i

(cid:16)

ˆHI − ∆E

|Φ0(cid:105)

(cid:33)

=(cid:104)Φ0|

ˆHI + ˆHI

( ˆHI − ∆E) + . . .

|Φ0(cid:105).

W0 − ˆH0

63

(2.179)

(2.180)

The operator ˆQ commutes with ˆH0 and since ∆E is a constant we obtain that

ˆQ∆E|Φ0(cid:105) = ˆQ∆E| ˆQΦ0(cid:105) = 0.

Inserting this result in the expression for the energy gives us

(cid:32)

∆E = (cid:104)Φ0|

ˆHI + ˆHI

ˆQ

W0 − ˆH0

ˆHI + ˆHI

ˆQ

W0 − ˆH0

( ˆHI − ∆E)

ˆQ

W0 − ˆH0

ˆHI + . . .

(2.181)

(cid:33)

|Φ0(cid:105).
(2.182)

We can now perturbatively expand this expression in terms of the interaction ˆHI , which

is assumed to be small. We obtain then

∆E =

with the following expression for ∆E(i)

∞(cid:88)

i=1

∆E(i),

∆E(1) = (cid:104)Φ0| ˆHI|Φ0(cid:105),

which is just the contribution to first order in perturbation theory,

∆E(2) = (cid:104)Φ0| ˆHI

ˆQ

W0 − ˆH0

ˆHI|Φ0(cid:105),

(2.183)

(2.184)

(2.185)

which is the contribution to second order. There exists a formal theory for the calculation of

each additional term, see for example Ref. [16], where a diagrammatic method is described to

generate any order of MBPT. Inserting in the ˆQ space operator and the energy denominators

in Eqn. (2.185) we get the expression for MBPT(2), the energy correction for many-body

64

perturbation theory to second order,

∆E(2) =

1
22

(cid:88)

(cid:88)

ij≤F

ab>F

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)
εi + εj − εa − εb

.

(2.186)

In the expressions for the various diagrams the quantity ε denotes the single-particle

energies defined by H0. Many-body perturbation theory is quite powerful, and can provide

corrections quite accurately for many systems. Unfortunately, most nuclear physics appli-

cations do not fall in this regime where MBPT is accurate due to that large short range

correlations of the nucleon-nucelon potential. Any system for which the interactions are too

large compared to the non-interacting mean-field model will not be meaningfully captured

by MBPT even at high orders of correction. This will be illustrated in Chapter 3, where the

simple pairing model is examined.

2.15

In-Medium Similarity Renormalization Group

So far we have covered three important methods in quantum many-body theory. First,

was Full configuration interaction (FCI), where the only approximation is the necessary

truncation to the single-particle basis. The many-body basis is then constructed from the

single-particle basis, but FCI does not make any additional truncations to the many-body

space. Next, was Hartree-Fock (HF) mean field theory, which optimizes the many-body

ground state energy in a single Slater determinant by performing a unitary transformation

on the single-particle basis. Last, was many-body perturbation theory (MBPT), which exists

between a simple mean field model and the full many-body solution. This is the regime

where the vast majority of many-body physics is done, as many interesting problems are

65

too computationally expensive for FCI, but too correlated for Hartree-Fock to be sufficient.

As a result, many different approximations to the many-body problem have been developed

with various strengths and weaknesses to target different applications. The strength of ab

initio many-body methods is that by maintaining the essential degrees of freedom, all of the

approximations can be extended to recover the full solution. Of course the trade-off is then

that these extensions towards the exact solutions again become prohibitively expensive, but

this allows for a choice in the trade-off between accuracy in computational cost. Additionally,

in nuclear physics the nuclear potentials can be developed in a similar way that allows for

increased accuracy (at increasing computational cost), but this work will focus on various

approximations to the many-body methods rather than the input potentials. In particular,

most of the calculations in this work are done with an ab initio many-body method called

coupled cluster (CC) theory, which is explained in detail in Chapter 4. Another ab initio

many-body method similar to coupled cluster is in-medium similarity renormalization group

(IM-SRG) [42, 43, 44, 45, 46, 47, 48, 3].

The renormalization group is a tool that has been used in physics for many decades,

which allows physical quantities of interest to be examined at different distance or energy

scales and has been essential in the development of quantum electrodynamics and quantum

chromodynamics. In nuclear physics most realistic nuclear potentials have a sharp repulsive

core which can lead to divergences in calculating matrix elements, generating strong off-

diagonal contributions as low momentum modes are coupled to high momentum modes.

However for certain physical quantities, like the ground state energy of an atomic nucleus,

the low energy of the system indicates that the nucleons should not probe the very short

range distance scales of this repulsive core. Similarity renormalization group (SRG) has

had success in “softening” the repulsive of the nuclear potential, by driving the momenum

66

space interaction matrix to a band diagonal form, decoupling the high momentum from low

momentum modes while maintaining accuracy of the target observables [4, 5].

In-medium similarity renormalization group (IM-SRG) takes this philosophy of decou-

pling distance scales and applies it to the “medium” of a particular reference state. The idea

is that for a matrix problem in the many-body basis, transforming the matrix to a form where

the ground state is decoupled from the rest of the matrix will give the ground state eigen-

value. In the case of IM-SRG, the Hamiltonian is normal ordered with respect to a reference

state and the ground state energy is isolated by a continuous unitary transformation.

A unitary transformation U is an isomorphism between two Hilbert spaces H1, H2 that

preserves the inner product,

U : H1 → H2,

(cid:104)U x, U y(cid:105)H1

= (cid:104)x, y(cid:105)H2

,

∀x, y ∈ H1. Similarly, for an antiunitary transformation,

(cid:104)U x, U y(cid:105) = (cid:104)x, y(cid:105)∗ = (cid:104)y, x(cid:105) .

Thus for a unitary transformation, in bra-ket notation,

(cid:104)U x|U y(cid:105) = (cid:104)x|U†U|y(cid:105) = (cid:104)x|y(cid:105) ,

=⇒ U†U = U U† = 1.

67

(2.187)

(2.188)

(2.189)

(2.190)

(2.191)

A continuous unitary transformation is a unitary transformation parametrized by some con-

tinuous parameter s, such that U (s)U (s)† = 1. This generates a unitarily transformed

Hamiltonian for all points s,

H(s) = U†(s)HU (s).

(2.192)

The transformation is implemented by solving a coupled set of flow equations for the matrix

elements for the Hamiltonian, which we can find by taking the derivative of Eqn. (2.192),

dH(s)

ds

=

dU†(s)

ds

HU (s) + U†(s)H

dU (s)

ds

,

and the derivative of Eqn. (2.191)

dU†(s)

ds

U (s) + U†(s)

dU (s)

ds

= 0.

From here, we write down the generator of the transformation η as

η(s) =

dU†(s)

ds

U (s) = −U†(s)

dU (s)

ds

,

which leads to the flow equations as

=(cid:2)η(s), H(s)(cid:3).

dH(s)

ds

(2.193)

(2.194)

(2.195)

(2.196)

For actual calculations, an explicit expression from the transformation U (s) is rarely written

out. Instead, the generator η defines the unitary transformation. To actually implement

68

this, we partition the Hamiltonian as

H(s) = Hd(s) + Hod(s),

(2.197)

where these are the diagonal and off-diagonal components of the matrix. The evolution

with the continuous flow parameter s is again The choice of the generator first suggested by

Wegner [49],

guarantees

η(s) =(cid:2)Hd(s), H(s)(cid:3) =(cid:2)Hd(s), Hod(s)(cid:3),
(cid:18)(cid:0)Hod(cid:1)2
= 2Tr(cid:0)η2(cid:1) = −2Tr(cid:0)η†η(cid:1)

(cid:19)

≤ 0,

d
ds

Tr

(2.198)

(2.199)

which demonstrates that Hod decays with increasing s which is precisely what is needed

to decouple the high and low momentum modes[50]. Analyzing the flow equations in the

eigenbasis of Hd(s) and defining Hd

ii(s) ≡ 
i one can show that

ij (s) ∼ e−s(
i−
j )2
Hod

Hod

ij (0).

(2.200)

However, this can lead to stiff ODE’s, so a more common generator is the White generator

[51]

which gives uniform surpression

ηij(s) =

Hod
ij (s)

i − 
j

,

Hod
ij (s) ∼ e−sHod

ij (0).

69

(2.201)

(2.202)

While SRG is typically used to soften nuclear potentials with a repulsive core, an alter-

native is to perform the SRG evolution in-medium (IM-SRG) for each A-body system of

interest. Starting from a general second-quantized Hamiltonian with two- and three-body

interactions

(cid:88)

qr

H =

Tqra†qar +

1
2!2

(cid:88)

qrst

(2)
qrsta†qa†ratas +
V

1
3!2

(cid:88)

qrstuv

(3)
qrstuva†qa†ra†savauat + . . .
V

(2.203)

All operators can be normal-ordered with respect to a finite-density Fermi vacuum |Φ(cid:105) (e.g.
the Hartree-Fock ground state), as opposed to the zero particle vacuum. Wick’s theorem

can then be used to exactly write H as

(cid:88)

qr

(cid:88)

qrst

H = E +

fqr{a†qar}+

1
4

Γqrst{a†qa†ratas}+

1
36

(cid:88)

qrstuv

Wqrstuv{a†qa†ra†savauat}, (2.204)

where strings of normal ordered operators obey

(cid:104)Φ|{a†q . . . ar}|Φ(cid:105) = 0,

(2.205)

and the terms in (2.204) are given by

(cid:88)

E =

Tqqnq +

q

fqr = Tqr +

(cid:88)
(cid:88)

s

u

Γqrst = V

(2)
qrst +

Wqrtsuv = V

(3)
qrstuv,

(cid:88)

qr

1
2

V

(2)
qsrsns +

V

(2)
qrqrnqnr +

(cid:88)

st

1
2

(cid:88)

qrs

1
6

(3)
qrsqrsnqnrns,

V

(2.206)

(3)
V
qstrstnsnt,

(2.207)

(2.208)

(2.209)

(3)
V
qrustunu,

70

where nq = θ(
F − 
q) are the occupation numbers in the reference state |Φ(cid:105). Notice
that the normal ordered 0-,1-, and 2-body terms include contributions from the three-body

interaction V (3) through sums over the occupied single-particle states in the reference state

|Φ(cid:105). Neglecting the residual three-body interaction leads to the normal-ordered two-body
approximation (NO2B) which has shown to be an excellent approximation in nuclear systems.

Truncating the in-medium SRG equations to normal-ordered two-body operators is denoted

IM-SRG(2). Using this normal ordered Hamiltonian and using Wick’s theorem on Eqn.

(2.196) with H(s) = E0(s) + f (s) + Γ(s) and truncating η(s) = η(1)(s) + η(2)(s) to two-body

yields the coupled IM-SRG(2) equations

dE0
ds

=

ηqrfrq(nq − nr) +

(cid:88)

qrst

1
2

ηqrstΓstqrnqnr ¯ns¯nt,

(2.210)

dfqr
ds

=

(1 + Pqr)ηqsfsr +

(cid:88)

st

(ns − nt)(ηstΓtqsr − fstηtqsr)

(nsnt¯nu + ¯ns¯ntnu)(1 + Pqr)ηuqstΓstur,

dΓqrst

ds

=

(1 − Pqr)(ηquΓurst − fquηurst)

(cid:88)

qr

(cid:88)
(cid:88)

s

+

stu

u

(cid:88)
(cid:88)
(cid:88)
(cid:88)

u
1
2

−

+

uv

−

uv

(2.211)

(2.212)

(1 − Pst)(ηusΓqrut − fusηqrut)

(1 − nu − nv)(ηqruvΓuvst − Γqruvηuvst)

(nu − nv)(1 − Pqr)(1 − Pst)ηvrutΓuqvs,

71

where the ¯nr ≡ (1 − nr), Pqr is the permutation operator and s dependence has been made
implicit to clear up visual clutter. The White generator is then

(cid:88)

ai

η =

(cid:88)

abij

fai

fa − fi{a†aai} +

1
4

Γabij

fa + fb − fi − fj {a†aa†bajai} − H.c.,

(2.213)

where fa = faa are the Møller-Plesset energy denominators. Unfortunately, these equations

can be very sensitive, as small amounts of numerical error can break the unitarity of the

transformation. This means that to solve these equations, a high-order differential equation

solver is typically needed. These solvers need to store many copies of the solution vector to

maintain accuracy, and these copies rapidly increase the storage requirements. Fortunately,

the Magnus expansion can help out here, ensuring that unitarity is preserved at every step

in the differential equation.

2.16 The Magnus Formulation of IM-SRG

The starting point of the Magnus formulation [50] of IM-SRG is once again taking the

derivative of the unitarity condition U (s)U†(s) = U†(s)U (s) = 1,

dU (s)

ds

U†(s) = −U (s)

dU†(s)

ds

.

(2.214)

and multiply Eqn. (2.214) on the right by U (s) to yield the

Now define η ≡ U (s)
differential equation

dU†(s)

ds

dU (s)

ds

dU†(s)

U†(s)U (s) = −U (s)
=⇒

ds
= −η(s)U (s),

dU (s)

ds

U (s),

(2.215)

(2.216)

72

with the boundary condition U (0) = 1. To get some intuition for this differential equation, we

look to a familiar unitary transformation, like the time evolution operator, the Hamiltonian.

U (t) = e−iHt.

Taking the time derivative yields

dU (t)

dt

= −iHe−iHt = −iHU (t).

(2.217)

(2.218)

This is true when H is independent of t and explains why the solution is so compact. If we

look at U (s) = e−ηs, the derivative would be

dU (s)

ds

= −ηe−ηs = −ηU (s),

(2.219)

which would be a nice solution to the differential equation. So it becomes clear that the

s dependence in η(s) makes things more complicated. If we had U (s) = e−η(s), then the

derivative would be

dU (s)

ds

dη(s)

ds

= −

e−η(s).

(2.220)

Thus to get the solution we want, we need the anti-derivative of η(s) to be exponentiated;

so something like Exp(−
matrix exponential, which is really just short hand for the polynomial series. So terms like

0 η(s(cid:48))ds(cid:48)). But here, there are all sorts of issues since this is a

(cid:82) s

(cid:90) s

(cid:90) s

1
n!

0 ···

0

η(s(cid:48)1) . . . η(s(cid:48)n)ds1 . . . dsn,

(2.221)

73

arise. And here, unless all of the η terms at any value of s commute, the order matters. This

can be formally integrated as the time-ordered exponential

(cid:8)e−

0 η(s(cid:48))ds(cid:48)(cid:9)
(cid:82) s

≡ 1 −

U (s) = Ts

(cid:90) s

0

(cid:90) s

(cid:90) ds(cid:48)

0

0

ds(cid:48)η(s(cid:48)) +

ds(cid:48)

ds(cid:48)(cid:48)η(s(cid:48))η(s(cid:48)(cid:48)) + . . .

(2.222)

This is not useful from a practical point of view. The Magnus expansion [52] is the statement

that given a few technical requirements on η(s), a solution of the form

U (s) = eΩ(s)

(2.223)

exists, where Ω†(s) = −Ω(s) and Ω(0) = 0. This lines up with the previously stated
boundary condition of U (0) = 1, which is satisfied by Ω(0) = 0. The anti-Hermitian property

of Ω is necessary since for any unitary operator U to be expressed as exponentiated operator

Ω requires that the exponentiated operator Ω be anti-Hermitian,

U U† = 1

= eΩeΩ† = eΩ+Ω†e[Ω,Ω†].
?

(2.224)

This expression will be satisfied as long as Ω† = −Ω, since

(cid:2)Ω, Ω†(cid:3) = ΩΩ† − Ω†Ω = −Ω2 + Ω2 = 0

(cid:2)Ω,Ω†(cid:3)

=⇒ eΩ+Ω†e

= e0e0 = 1.

(2.225)

This is why the time evolution operator eiHt has the characteristic phase i. Since H is

Hermitian, the i is needed in the exponential to make the exponent anti-Hermitian overall

to ensure the unitarity of the transformation.

74

In previous applications of the Magnus expansion, Ω(s) is expanded in powers of η(s) as

∞(cid:88)

n=1

Ω =

Ωn

(2.226)

where

(cid:90) s
(cid:90) s

0

0

Ω1(s) = −
1
2

Ω2(s) =

...

ds1η(s1)

(cid:90) s1

0

ds1

ds2

(cid:2)η(s1), η(s2)(cid:3).

(2.227)

(2.228)

(2.229)

Here, the complications of the time-ordered exponential are moved inside the exponential.

The advantage of this is that truncating Ω at any order will still be anti-Hermitian, and

thus result in a unitary transformation. This is unlike truncating (2.222) which is not

guaranteed to be unitary if any truncations are made. Let’s quickly check that truncating Ω

will still be anti-Hermitian. First, check that η is anti-Hermitian. Starting from U (s)

dU†(s)

ds =

− dU (s)

ds U†(s) with the fact that (AB)† = B†A†

(cid:16)

(cid:17)† =

η†(s) =

U (s)

dU†(s)

ds

dU (s)

ds

U†(s) = −η(s)

(2.230)

which shows that η is indeed anti-Hermitian. To check that the commutators in a term like

75

Ω2 are anti-Hermitian, let’s call A =(cid:2)η(s1), η(s2)(cid:3). Then

A† =(cid:2)η(s1), η(s2)(cid:3)† =

(cid:16)

(cid:17)†

η(s1)η(s2) − η(s2)η(s1)
= η(s2)†η(s1)† − η(s1)†η(s2)†
= (−1)2η(s2)η(s1) − (−1)2η(s1)η(s2)
= −
= −A.

(cid:0)η(s1)η(s2) − η(s2)η(s1)(cid:1)

This proves that the commutator of any two anti-Hermitian operators is itself anti-Hermitian.

Therefore, every term of the Magnus expansion is anti-Hermitian, so truncating at any level

ensures that Ω is anti-Hermitian.

To demonstrate the SRG, let’s consider a small two-level system, represented by the

(2.231)

(2.232)

(2.233)

 .

1

1

1 −1

initial Hamiltonian

H = T + V =

76

Let’s try to diagonalize H using the Wegner generator η(s) = [T, H(s)],

1



1

1

0



1 −1

0 −1

η(0) = T H − HT

1

1 −1


1
 −
1 −1
 −



1

1

1

2

1
 1
 0

=

=

=

0

0 −1

−1 1

−2 0

= 2iσ2,

(2.234)

and by definition, Ω(0) = 0. Looking at the recursively defined derivative of Ω

∞(cid:88)

k=0

dΩ
ds

=

Bk
k!

adk

Ω(η),

ad0

adk

Ω(η) = η,

Ω(η) =(cid:2)Ω, adk−1
Ω (η)(cid:3).

(2.235)

At s = 0 we have dΩ

ds |s=0 = η(0), since ad1

Ω = [0, η(0)] = 0. In general, the next step is to

calculate Ω(s) by integrating Eqn. (2.235), and then find the transformed Hamiltonian as

H(s) = eΩ(s)H(0)e−Ω(s)

(2.236)

by using the Baker-Campbell-Hausdorf expansion. However, in this simple model, we can just

take the exponential of the Pauli matrices rather than doing a truncated BCH expansion.

77

With this example η and Ω truncate after one term, and will always be antisymmetric

matrices, that is

η(s) = igη(s)σ2

Ω(s) = igΩ(s)σ2.

(2.237)

(2.238)

With this form for Ω, we can look at the exact BCH expansion for H(0) = σ1 + σ3, using

the matrix exponential of a Pauli matrix. In the case where (cid:126)a = aˆn, we have

eia(ˆn·(cid:126)σ) = 1cos(a) + i(ˆn · (cid:126)σ)sin(a).

(2.239)

In our case, we want eΩ = eigσ2 so to get this, ia(ˆn · (cid:126)σ) = igσ2, therefore a = g and ˆn = ˆy.
This leads to

eigσ2 = 1cos(g) + iσ2sin(g)

(2.240)

and for e−Ω just take g → −g. Then use cos(−g) = cos(g) and sin(−g) = sin(g). Thus the

78

transformed Hamiltonian is

H(s) = eigσ2(σ1 + σ3)e−igσ2

=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)σ1 + σ3)(1cos(g) − iσ2sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)σ1cos(g) − iσ1σ2sin(g) + σ3cos(g) − iσ3σ2sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)σ1cos(g) − i2σ3sin(g) + σ3cos(g) + i2σ1sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)σ1(cos(g) − sin(g)) + σ3(cos(g) + sin(g))(cid:3)

= σ1(cos(g) − sin(g))cos(g) + σ3(cos(g) + sin(g))cos(g)
+ iσ2σ1(cos(g) − sin(g))sin(g) + iσ2σ3(cos(g) + sin(g))sin(g)
= σ1(cos(g) − sin(g))cos(g) + σ3(cos(g) + sin(g))cos(g)
− i2σ3(cos(g) − sin(g))sin(g) + i2σ1(cos(g) + sin(g))sin(g)
= σ1

(cid:2)cos2(g) − sin2(g) − 2cos(g)sin(g)(cid:3)
(cid:2)cos2(g) − sin2(g) + 2cos(g)sin(g)(cid:3)
(cid:2)cos(2g) − sin(2g)(cid:3) + σ3

(cid:2)cos(2g) + sin(2g)(cid:3),

+ σ3

= σ1

where we used σaσb = δab1 + iεabcσc as well as the trig identity cos2(g) − sin2(g) = cos(2g)
and sin(2g) = 2sin(g)cos(g). This can be generalized slightly for a Hamiltonian of the form

79

H(0) = dσ3 + vσ1, where the full transformed Hamiltonian is

H(s) = eigσ2(vσ1 + dσ3)e−igσ2

=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(vσ1 + dσ3)(cid:2)1cos(g) − iσ2sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)vσ1cos(g) − ivσ1σ2sin(g) + dσ3cos(g) − idσ3σ2sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)vσ1cos(g) − i2vσ3sin(g) + dσ3cos(g) + i2dσ1sin(g)(cid:3)
=(cid:2)1cos(g) + iσ2sin(g)(cid:3)(cid:2)σ1(vcos(g) − dsin(g)) + σ3(dcos(g) + vsin(g))(cid:3)

= σ1(v ∗ cos(g) − d ∗ sin(g))cos(g) + σ3(d ∗ cos(g) + v ∗ sin(g))cos(g)
+ iσ2σ1(v ∗ cos(g) − d ∗ sin(g))sin(g) + iσ2σ3(d ∗ cos(g) + v ∗ sin(g))sin(g)
= σ1(v ∗ cos(g) − d ∗ sin(g))cos(g) + σ3(d ∗ cos(g) + v ∗ sin(g))cos(g)
− i2σ3(v ∗ cos(g) − d ∗ sin(g))sin(g) + i2σ1(d ∗ cos(g) + v ∗ sin(g))sin(g)
= σ1

(cid:2)v ∗ cos2(g) − v ∗ sin2(g) − 2d ∗ cos(g)sin(g)(cid:3)
(cid:2)d ∗ cos2(g) − d ∗ sin2(g) + 2v ∗ cos(g)sin(g)(cid:3)
(cid:2)v ∗ cos(2g) − d ∗ sin(2g)(cid:3) + σ3

(cid:2)d ∗ cos(2g) + v ∗ sin(2g)(cid:3).

+ σ3

= σ1

(2.241)

With these exact results derived, we can compare the direct SRG integration against

the Magnus formulation. This is shown in figure 2.1, and it is clear how much numerical

stability is gained from ensuring the unitarity each step with the Magnus expansion. The

Magnus formulation relatively quickly reaches the “machine precision” of the finite precision

floating point variables in the calculation, while the direct integration method has error that

is highly dependent on the “time step” δs of the integration.

This 2×2 toy example hides two sources of error that would exist in an actual calculation.
This first is that the expressions truncate naturally for η and Ω, which will not happen in

80

Figure 2.1: SRG with direct integration and with the Magnus expansion.

a many-body physics problem, and the second source of error is that the BCH expansion

is done to infinite order in Eqn. (2.241). In a realistic many-body calculation, the terms

in these series are observed to decrease monotonically, and a cutoff tolerance can be used

to minimize the numerical error from truncation. To gain insight into the error generated

from the BCH expansion truncation, the nested commutators can be compared against the

exact expression derived in Eqn. (2.241). In figure 2.2 this source of truncation error goes

back down to machine precision after about 20 terms, which is pretty substantial, although

machine precision is rarely necessary.

Rather than setting a fixed number of terms in the expansion, it is more typical to specify

an error tolerance so that the BCH expansion can be assuredly not the primary source of

error. This is what is done in figure 2.3, where the tolerance 
 here is defined to be greater

than the row 0 column 0 element of the next term in the BCH expansion.

By setting 
 = 10−8, we are asserting that we want the smallest eigenvalue (the (0,0)

element of the matrix) to be changing by not more than 10−8 for the next term in the BCH

81

Figure 2.2: Magnus SRG with the exact unitary transformation and with a BCH expansion
truncated after a fixed number of terms

series. There are other tolerances that can be chosen, but looking at the plot, it seems to

work well. As a note, the three tolerances of (10−8, 10−12, 10−16) used (12, 15, 18) terms in

their expansions.

The Magnus expansion has shown to be a great tool for these calculations, but potentially

the greatest gain from this is that any operator can also be computed after the flow. Once
the flow has finished, it only costs an exact BCH to compute O(s) = eΩOeΩ† along with
H(s) = eΩHeΩ†.

While the 2x2 example was not an exercise in many-body physics, IM-SRG and the use of

the Magnus expansion have seen great success in the calculation of many interesting nuclear

systems [42, 43, 44, 45, 46, 47, 48, 3]. The many-body physics results of this work will focus

on the use of coupled cluster theory which is detailed in Chapter 4.

82

Figure 2.3: Magnus SRG with the exact unitary transformation and with a BCH expansion
truncated after a fixed tolerance is met

83

Chapter 3

Physical Systems

With the mathematical framework of quantum many-body theory as a foundation any

quantum system can be investigated numerically. The many-body Schr¨odinger equation has

proven to be an excellent model for studying nearly any physical system for which the parti-

cles are traveling sufficiently slower than the speed of light. A wide range of fields including

atomic physics, quantum chemistry, materials science, and nuclear physics greatly bene-

fit from these theoretical tools, which make studying the mathematical and computational

methods surrounding many-body physics a worthwhile endeavor per se. While the physi-

cal systems introduced in this chapter have applications in answering real world questions,

much of the interest in these systems is theoretical. The pairing model is a simple quantum

system which can be studied analytically and exactly. It is therefore an excellent testing

ground for properties of various many-body methods, and as a system to validate numerical

implementations.

Infinite fermionic matter is important for studying valence electrons in

metals [23], and also for studying the volumetric bulk of neutrons thought to constitute the

crust of neutron stars, or as a model for dense nuclear matter [2, 1, 22].

84

3.1 Pairing Model

The pairing model Hamiltonian ˆH = ˆH0 + ˆV is defined as

(cid:88)
(p − 1)a†pσapσ

pσ

ˆH0 = δ

(cid:88)

pq

ˆV = −

1
2

g

a†p+a†p−, aq−aq+

(3.1)

(3.2)

which represents a quantum system with p levels, each having a spin degeneracy of two.

A common choice for single-particle states are eigenstates of the Hartree-Fock operator,

(ˆu + ˆuHF)|p(cid:105) = 
p |p(cid:105). In the pairing model, this condition is already fulfilled. We define
the states below the Fermi level as holes and redefine the single-particle energies,

(cid:88)

i


q = hqq +

(cid:104)qi|ˆv|qi(cid:105) .

(3.3)

To be more specific, let us look at the pairing model with four particles and eight single-

particle states. These states (with δ = 1.0) could be labeled as shown in Table 3.1. The

Hamiltonian matrix for this four-particle problem with no broken pairs is defined by six

possible Slater determinants, one representing the ground state and zero-particle-zero-hole

excitations 0p− 0h, four representing various 2p− 2h excitations and finally one representing
a 4p − 4h excitation. Ignoring Slater determinants with broken pairs, this problem is then

85

Table 3.1: Single-particle states and their quantum numbers and their energies from
Eq. (3.3). The degeneracy for every quantum number p is equal to two due to the two
possible spin values.

State Label p 2sz E
0
1
2
3
4
5
6
7

1
-1
1
-1
1
-1
1
-1

type
hole
-g/2
hole
-g/2
1-g/2 hole
1-g/2 hole
2
2
3
3

1
1
2
2
3
3
4
4

particle
particle
particle
particle

represented by the Hamiltonian matrix



H =

0

4δ − g −g/2 −g/2

2δ − g −g/2 −g/2 −g/2 −g/2
−0
−g/2
−g/2
−g/2 −g/2
−g/2 −g/2
6δ − g −g/2 −g/2
−g/2 −g/2
−g/2
8δ − g −g/2
10δ − g

−g/2 −g/2 −g/2 −g/2

−g/2 −g/2

6δ − g

0

0

0

0



.

(3.4)

Here, the exact eigenvalues can be found by diagonalizing this small matrix. Additionally, it

is easy to calculate low orders of many-body perturbation theory analytically. This is a very

useful check of the numerical implementation since this analytical expression can also be

used to check our coupled cluster implementation as described in Chapter 4. As a reminder,

the expression for the correlation energy for MBPT(2) is

∆EM BP T 2 =

1
4

(cid:88)

abij

(cid:88)

a<b,i<j

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)


ab
ij

=

86

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)


ab
ij

.

(3.5)

Additionally, we look at many-body pertubation theory at third order (MBPT(3)) which is

given by the expression

∆EM BP T 3 = ∆EM BP T 2+

(cid:88)

abcdij

1
8

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|cd(cid:105)(cid:104)cd|ˆv|ij(cid:105)


ab
ij 
cd
ij

(cid:88)

abijkl

+

1
8

(cid:104)ab|ˆv|kl(cid:105)(cid:104)kl|ˆv|ij(cid:105)(cid:104)ij|ˆv|ab(cid:105)


ab
ij 
ab
kl

.

For our pairing example we obtain the following result

∆EM BP T 2 = (cid:104)01|ˆv|45(cid:105)2


45
01

+ (cid:104)01|ˆv|67(cid:105)2


67
01

+ (cid:104)23|ˆv|45(cid:105)2


45
23

+ (cid:104)23|ˆv|67(cid:105)2


67
23

,

which translates into

∆EM BP T 2 = −

(cid:18) 1

4 + g

g2
4

(cid:19)

.

+

1

6 + g

+

1

2 + g

+

1

4 + g

(3.6)

(3.7)

(3.8)

Figure 3.1 shows the resulting correlation energies for the exact case, MBPT2 and MBPT3.

In Fig. 3.1 we see that the approximation to both second and third order are very good

when the interaction strength is small and contained in the interval g ∈ [−0.5, 0.5], but as the
interaction gets stronger in absolute value the agreement with the exact reference energy for

MBPT2 and MBPT3 worsens. We also note that the third-order result is actually worse than

the second order result for larger values of the interaction strength, indicating that there is

no guarantee that higher orders in many-body perturbation theory may reduce the relative

error in a systematic way. The disagreement when the interaction strength increases hints

at the possibility that many-body perturbation theory may not converge order by order.

Also note the non-variational character of many-body perturbation theory, with results at

different levels of many-body perturbation theory either overshooting or undershooting the

87

Figure 3.1: Correlation energy for the pairing model with exact diagonalization, MBPT2 and
perturbation theory to third order MBPT3 for a range of interaction values. A canonical
Hartree-Fock basis has been employed in all MBPT calculations.

88

1.00.50.00.51.0Interaction strength, g0.50.40.30.20.10.0Correlation energyExactMBPT2MBPT3true ground state correlation energy.

3.2 Single-Particle Basis for Infinite Fermionic Matter

Neutron stars are several kilometers across, but supported by the purely quantum phe-

nomenon of Fermi statistics. Studying systems that span from 104m to 10−15m [1] is a

task that is certainly impossible from an ab initio perspective. However, the short range of

the nucleon-nucleon interaction allows us to study a small slab of this matter to determine

properties of the bulk. This matter is self bound, but unlike atomic nuclei, the nucleons are

bound gravitationally which presents a considerable problem as a quantum theory of gravity

is debatably the largest unsolved problem in physics. However, we can work around this

problem by forcing the nucleons together via an external density (ρ) parameter. Once the

neutrons are fixed to a particular density to simulate the gravitational environment, there

is no external potential. This means that the one-body piece of the Hamiltonian is just

the kinetic energy operator ˆp2/ 2m, for which the eigenstates are free particles, represented

mathematically by plane waves. These basis states are infinite in their spatial extent, mak-

ing them difficult to work with, so the plane waves are put into a finite box, discretizing

the spectrum. These one-particle wave functions are normalized to a volume Ω for a box

with length L (the limit L → ∞ is to be taken after we have computed various expectation
values)

ψkσ(r) =

1
√Ω

exp (ikr)ξσ,

(3.9)

89

where k is the wave number and ξσ is the spin function for either spin up or down nucleons

 1

0

 ξσ=−1/2 =

 .

 0

1

ξσ=+1/2 =

(3.10)

We assume that we have periodic boundary conditions (Ψ(0) = eiθΨ(L)) which limit the

allowed wave numbers to

ki =

2πni

L

, i = x, y, z,

ni ∈ Z.

The operator for the kinetic energy can be written as

(cid:88)

pσp

ˆT =

2k2
P
2m

a†pσpapσp.

(3.11)

(3.12)

When using periodic boundary conditions, the discrete-momentum single-particle basis func-

tions (excluding spin and/or isospin degrees of freedom) result in the following single-particle

energy

(cid:18)2π

(cid:19)2(cid:16)

L

2
2m

(cid:17)

(cid:16)

2
2m

=

n2
x + n2

y + n2
z

(cid:17)

,

k2
nx + k2

ny + k2
nz

εnx,ny,nz =

for a three-dimensional system with

kni =

2πni

L

, ni ∈ Z.

(3.13)

We will select the single-particle basis such that both the occupied and unoccupied single-

particle states have a closed-shell structure. This means that all single-particle states cor-

90

responding to energies below a chosen cutoff are included in the basis. With the kinetic

energy rewritten in terms of the discretized momenta we can set up a list (assuming identi-

cal particles and including spin up and spin down solutions) of single-particle energies with

momentum quantum numbers such that n2

x + n2

y + n2

z ≤ 3, as shown, for example, in Table

3.2.

Continuing in this way we get for n2

x +n2

y +n2

z = 4 a total of 12 additional states, resulting

in 66 as a new magic number. For the lowest six energy values the degeneracy in energy

gives us 2, 14, 38, 54, 66 and 114 as magic numbers. Each many-body calculation has an

energy cutoff and a magic number determining how many particles are in the simulation,

and a second magic number determining how many unoccupied single-particles states span

the finite Hilbert space. If we wish to study infinite nuclear matter with both protons and

neutrons, the above magic numbers become 4, 28, 76, 108, 132, 228, . . . .

Once the number of particles in the simulation are determined and a density ρ has been

selected the Fermi momentum kF of the system is determined via

ρ = g

k3
F
6π2 ,

(3.14)

where g is the degeneracy, which is two for one type of spin-1/2 particles and four for

symmetric nuclear matter. From here we can define the length L of the box used with

periodic boundary contributions via the relation

V = L3 =

A
ρ

,

(3.15)

where A is the number of nucleons. If we deal with the electron gas only, this needs to be

91

Table 3.2: Total number of particle filling N↑↓ for various n2
z values for one spin-1/2
fermion species. Borrowing from nuclear shell-model terminology, filled shells correspond to
all single-particle states for one n2
z value being occupied. For matter with both
protons and neutrons, the filling degree increased with a factor of 2.

x + n2

y + n2

x +n2

y +n2

z nx ny nz N↑↓
2

0
0
0
-1
1
0
0
-1
1
-1
1
0
0
0
0
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1

0
0
0
0
0
-1
1
0
0
0
0
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1

14

38

54

n2
x + n2
y + n2
0
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3

0
-1
1
0
0
0
0
-1
-1
1
1
-1
-1
1
1
0
0
0
0
-1
-1
-1
-1
1
1
1
1

92

replaced by the number of electrons N .

3.3 Two-Nucleon Interaction

As mentioned above, we will employ a plane wave basis for our calculations of infinite

matter properties. With a Cartesian basis we can directly calculate the various matrix

elements. However, a discrete and finite Cartesian basis represents an approximation to the

thermodynamical limit. In order to compare the stability of our basis with results from the

thermodynamical limit, it is convenient to rewrite the nucleon-nucleon interaction in terms

of a partial wave expansion. This will allow us to compute the Hartree-Fock energy of the

ground state in the thermodynamical limit (with the caveat that we need to limit the number

of partial waves). In order to find the expressions for the Hartree-Fock energy in a partial

wave basis, we will find it convenient to rewrite our two-body force in terms of the relative

and center-of-mass motion momenta.

The direct matrix element, with single-particle three-dimensional momenta kp, spin σp

and isospin τp, is defined as

(cid:104)kpσpτpkqσqτq|ˆv|krσrτrksσsτs(cid:105),

(3.16)

or in a more compact form as (cid:104)pq|ˆv|rs(cid:105) where the boldfaced letters p etc represent the
Introducing the relative
relevant quantum numbers, here momentum, spin and isospin.

momentum

(cid:0)kp − kq

(cid:1) ,

1
2

k =

93

(3.17)

and the center-of-mass momentum

we have

K = kp + kq,

(3.18)

(cid:104)kpσpτpkqσqτq|ˆv|krσrτrksσsτs(cid:105) = (cid:104)kKσpτpσqτq|ˆv|k(cid:48)K(cid:48)σrτrσsτs(cid:105).

(3.19)

The nucleon-nucleon interaction conserves the total momentum and charge, implying that

the above uncoupled matrix element reads

(cid:104)kKσpτpσqτq|ˆv|k(cid:48)K(cid:48)σrτrσsτs(cid:105) = δTz,T(cid:48)z

δ(K− K(cid:48))(cid:104)kTzSz = (σa + σb)|ˆv|k(cid:48)TzS(cid:48)z = (σc + σd)(cid:105),

(3.20)

where we have defined the isospin projections Tz = τp + τq and T(cid:48)z = τr + τs. Defining

ˆv = ˆv(k, k(cid:48)), we can rewrite the previous equation in a more compact form as

δTz,T(cid:48)z

δ(K−K(cid:48))(cid:104)kTzSz = (σp+σq)|ˆv|k(cid:48)TzS(cid:48)z = (σr+σs)(cid:105) = δTz,T(cid:48)z

δ(K−K(cid:48))(cid:104)TzSz|ˆv(k, k(cid:48))|TzS(cid:48)z(cid:105).

(3.21)

These matrix elements can in turn be rewritten in terms of the total two-body quantum

numbers for the spin S of two spin-1/2 fermions as

(cid:88)

SS(cid:48)

(cid:104)kTzSz|ˆv(k, k(cid:48))|k(cid:48)TzS(cid:48)z(cid:105) =

1
2

(cid:104)

σp

1
2

σq|SSz(cid:105)(cid:104)

1
2

σr

1
2

σs|S(cid:48)S(cid:48)z(cid:105)(cid:104)kTzSSz|ˆv(k, k(cid:48))|kTzS(cid:48)S(cid:48)z(cid:105).

(3.22)

The coefficients (cid:104) 1
2 σq|SSz(cid:105) are so-called Clebsch-Gordan recoupling coefficients. We will
assume that our interactions conserve charge. We will refer to Tz = 0 as the pn (proton-

2 σp

1

94

neutron) channel, Tz = −1 as the pp (proton-proton) channel and Tz = 1 as the nn (neutron-
neutron) channel.

The nucleon-nucleon force is often derived and analyzed theoretically in terms of a partial

wave expansion. A state with linear momentum k can be written in terms of spherical

harmonics Ylm as

∞(cid:88)

l(cid:88)

l=0

m=−l

|k(cid:105) =

ılYlm(cid:104)ˆk|klml(cid:105).

(3.23)

In terms of the relative and center-of-mass momenta k and K, the potential in momentum

space is related to the nonlocal operator V (r, r(cid:48)) by

(cid:90)

(cid:104)k(cid:48)K(cid:48)|ˆv|k(cid:48)K(cid:105) =

drdr(cid:48)e−ık(cid:48)r(cid:48)V (r(cid:48), r)eıkrδ(K, K(cid:48)).

(3.24)

We will assume that the interaction is spherically symmetric and use the partial wave expan-

sion of the plane waves in terms of spherical harmonics. This means that we can separate

the radial part of the wave function from its angular dependence. The wave function of the

relative motion is described in terms of plane waves as

eıkr = (cid:104)r|k(cid:105) = 4π

ıljl(kr)Y ∗lm(ˆk)Ylm(ˆr),

(3.25)

(cid:88)

lm

where jl is a spherical Bessel function and Ylm the spherical harmonic. This partial wave basis

is useful for defining the operator for the nucleon-nucleon interaction, which is symmetric

with respect to rotations, parity and isospin transformations. These symmetries imply that

the interaction is diagonal with respect to the quantum numbers of total angular momentum

J, spin S and isospin T . Using the above plane wave expansion, and coupling to final J, S

95

and T we get

(cid:104)k(cid:48)|V |k(cid:105) = (4π)2(cid:88)

JM

(cid:88)

(cid:88)

lm

l(cid:48)m(cid:48)

ıl+l(cid:48)Y ∗lm(ˆk)Yl(cid:48)m(cid:48)(ˆk(cid:48))Cl(cid:48)SJ

m(cid:48)MS MClSJ

mMS M(cid:104)k(cid:48)l(cid:48)ST JM|V |klST JM(cid:105),

(3.26)

where we have defined

(cid:104)k(cid:48)l(cid:48)ST JM|V |klST JM(cid:105) =

(cid:90)

jl(cid:48)(k(cid:48)r(cid:48))(cid:104)l(cid:48)ST JM|V (r(cid:48), r)|lST JM(cid:105)jl(kr)r(cid:48)2

dr(cid:48)r2dr.

(3.27)

We have omitted the momentum of the center-of-mass motion K and the corresponding

orbital momentum L, since the interaction is diagonal in these variables.

The interaction we will use for these calculations is a semirealistic nucleon-nucleon po-

tential known as the Minnesota potential [53] which has the form, Vα (r) = Vα exp (−αr2).
The spin and isospin dependence of the Minnesota potential is given by,

(cid:18)

(cid:19)

V (r) =

1
2

VR +

1
2

(1 + P σ

12) VT +

1
2

(1 − P σ

12) VS

(1 − P σ

12P τ

12) ,

(3.28)

12 = 1

where P σ

2 (1 + τ1 · τ2) are the spin and isospin exchange
operators, respectively. A Fourier transform to momentum space of the radial part Vα (r)

2 (1 + σ1 · σ2) and P τ

12 = 1

is rather simple, since the radial depends only on the magnitude of the relative distance

and thereby the relative momentum (cid:126)q = 1
2

. Omitting spin and isospin

dependencies, the momentum space version of the interaction reads

(cid:17)

(cid:16)(cid:126)kp − (cid:126)kq − (cid:126)kr + (cid:126)ks
(cid:17)3/2
(cid:16) π

exp (−q2
4α

(cid:104)kpkq|Vα|krks(cid:105) =

Vα
L3

α

)δ(cid:126)kp+(cid:126)kq,(cid:126)kr+(cid:126)ks

.

(3.29)

The various parameters defining the interaction model used in this work are listed in Table

96

Table 3.3: Parameters used to define the Minnesota interaction model [53].

α Vα in MeV κα in fm−2
R 200
T
178
S 91.85

1.487
0.639
0.465

3.3.

3.4 Homogeneous Electron Gas

From a numerical calculation perspective, once the tools have been developed to compute

properties it is not too difficult to compute properties of other infinite matter systems.

In this case, properties of the homogeneous electron gas (HEG) can be examined using a

similar prescription as infinite nuclear matter. The plane wave basis and periodic boundary

conditions can once again be used to simulate an infinite gas of electrons interacting in a

uniform positive background charge to keep the system charge neutral on average. There

are a few differences, the first of which is that the mass of the particles is different and so

the single particle energies will differ. Next is more of a cultural shift, in that it is typical to

describe the density of electrons in terms of the dimensionless Wigner-Seitz radius rs as an

input parameter defined by

rs =

r0
rb

,

(3.30)

where rb = / mecα is the Bohr radius and r0 can be used to define the box size L by

4
3

πr3

0 =

N
L3 .

97

(3.31)

The electron-electron interaction is given by the Coulomb interaction which conserves total

linear momentum and total spin projection such that

(cid:126)q = (cid:126)kp + (cid:126)kq = (cid:126)kr + (cid:126)ks,

szp + szq = szr + szs.

(3.32)

Any matrix element which breaks these symmetries must be zero, otherwise the matrix

elements expressed in momentum space are given by

(cid:104)pq|ˆv|rs(cid:105) =

e2
L3

1
q2 .

(3.33)

The divergence at q = 0 is avoided by clever cancellation with the energy of the uniform

positive background charge, as shown in reference [30].

98

Chapter 4

Coupled Cluster

4.1 Prologue to Coupled Cluster

The previous chapter laid out much of the machinery that is useful for quantum many-

body theory, and a few many-body methods like full configuration interaction (FCI)[8, 9, 10,

11], Hartree-Fock (HF) [35, 36], and in-medium similarity renormalization group (IM-SRG)

[50, 42, 43, 44, 45]. With this groundwork laid we are in a good position to derivate coupled

cluster (CC) theory. Coupled cluster theory is another approach to solving the many-body

Schr¨odinger equation in the same vein as configuration interaction from the previous chapter,

except with a different scheme for organizing the excitations. In the FCI scheme, any many-

body state can be written with respect to a reference state as:

(cid:88)

i,a

(cid:88)

i,j,a,b

(cid:88)

C
i1,i2,...,iA
a1,a2,...aA

|ΨF CI(cid:105) = C0 |Φ0(cid:105) +

Ca
i |Φa

i (cid:105) +

Cab
ij |Φab

ij (cid:105) + ··· +

a1a2...aA
i1i2...iA |Φ

a1a2...aA
i1i2...iA (cid:105) ,

where the overlap coefficients are defined as

C

a1a2...aA
i1i2...iA

= (cid:104)ΨF CI|Φ

a1a2...aA
i1i2...iA (cid:105) .

99

(4.1)

(4.2)

This is really just the statement that any state can be represented as a linear combination

of Slater determinants, which is a complete A-body basis as discussed last chapter. The first

approximation to this complete solution is to find a finite cutoff M for the infinite single-

particle basis. This must always be done, as there is no way to finish this calculation on a

computer if the particle state index a is summing to infinity. Once the single-particle basis

has been truncated, the above expression gives the exact answer in this subspace, as FCI

includes all Slater determinants up to A-body excitations, at which point the series naturally

truncates. However, this is prohibitively expensive, scaling factorially with respect to the

single-particle basis since there are(cid:0)M

(cid:1) Slater determinants in this many-body basis. The

A

next natural approximation is to start excluding certain Slater determinants. For example,

if only singles and doubles excitations are included, then we get the configuration interaction

singles doubles (CISD) approximation, which is expressed as

|ΨCISD(cid:105) = C0 |Φ0(cid:105) +

i |Φa
Ca

i (cid:105) +

ij |Φab
Cab
ij (cid:105) .

(4.3)

(cid:88)

i,a

(cid:88)

i,j,a,b

While this is a conceptually nice place to truncate the series, it turns to be a pretty poor

way to include correlations into the target many-body state. This is where coupled cluster

(CC) theory has made its mark in the many-body community. Coupled cluster theory

is a way to organize the many-body basis such that the natural truncations lead to very

accurate calculations for relatively low computational cost. Originally developed in the

1950’s by Coester [17, 18] and K¨ummel [19], CC saw some success solving problems for

nuclear physics. Unfortunately, the nucleon-nucleon interactions of the 50’s required large

single-particle bases to converge, and the computers of the day weren’t powerful enough for

CC to find much success. However, CC was reformulated in the 60’s by ˇC´ıˇzek et al., for use in

100

electronic systems [12, 13, 14]. This proved to work magnificently, as coupled cluster theory

saw enormous success with the interaction of electrons and underwent rapid development

over the next several decades [54, 55, 15]. Fast forward to today, CC is referred to as the

“gold standard” for precise quantum chemistry calculations.

After the developmental boom in quantum chemistry, nucleon-nucleon interactions gradu-

ally became more suitable to ab initio methods leading to its readoption in nuclear physics in

the early to mid-2000’s [56, 57, 58, 59]. Today, improved nuclear forces softened by similarity

renormalization group (SRG) [4, 5], greater computational power, and improved many-body

techniques have created an environment for CC theory to thrive in nuclear physics.

To first understand coupled cluster theory, it might help to first look at the Thouless

theorem. The Thouless theorem states that any Slater determinant |Φ0(cid:105) can be transformed
to any other Slater determinant |Φ(cid:105) that isn’t orthogonal to the original by

|Φ(cid:105) = e
ˆT1 =

(cid:88)
ˆT1 |Φ0(cid:105)
ta
i ˆa†ˆi,

i,a

(4.4)

(4.5)

where the coefficients ta

i are uniquely determined [60]. The exponential of this one-particle

one-hole operator ˆT1 is referred to as the operator exponential, and is written out as the

infinite series

∞(cid:88)

k=0

1
k!

ˆT1 =

e

ˆT k
1 = 1 + ˆT1 +

1
2

ˆT 2
1 + . . . ,

(4.6)

where ˆT 2

1 = ˆT1 ˆT1 is just the repeated action of the operator on a state. As a notational

remark, in this chapter ket states with the capital Greek letter |Φ(cid:105) will refer only to a single
Slater determinant, but ket states with the capital Greek letter |Ψ(cid:105) can refer to any many-

101

body state. Also, matrix elements will be written both as ta

i and (cid:104)a|ˆt|i(cid:105) whenever either is

more convenient, but they refer to the same object.

The Thouless theorem is very powerful, but is limited in that this exponentiated operator

can only take a single product state to another product state. If we expand this operator to

take into account higher order excitations, we can generate higher order correlations. This

is the famous exponential ansatz of coupled cluster theory:

ˆT |Φ(cid:105)
A(cid:88)
ˆTn
(cid:88)

n=1
1

|Ψ(cid:105) = e

ˆT (A) =

ˆTn =

a1a2...aA
t
i1i2...iA

(A!)2

i1,i2,...,iA
a1,a2,...aA

(4.7)

(4.8)

(4.9)

ˆa1† ˆa2† . . . ˆaA† ˆiA . . . ˆi2 ˆi1.

By including every excitation up to A-body excitations, any many-body state |Ψ(cid:105) can be
generated by finding the appropriate operator coefficients t applied to a reference state |Φ(cid:105)
which is non-orthogonal to the target state. If any non-orthogonal many-body state can be

found, then this must be quite similar to the FCI statement which is similarly just a linear

combination of excitations of some reference state. Since this is just another complete many-

body space, it is natural to ask why this formulation is any better than FCI. The answer lies

in the fact that truncating CC theory at a given order brings in many additional many-body

correlations for a similar cost than FCI at the corresponding CI truncation. This is due to

the rich structure generated by the cross terms of the exponential,

ˆT = 1 + ( ˆT1 + ˆT2 + . . . ) +
e
1
2

= 1 + ˆT1 + ˆT2 + ··· +

1
2
ˆT 2
1 + ˆT1 ˆT2 +

( ˆT1 + ˆT2 + . . . )2 + . . .

1
2

ˆT 2
2 + . . .

102

where terms like ˆT1 ˆT2 do not appear in CI with only singles and doubles excitations. By

comparing excitation levels, we can write down what various levels of CI equal in CC theory

ˆC1 = ˆT1,

ˆC2 = ˆT2 +

1
2

ˆT 2
1 ,

ˆC3 = ˆT3 + ˆT1 ˆT2 +

1
6

ˆT 3
1 .

(4.10)

(4.11)

(4.12)

This shows which terms of CC theory CISDT (single, doubles and triples excitations) re-

covers, whereas just CCD accounts for ˆT2, ˆT 3

2 , ˆT 4

2 . . . excitations up to infinite order. These

non-linear contributions of the excitation operators generally leads CC (for example CCSD)

to recover more correlations than CI at the same level of excitation (for example CISD). The

full details of how coupled cluster theory works will not be explained in this chapter, but

given a few assumptions, we can derive a working set of equations with which to calculate

properties of many-body systems [16].

4.2 Coupled Cluster Theory

The normal-ordered many-body Schr¨odinger equation can be expressed using the expo-

nential ansatz as

ˆHN |Ψ(cid:105) = ˆHN e

ˆT |Φ0(cid:105) = ∆Ee

ˆT |Φ0(cid:105) ,

(4.13)

where |Φ0(cid:105) again is our reference Slater determinant, and |Ψ(cid:105) is the ground state eigenstate
of an A-body Hamiltonian of interest. Then we can get the ground state energy by projecting

103

Eqn. (4.13) onto (cid:104)Ψ| which gives

∆E0 = (cid:104)Ψ| ˆHN|Ψ(cid:105) = (cid:104)Φ0|e− ˆT ˆHN e

ˆT|Φ0(cid:105) = (cid:104)Φ0|HN|Φ0(cid:105) ,

(4.14)

where we have defined the normal-ordered coupled cluster effective Hamiltonian

H ≡ e− ˆT ˆHN e
ˆT .

(4.15)

If we can find a way to determine the coefficients of ˆT , then we immediately have a pre-

scription for finding the group state energy. An important aspect of CC theory is that the

ˆT
excitation operator ˆT , is not Hermitian, which as explained in Chapter 2 ensures that e

is not a unitary operator. This means that Eqn. (4.15) describes a non-unitary similarity

transformation. This has some inconvenient consequences, such that CC theory is not varia-

tional (where the approximate ground state energy always approaches the true ground state

energy from above), but for much of the work presented here, this is not a problem, as a

similarity transformation preserves the eigenvalue spectrum of the operator, which is what

we are after.

We can apply the Baker-Campbell-Hausdorff expansion to the similarity transform to

gain an explicit expression for H as

H = e− ˆT ˆHN e

ˆT = ˆHN +

ˆHN , ˆT

(cid:104)

(cid:105)

(cid:104)(cid:104)

+

1
2

ˆHN , ˆT

(cid:105)

(cid:105)

, ˆT

+

1
3!

(cid:104)(cid:104)(cid:104)

(cid:105)

(cid:105)

, ˆT

(cid:105)

, ˆT

ˆHn, ˆT

+ . . . (4.16)

One can show that this series naturally terminates [16]. The basic reason is that applying

the generalized Wick’s theorem to this expression cancels out the vast majority of the terms.

104

Only terms which start with ˆHN on the left, and are contracted with this term survive.

H = e− ˆT ˆHN e

ˆT

= ˆHN + ˆHN ˆT + ˆHN ˆT ˆT + ˆHN ˆT ˆT ˆT + ˆHN ˆT ˆT ˆT ˆT

(cid:16)

ˆT(cid:17)

=

ˆHN e

,

C

(4.17)

where the contraction symbol represents the sum over all the ways each of the operators can

be contracted together and the C subscript stands for “connected”, meaning that only terms

which connect to ˆHN via a contraction survive. The reason that this expression terminates at

four contractions, is that we are assuming the Hamiltonian to have at most two-body forces,

and thus only two creation and two annihilation operators. This leaves only four operators to

contract with, so there can be no further terms. However, in nuclear theory, three-body and

higher-body forces are often needed, which will add to the amount of contractions available.

This is quite amazing, since the exponential of the excitation operator has contributions up

to infinite order of powers of ˆT , and naturally terminates without any approximation. Other

approximations will need to be made down the road, but this fact is where much of the

power of CC theory resides.

We can now write the CC correlation energy as the connected form of the Schr¨odinger

equation

(cid:104)Φ0|( ˆHN e

ˆT )C|Φ0(cid:105) = ∆E,

(4.18)

by projecting onto the reference state (cid:104)Φ0|. We can also project the Schr¨odinger equation
onto any of the orthogonal excitations from the reference state

ˆT )C|Φ0(cid:105) = 0.

(cid:104)Φab...

ij...|( ˆHN e
105

(4.19)

Since we need the t-amplitudes of the excitation operator to calculate the correlation energy,

we can project onto as many excitations as necessary to generate as many equations as we

have unknowns.

4.3 Coupled Cluster Diagrams

To actually generate the CC equations from Eqn. (4.18) and Eqn. (4.19) would require

an enormous amount of algebra. Even with the incredible reduction in complexity of the

generalized Wick’s theorem, carrying out these operations is very time consuming and prone

to error. In Quantum Field Theory, the use of pictorial Feynman diagrams are an essential

tool for handling otherwise unwieldy algebraic expressions of Wick’s theorem in a time-

dependent framework. In CC theory, the use of time-independent Brandow diagrams are

used in a similar way to handle the Wick’s theorem expressions necessary to generate the

CC equations.

Plugging in a normal-ordered one and two-body force for the Hamiltonian ˆHN = ˆFN + ˆVN

into the energy equation (4.18) yields

∆E = (cid:104)Φ0|(FN T1)c|Φ0(cid:105) + (cid:104)Φ0|(VN T2)c|Φ0(cid:105) +

1

2 (cid:104)Φ0|(VN T 2

1 )c|Φ0(cid:105) .

(4.20)

To represent these expressions diagrammatically, we start with the reference state |Φ0(cid:105) as
our blank canvas since all of the necessary expressions are being applied to this state. Next,

excitations applied to the reference state are drawn as lines, with upward lines representing

particle states and downward lines representing hole states

106

a

i

|Φa
i (cid:105) =

|Φab
ij (cid:105) =

a

i

j

b

(4.21)

where these diagrams are read from bottom to top. So the reference state |Φ0(cid:105) is at the
bottom, and undergoes a one-particle one-hole excitation moving towards the top of the

diagram. The notation of the arrows is borrowed from Quantum Field Theory where anti-

particles can be thought of as the time-reversal form of their corresponding particle, and so

move backwards in time. Here, the hole states can be thought of as the anti-particle to the

particle states, and the arrow direction is reversed. Next, the one-body excitation operator

ˆT1 is drawn as a one-particle one-hole excitation originating from an open circle, and the

two-body two-hole excitation operator ˆT2 is drawn as black line from which the particle and

hole lines come from

(cid:88)

i,a

ˆT1 =

(cid:104)a|ˆt|i(cid:105) ˆa†ˆi :

i

a

(cid:88)

ijab

ˆT2 =

(cid:104)ab|ˆt|ij(cid:105) ˆa†ˆb†ˆjˆi :

a

bi

j

.

(4.22)

These lines that extend from an operator and exit via the top of the diagram are called

“external” lines, and indicate “live” operators which can be connected (contracted) against

other operators above them. The fact that the ˆT operator’s lines only point up is capturing

the fact that this is an excitation operator, and thus can only connect to operators above

107

it in the diagram, or algebraically, can only contract with operators applied after it (like

ˆHN ˆT ). First, let’s look at the one-body piece of the Hamiltonian ˆFN . Since this operator

has one creation operator and one annihilation operator, it must have two lines associated

with it. The diagrammatic symbol for the one-body operator will be two lines attached to a

dark X by a dotted line

(cid:88)
pq (cid:104)p| ˆf|q(cid:105)

ˆFN =

(cid:110)

(cid:111)

ˆp† ˆq

=

.

(4.23)

The lines here do not have arrows, since p and q do not have a fixed particle or hole natures.

This means that there are four different orientations of this diagram.

(cid:110)

(cid:88)

ab

(cid:104)a| ˆf|b(cid:105)

(cid:111)

ˆa†ˆb

a

b

=

(cid:88)

ij

(cid:104)i| ˆf|j(cid:105)

(cid:110)

(cid:111)

ˆi†ˆj

j

i

=

(4.24)

(cid:110)

(cid:88)

ai

(cid:104)a| ˆf|i(cid:105)

(cid:111)

ia

=

(cid:88)

ai

(cid:104)i| ˆf|a(cid:105)

(cid:111)

(cid:110)

ˆi†ˆa

= a i

.

(4.25)

ˆa†ˆi

Only looking at the diagram, it might seem difficult to recover the algebraic expression, but

there is a unique mapping looking at how the lines enter and exit the vertex. The matrix

element associated with the one-body vertex will always be written as

(cid:104)index exiting| ˆf|index entering(cid:105)

(4.26)

108

and the corresponding operators are

(cid:110)

(cid:111)

(index exiting)†(index entering)

(4.27)

which provides a unique description of the operator. The sums are implied via the normal

Einstein summation rules for repeated indices. Note that these terms can be described by

whether they are an excitation, a de-excitation, or neither. The two terms in Eqn. (4.24)

are neither excitations or de-excitations since the creation and annihilation operators do not

change the particle-hole nature of the state. The first term in Eqn. (4.25) is an excitation

operator, with lines extending out of the top of the diagram (like ˆT1), and the second term

is a de-excitation operator, with lines extending out of the bottom of the diagram.

The last operator we need is the two-body piece of the Hamiltonian ˆVN , which is expressed

diagrammatically as two vertices connected via a dotted line, with each vertex having one

creation and one annihilation operator

(cid:88)
pqrs(cid:104)pq|ˆv|rs(cid:105) ˆp† ˆq†ˆsˆr =

ˆVN =

.

(4.28)

The rules of this operator are very similar to ˆFN , except now for four indices, they are

uniquely mapped based on whether the lines are attached to the left vertex or the right

vertex as

(cid:104)left-out right-out|ˆv|left-in right-out(cid:105) .

(4.29)

109

Similarly, the operators are

(cid:110)

(left-out)†(right-out)†(right-in)(left-in)

(cid:111)

.

(4.30)

The last ingredient needed to diagrammatically perform Wick’s theorem, is that of a

contraction operation. Diagrammatically, this corresponds to joining the line from one dia-

gram with a line from another diagram. This leads to a very intuitive set of consequences

that exactly match up with Wick’s theorem. Lines of opposite orientation cannot be joined,

and an operator with k external lines can at most contract k times with other operators.

Diagrams can be stretched and manipulated, while the indices and the entering and exiting

line rules above keep track of just about everything. The part which is not so intuitive,

which will not be derived here, is how to keep track of phases and weights. This will be

detailed in a table below.

Let’s look at a quick example of how to perform a contraction between the operators

ˆFN and ˆT1. Since ˆT1 only has lines from above (due to it being an excitation operator),

the operator ˆFN must be placed above the operator. This corresponds algebraically to this

operator being placed afterwards as ˆFN ˆT1. Since these operators both have two external

lines, either one or two connections can be made, corresponding to how only one or two

contractions can be made between their creation and annihilation operators. There are

110

three topologically distinct ways in total to do this

.

(4.31)

Here, the first two terms would have the same topology if the orientation of the lines was

not fixed by the particle or hole arrow. Regardless of how the arrows are oriented, the third

term produces the same expression. In general, these techniques refer to what are called

non-oriented Hugenholtz diagrams, and the oriented Brandow diagrams [33, 32, 61]. The

Hugenholtz diagrams are useful for getting a grasp on the unique topologies available by

connecting lines. From there, all of the various particle-hole orientations can be drawn to

find the actual algebraic expressions.

Now that we have established a rough intuition for the operators and how to connect

them, we list a consistent set of rules for reading the algebraic expressions generated from

the diagrammatic expressions for Wick’s theorem [16]:

1. Label external lines with hole (i, j, k) and particle (a, b, c) target indices. These corre-

spond to the bra state indicies.

2. Label internal lines with hole (l, m, n, . . . ) and particle (d, e, f, . . . ) and sum over these

indices.

3. Every one-body interaction vertex picks up a factor of (cid:104)out| ˆf|in(cid:105) = f out
in .

4. Every two-body interaction vertex picks up a factor of (cid:104)left-out right-out|ˆv|left-in right-in(cid:105).

5. Every ˆTm vertex picks up an amplitude tab...
ij...

111

6. Each pair of equivalent internal lines picks up a factor of 1

2. Two lines are considered

equivalent if they have the same starting and ending vertices.

7. Each pair of equivalent ˆTm vertices picks up a factor of 1

2. Two ˆTm vertices are

considered equivalent if they connect to the interaction vertex in the same way.

8. The sign (±) of a diagram term is (−1)h+l, where h is the number of hole lines and l

is the number of loops.

9. Each pair of unique external particle (or hole) lines not connected to the same inter-

action adds a permutation factor ˆP (l1, l2), where l1 and l2 refer to the labels of the

equivalent lines.

While working with just a two-body force, ˆHN only has four legs, and thus can connect

to a maximum of four other ˆTm diagrams. To keep track of all of the ways that the diagrams

can connect, we will use the sign table method. In this method, each interaction and cluster

operator is assigned a set of plus signs and minus signs. These assignments label the number

of lines extending below interaction vertices, and lines above cluster operators. A plus sign

is used for each particle line and a minus sign for each hole line. Let us list out the relevant

operators for CCD. First, ˆFN :

+

−

0

+−

(4.32)

Next, ˆVN :

112

+

−

++

−−

+−

(4.33)

+ + −

+ − −

+ + −−

0

Lastly, the cluster operators can only connect upwards, so ˆT2 and ˆT3:

+ + −−

+ + + − −−

(4.34)

Repeated cluster operators are separated by a vertical line:

+ + − − | + + − −

(4.35)

113

Let’s look at an example term of the form ( ˆVN ˆT 2

2 )C , which would arise from the equation

(cid:104)Φab

ij |( ˆHN e

ˆT )C|Φ0(cid:105) .

(4.36)

To carry this out, we set up the interaction vertices ˆVN above two ˆT2 operators like so

+ + −−

(4.37)

+ + − − | + + − −

.

The C subscript is a reminder that any disconnected terms vanish, so these three operators

must all be connected. Lastly, to project onto the doubly excited bra state (cid:104)Φab
ij | we need to
connect the operators with two external hole lines and two external particle lines remaining.

The sign table method helps us determine how many unique diagrams we have, and which

diagrams are left with the four external lines that are needed. They must come from the

cluster operators, as if one of the lines of the interaction matrix is an external line, there will

be no way to connect the diagrams and have only four external lines total. However, even

with this constraint, there are several ways to do so. These four external lines correspond

to two + and two - labels. To ensure the diagram is connected, each ˆT2 can only connect 1,

114

2, or 3 of its lines to ˆVN , as 0 would create leave it disconnected, and 4 would use up all of

ˆVN ’s lines, leaving the other ˆT2 operator disconnected. In the sign table notation, these are

written as

+ + | − − + −| + − + + − | − + − −|+

which is organized into Table 4.1. Since the order of + and - is irrelevant, we choose to

Table 4.1: Sign table for the four terms of eq. (4.37).

ˆT2
ˆT2
+−
+−
+ + − −
− + + −
−−
++

always list + signs first. Secondly, since the two ˆT2 operators are equivalent, we do not need

to list terms that are symmetric about the bar. For example, + + | − − is equivalent to
− − | + +, and + − −|+ is equivalent to +| + −−. Using the sign table, we generate the

115

following four diagrams and their algebraic expressions

(

1
2

ˆVN ˆT 2

2 )C →

+

+

+

(cid:88)

lmde

1
2

(cid:88)

lmde

(cid:88)

lmde

(4.38)

(cid:88)

lmde

1
2

(cid:104)lm|ˆv|de(cid:105) tad

ij teb
lm

= ˆP (ij) ˆP (ab)

(cid:104)lm|ˆv|de(cid:105) tad

il teb

mj + ˆP (ab)

+ ˆP (ij)

1
2

(cid:104)lm|ˆv|de(cid:105) tde

il tab

mj +

1
4

(cid:104)lm|ˆv|de(cid:105) tde

ij tab
lm.

One may note how the two ˆT2 operators are counted as equivalent, omitting configurations

that already exist if we were to switch place between the two operators, i.e. + + | − − is
the same as − − | + +, thus counted only once. Also, once the particle-hole orientation of
the arrows has been selected, the assignment of the index labels can vary, but will always

generate equivalent expressions. For this reason, it is common to omit the labels in the

diagrams.

116

4.4 Diagrammatic Derivation of the Coupled Cluster

Equations

All of the ingredients are now in place to derive the coupled cluster equations. As a

reminder the general expressions for coupled cluster equations are generated by projecting

the coupled cluster effective Hamiltonian onto the reference state, and onto an excitation of

the reference state,

(cid:104)Φ0|( ˆHN e

ˆT )C|Φ0(cid:105) = ∆E,
ˆT )C|Φ0(cid:105) = 0.

(cid:104)Φab...

ij...|( ˆHN e

Starting with the energy equation, projecting onto the reference state is equivalent to finding

the connected diagrams with no external lines. Truncating the Hamiltonian to only two-body

forces again, and expanding the exponential yields

(cid:2) ˆFN (1 + ˆT1 + ··· )(cid:3)

∆E = (cid:104)Φ0|

(cid:2) ˆVN (1 + ˆT1 + ··· )(cid:3)

C|Φ0(cid:105) + (cid:104)Φ0|

C|Φ0(cid:105) .

(4.39)

We see that even though the exponential in general has infinite terms, the only ways to

generate connected closed diagrams are the one-body force ˆFN connected with a singles

excitation ˆT1, the two-body interaction ˆVN connected with a doubles excitation ˆT2, and the

two-body interaction connected with two singles excitations. Let’s first look at the unique

topologies via non-oriented Hugenholtz diagrams [33].

117

( ˆFN ˆT1)C :

( ˆVN ˆT2)C :

( ˆVN ˆT 2

1 )C :

(4.40)

It turns out orienting these lines to have particle-hole character only generates one Brandow

diagram [61] for each of them. The diagrams and their corresponding expressions are

( ˆFN ˆT1)C :

i

a

( ˆVN ˆT2)C :

i

a

j

b

=

(cid:88)

i,a

=

(cid:88)

i,j,a,b

1
4

(cid:104)i| ˆf|a(cid:105)(cid:104)a|ˆt1|i(cid:105)

(4.41)

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆt2|ij(cid:105)

(4.42)

1
2

( ˆVN ˆT 2

1 )C :

i

a

j

b

=

(cid:88)

i,j,a,b

1
2

(cid:104)ij|ˆv|ab(cid:105)(cid:104)a|ˆt1|i(cid:105)(cid:104)b|ˆt1|j(cid:105)

(4.43)

At long last we have derived the expression for the CC correlation energy for a Hamiltonian

with one and two-body forces

∆ECC = f i

ata

i +

1
4

ij
abtab
v

ij +

1
2

ij
i tb
abta
v
j,

(4.44)

118

where the line labels in this case are all arbitrary indices since they are completely summed

over. The positions of the labels in the algebraic expressions are determined by the indices

of the incoming and outgoing lines. For bra-ket notation, (cid:104)out1out2|ˆv|in1in2(cid:105) for tensor
notation, v
becomes an unambiguous mapping. This expression for the coupled

out1out2
in1in2

cluster correlation energy can be further simplified. For the near future, we will assume

that we are working in a Hartree-Fock basis which by construction zeros out all of the hole-

particle one-body terms (cid:104)i| ˆf|a(cid:105) = 0. The next assumption we will make is that we are using
the CCD approximation of coupled cluster. In this approximation, ˆT ≈ ˆT2. This is the first
non-trivial approximation to CC, since CCS would just recover a single optimized Slater

determinant as shown by the Thouless theorem. As a note, the next approximation would

be CCSD, but for the pairing model with no broken pairs and infinite matter systems, there

are no singles excitations and so for the systems in this work CCSD is equal to CCD.

These assumptions allow us to discard terms with f i

a and terms with singles excitations,

and we arrive at the expression for the CCD correlation energy,

∆ECCD =

1
4

ij
abtab
ij .
v

(4.45)

Since ˆVN is an input from the Hamiltonian, we only need to find the tab

ij amplitudes to

calculate the CCD correlation energy. This means we have N 2

p N 2

h unknowns, and so we

need an equivalent amount of constraints to pin these values down. This is typically done

by projecting onto the excited reference state

a1...an
i1...in |(HN eT )c|Φ0(cid:105) = 0

(cid:104)Φ

(4.46)

119

or in the case of CCD:

(cid:104)Φab

ij |( ˆHN e

ˆT2)C|Φ0(cid:105) = 0.

(4.47)

Projecting on the bra state (cid:104)Φab
ij | means that we need to find all of the connected diagrams
with four external lines a, b, i, j. Since ˆFN only has two lines to connect, and ˆHN only has

four lines to connect, the only possible connected diagrams are given by

(cid:20)

(cid:21)
C|Φ0(cid:105) = 0.

(4.48)

(cid:104)Φab

ij |( ˆFN ˆT2)C|Φ0(cid:105) + (cid:104)Φab
ij |

ˆVN (1 + ˆT2 +

1
2

ˆT 2
2 )

Anything beyond ˆT 2

2 would have too many lines to be able to fully contract with ˆHN . Let’s

go through this expression term by term.

First, the contribution from the one-body operator

( ˆFN ˆT2)C →

+

(4.49)

(cid:88)

d

= ˆP (ab)

(cid:104)b| ˆf|d(cid:105) tad

ij − ˆP (ij)

(cid:88)

l

(cid:104)l| ˆf|i(cid:105) tab
lj ,

where as a reminder, in the first term, the ˆP (ab) operator is the permutation operator

1 − ˆPab, which comes from the fact that the two external particle lines in the first term
connect to different operators, and thus the permutation operator is necessary to recover the

antisymmetry of (cid:104)Φab
ij |.

120

Next is the very simple term from the Hamiltonian itself

( ˆVN )C →

=(cid:104)ab|ˆv|ij(cid:105) .

(4.50)

Then the two-body interaction connected to one ˆT2 operator

( ˆVN ˆT2)C →

+

+

(cid:88)

de

=

1
2

(cid:104)ab|ˆv|de(cid:105) tde

ij +

1
2

(cid:88)

lm

(cid:104)lm|ˆv|ij(cid:105) tab

lm + ˆP (ij) ˆP (ab)

(4.51)

(cid:88)

ld

(cid:104)lb|ˆv|dj(cid:105) tad
il .

The final term is 1

2( ˆVN ˆT 2

2 )C , which we have already done when describing how to use the

sign table in Eqn. (4.38). Putting all of the algebraic expressions together, we get the N 2

p N 2
h

constraints for tab

ij known as the CCD equations, here written in the more compact tensor

121

notation,

(cid:88)

d

0 = vab

ij + ˆP (ab)

(cid:88)
(cid:88)

de

1
2

1
4

1
2

+

+

−

lmde

ˆP (ij)

(cid:88)

lmde

(cid:88)

vab
detde

ij +

1
2

fbdtad

(cid:88)

lm

(cid:88)

ld

fkltab
il

l

ij tab
vlm

ij − ˆP (ij)
lm + ˆP (ab|ij)
(cid:88)
(cid:88)

vlm
de tad

ˆP (ab)

lmde
1
2

lmde

il tbe
jm

vlm
de ted

il tab

mj −

vlm
de tde

ij tab

lm + ˆP (ij)

vlb
djtad
il

vlm
de tad

mlteb
ij ,

(4.52)

where ˆP (ab|ij) ≡ ˆP (ab) ˆP (ij). Finding the set of amplitudes tab
satisfied may seem challenging at first, but we can see a path forward after rewriting these

ij for which this equation is

equations a little more. These calculations are frequently done in the Hartree-Fock basis, in

which case the one-body terms are diagonal fpq = fppδpq, where fpp = εp. This assumption

is not necessary, however the general CC strategy is more clear if we take this as true for

now. This simplification, along with the antisymmetry of tab

ij means we can rewrite the terms

which include fpq as

(cid:88)

d

ˆP (ab)

fbdδbdtad

ij − ˆP (ij)

(cid:88)

l

fljδljtab
il

= ˆP (ab)εbtab

= ˆP (ab)fbbtab

ij

ij

ij − ˆP (ij)fjjtab
ij − ˆP (ij)εjtab
ij − εjtab
ij − εjtab
ij (εa + εb − εi − εj)

ij − εatba
ij + εatab

ij + εitab
ji
ij − εitab

ij

= εbtab

= εbtab

= tab

122

(4.53)

If we rewrite the CCD equations, Eqn. (4.52), subtracting these terms to the left hand side,

and defining εab

ij = εi + εj − εa − εb we get:

ab(new)
t
ij

ij = vab
εab

ij +

vab
det

de(old)
ij

(cid:88)

de

1
2

(cid:88)
(cid:88)

lm

1
2

1
4

1
2

+

+

−

lmde

ˆP (ij)

vlm
de t

(cid:88)

lmde

vlm
ij t

ab(old)
lm

+ ˆP (ab|ij)

(cid:88)

ld

ad(old)
vlb
djt
il

(cid:88)

de(old)
ij

ab(old)
t
lm

+ ˆP (ij)

ad(old)
vlm
de t
il

be(old)
t
jm

ed(old)
vlm
de t
il

ab(old)
mj

t

lmde
1
2

ˆP (ab)

−

(cid:88)

lmde

vlm
de t

ad(old)
ml

eb(old)
t
ij

.

(4.54)

Here the (new) and (old) superscripts have been added to show that the CCD equations can

be solved iteratively. Starting with some guess for tab

ij in the right hand side of Eqn. (4.54),

the sums are carried out and stored as t

ab(new)
ij

, which can then be used as the guess for the

next iteration. This process is solved iteratively, and in most cases converges. The conditions

for stability of the convergence can be jeopardized when the gap between the unoccupied

and occupied single-particles states is small, causing the energy denominator 
ab

ij to approach

zero [62]. This manifests for systems with strong many-body correlations, and low order

truncations of CC theory are insufficient to capture the physics. To check convergence, after

each iteration the new t-amplitudes can be plugged into the ∆ECCD equation to see how

much the energy has changed compared to the previous iteration. Numerically, an iteration

tolerance can be set, ending the iteration loop once the energy is changing by amounts

smaller than the tolerance.

Iterative convergence for CC has been studied in detail providing sophisticated ways to

accelerate convergence [63], but for now we will keep this iteration simple. One small change

that can be made is to add a linear mixing parameter α, such that the new t-amplitudes for

123

the i-th iteration are

(i)
t

mixing = αt(i) + (1 − α)t(i−1).

(4.55)

For α = 0.5, this means that the next iteration will only use half of the t-amplitudes just

calculated, and half from the previous iteration. This can help convergence, especially in

situations where the iterations oscillate back and forth between two values. Adding in the

mixing parameter can help damp out these large steps in the wrong direction.

All iterative methods need a starting point, so the initial guess of tab

ij = 0 is typically

used. This actually leads to a second iteration that is tab

ij , which when plugged

ij = vab

ij /εab

into the energy equation gives

∆E =

1
4

|vab
ij |2
εab
ij

,

which is the MBPT(2) result. This is quite exciting, as we have recovered another many-body

method with CC theory, and all iterations beyond the first add more and more many-body

correlations into the t-amplitudes.

Despite ˆT ≈ ˆT2 seeming like a harsh approximation truncating many terms, CCD turns
out to provide surprisingly accurate results for many systems. The application of CCD to

quantum many-body systems can be seen in Chapters 5 and 6, where for some small models

the CCD results can be compared to the exact result. Of course CCD has its limitations.

Many systems, especially systems for which the reference state is a poor starting point, CCD

can fail to converge. Additional correlations can be included, but this quickly increases the

computational scaling of CC theory.

For infinite matter calculations, there are no many-body contributions from the single

excitations, therefore the next level of correlations to include are the triples excitations

ˆT ≈ ˆT2 + ˆT3. This addition does not change the CC energy expression, but the doubles

124

equations for the (cid:104)Φab

ij | projection are now

(cid:17)

C |Φ(cid:105) = 0,

(4.56)

(cid:104)Φab
ij |

ˆHN e

ˆT2+ ˆT3

(cid:16)

(cid:16)

(cid:17)

and we now also need to include the triples equations by projecting onto (cid:104)Φabc

ijk| as well

(cid:104)Φabc
ijk|

ˆHN e

ˆT2+ ˆT3

C |Φ(cid:105) = 0.

(4.57)

The doubles equations now have additional diagrams from ( ˆHN ˆT3)C which are

( ˆFN ˆT3)C →

(cid:88)
me (cid:104)m| ˆf|e(cid:105) tabe

ijm

=

and

( ˆVN ˆT3)C →

+

(4.58)

(4.59)

(cid:88)

mef

=

1
2

ˆP (ab)

(cid:88)
mne(cid:104)mn|ˆv|je(cid:105) tabe
imn.

ˆP (ij)

1
2

(cid:104)bm|ˆv|ef(cid:105) t

aef

ijm −

125

where again we are going to assume that we have Hartree-Fock single-particle states which

gives the condition (cid:104)m| ˆf|e(cid:105) = 0 and so the term in Eqn. (4.58) is zero.

Next, are the equations for the triples amplitudes. The full triples equations are not the

focus of this thesis, but we will examine the terms which contribute at leading order to the

CC energy. They correspond to

( ˆVN ˆT2)C →

+

(cid:88)

d

= ˆP (k/ij|a/bc)

(cid:104)bc|ˆv|dk(cid:105) tad

ij − ˆP (i/jk|c/ab)

(cid:88)

l

(cid:104)lc|ˆv|jk(cid:105) tab

il

and

( ˆFN ˆT3)C →

+

(4.60)

(4.61)

(cid:88)

d

= ˆP (c/ab)

(cid:104)c| ˆf|d(cid:105) tabd

ijk − ˆP (k/ij)

(cid:88)

l

(cid:104)l| ˆf|k(cid:105) tabc

ijl

where we have a new type of permutation operation ˆP (a/bc) = 1 − Pab − Pac. Were again
we assume that we are in the Hartree-Fock basis (cid:104)p| ˆf|q(cid:105) = εpδpq is diagonal, and as in Eqn.

126

(4.53), we rewrite Eqn. (4.61) as

(cid:88)

d

ˆP (c/ab)

(cid:104)c| ˆf|d(cid:105) tabd

ijk − ˆP (k/ij)

(cid:88)

l

(cid:104)l| ˆf|k(cid:105) tabc

ijl = −εabc

ijktabc
ijk.

(4.62)

Just as in the CCD equations, we can move this term to the left-hand side, and divide by the

single-particle energy denominator εabc

ijk. In doing so, we now have a minimal set of diagrams

for triples excitations in CC theory. We can set up the iterative equations as

εabc
ijktabc

ijk = ˆP (k/ij|a/bc)

(cid:104)bc|ˆv|dk(cid:105) tad

ij − ˆP (i/jk|c/ab)

(cid:104)lc|ˆv|jk(cid:105) tab
il ,

(4.63)

(cid:88)

d

(cid:88)

l

and an initial guess for the t2-amplitudes of zero on the right-hand side will generate an

initial guess for the t3-amplitudes on the right-hand side. These t3-amplitudes on the left-

hand side of Eqn. (4.63) are then plugged back into the double excitation equations in Eqn.

(4.59), completing the iterative scheme. This is referred to as the CCDT-1 approximation

to the full CCDT [64].

4.5 Computational Scaling of Coupled Cluster Theory

Now that the equations have been derived, let’s look at the computational effort required

for a single CCD iteration in Eqn. (4.54). Due to the symmetries of the t-amplitudes, the

CCD equations must be solved for (i > j) and (a > b), however it is often convenient to

just compute these for all i, j, a, b for reasons that will be explained later. For now, let’s just

get a handle on the scaling. If these equations are solved for i, j, a, b that already brings a

computation complexity of O(N 2
number of occupied single-particle states (holes) relative to the Fermi energy, and Np is the

p ) to just loop through the entries of tab

ij , where Nh is the

hN 2

127

number of unoccupied single-particle states. This “big O” notation is used to get a handle on

the scaling of the most expensive term, without worrying about constants of multiplication

or non-leading terms. Within each t-amplitude equation there are some heavy sums, like

(cid:88)

lmde

1
4

de tde
vlm

ij tab
lm,

(4.64)

which brings an additional O(N 2
it is common to ignore the difference between the number of hole states and particle states

p ) leading to a cost of O(N 4

p ). With the big O notation,

hN 4

hN 2

and to just use N ≈ Nh ≈ Np, and to say that CCD written in this form is an O(N 8) theory
for calculating the ground state energy.

For memory requirements, the primary objects that need to be stored are tab

ij and vab
cd,

which require N 2

hN 2

p and N 4

p number of elements to be stored. In many realistic calculations,

Np ≈ 10 ∗ Nh or even Np ≈ 100 ∗ Nh, meaning that vab
by far the largest object that needs to be stored in this theory.

cd with four particle state indices is

If there are three-body

forces in the calculation, the object wabc

def needs to be stored as well, placing some serious

memory requirements on the calculation. A discussion of the memory requirements and data

structures to handle the requirements of CC are explained in detail in Chapter 5.

It turns out that by reorganizing some terms, we can reduce the computational complexity

of the CCD equations. For example, if we define the intermediate term

Xlm

ij =

vlm
de tde
ij

(4.65)

(cid:88)

de

128

the term in Eqn. (4.64) can be rewritten with the intermediate term as

(cid:88)

lm

1
4

Xlm

ij tab

lm =

1
4

(cid:88)

lmde

vlm
de tde

ij tab
lm.

(4.66)

By computing the sum over the de indices first, storing the intermediate result, then after-

wards computing the sum over the lm indices, we have gone from a scaling of O(N 2

hN 2

p ) to

O(N 2

h + N 2

p ), which is a huge advantage! The only downside to this is that extra storage

must be used to store X, but this is typically small compared to the other storage require-

ments. This means that using intermediates, CCD is actually only a O(N 6) theory, which
is extemely cheap for the accuracy it brings. A systematic way of generating these interme-

diates is to follow the development of diagrams for the coupled cluster effective Hamiltonian

H ≡ ( ˆHN e

ˆT )C as outlined by Shavitt and Bartlett [16].

Any term which is quadratic in t can be done in two steps with an intermediate. However,

by examining which terms can be grouped into their own operators in the CC effective Hamil-

tonian, we can occasionally reuse terms for additional efficiency. We define the intermediates

as

Xa

b = f a

b −

1
2

lmd
1
2

(cid:88)
(cid:88)
(cid:88)
(cid:88)

1
2

del

de

1
2

dl

bd tad
vlm
lm,

detde
vil
jl ,

ij
detde
kl ,
v

dbtda
vil
jl ,

Xi

j = f i

j +

X

ij
kl = v

ij
kl +

Xia

jb = via

jb −

129

(4.67)

(4.68)

(4.69)

(4.70)

and then we can rewrite the CCD equation as

0 =vab

ij + ˆP (ab)

Xa

(cid:88)

d

vab
detde

ij +

(cid:88)

de

1
2

−

(cid:88)

l

Xl

itab
lj

lm − ˆP (ab|ij)

d tdb
ij − ˆP (ij)
(cid:88)

Xlm

ij tab

1
2

lm

(4.71)

(cid:88)

ld

Xlb

idtad
lj ,

which is exactly equivalent to the O(N 8) equations, but reduced down to O(N 6) operations
with minimal additional memory requirements.

The computational details for quantum many-body methods is the primary focus of this

work and specifically Chapter 5. As a method with polynomial scaling coupled cluster theory

is a great method to investigate computational challenges, since improvements in the data

structures and algorithms implementing these equations can greatly expand what can be

calculated with the method. This is in contrast to exact methods like full configuration

interaction, where it will never outrun the factorial scaling as the problem size increases.

Calculations with a large number of particles (Nh ∼ 100), and a very large basis (Np ∼
104−105) are frequently needed for calculations of interesting physical systems, which require
great care in the performance of the CC implementation. Alternatively, for systems without

as extreme basis size requirements, it may be necessary to use triples (CCDT) ˆT ≈ ˆT2 + ˆT3
or three-body forces ˆWN to achieve the accuracy desired for a calculation, but these bring a

heavy cost.

For a calculation of CCDT with three-body forces, one term for the (cid:104)Φabc

ijk| projected

equations is

1
8

(cid:88)

ef g

wabc
ef gt

ef g
ijk ,

(4.72)

which will result in an O(N 9) scaling theory. This is simply too expensive for anything but

130

the smallest systems. Three-body forces are brought up here to bring attention to other

ways which CC theory can grow prohibitively expensive along with increasing basis size N .

This is a pattern in any many-body method, that increasing the accuracy of a calculation

always has a cost, which dictates which physical systems can be studied and which cannot.

Big O notation can show asymptotically how CC theory will grow, but to consider how

expensive a particular calculation will be, it lacks predictive power. Here the multiplicative

constants (c ∗ N 9) and non-leading terms can be quite important. Chapter 5 will show how
the implementation of the CC equations into a code can vary the multiplicative constant by

up to five orders of magnitude. This swing in cost is nearly impossible to see just from the

CC equations alone, and so great care should be taken when writing a computer program as

it can determine the viability of a many-body method.

131

Chapter 5

Computational Methodology

The previous chapters have stressed how large and unwieldy a many-body calculation can

be. Even for a modest single-particle basis, the factorial growth of the full Slater determinant

basis becomes quickly impossible for even the largest computers in the world. It was shown in

Chapter 4 that coupled cluster (CC) theory generates expressions for the approximate ground

state energy which have polynomial scaling. This allows CC theory to compute properties

of much larger systems by sacrificing some accuracy of the solution. However, even with

efficient implementations of these equations with intermediate diagrams [65], coupled cluster

theory is still computationally expensive and runs into computational limits for all but very

small physical systems. Modern many-body physics necessarily becomes a computationally

challenging field just by the very scale of the problems at hand. This chapter will detail

how the same mathematical expressions on paper can take centuries to compute or seconds

to compute depending on the choice of data structures and algorithms implemented in the

code.

5.1 Code Validation

Before these optimizations are implemented, it is useful to first implement the many-body

method equations into code in the most direct translation from mathematics as possible.

Optimizing the code to run faster and compute larger basis sets will increase the number

132

of lines of code substantially, increasing the chance of human error. Once an inefficient,

but correct version of the code is finished, incremental optimizations moving forward can be

compared to the previously validated solution.

5.1.1 Pairing Model

Here, a simple system like the pairing model described in Chapter 3 is an excellent small

system to check the numerical results. In the case of the simple pairing model it is easy

to calculate ∆EM BP T 2 analytically from Eqn. (2.186), where MBPT2 refers to many-body

perturbation theory that was described in Chapter 2. This is a very useful check of our codes

since this analytical expression can also be used to check our first CCD iteration. We restate

this expression here but restrict the sums over single-particle states

∆EM BP T 2 =

1
4

(cid:88)

abij

(cid:88)

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)


ab
ij

=

a<b,i<j

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)


ab
ij

.

For our pairing example we obtain the following result

∆EM BP T 2 = (cid:104)01|ˆv|45(cid:105)2


45
01

+ (cid:104)01|ˆv|67(cid:105)2


67
01

+ (cid:104)23|ˆv|45(cid:105)2


45
23

+ (cid:104)23|ˆv|67(cid:105)2


67
23

,

which translates into

∆EM BP T 2 = −

(cid:18) 1

4 + g

g2
4

(cid:19)

.

+

1

6 + g

+

1

2 + g

+

1

4 + g

This expression can be used to check the results for any value of g and therefore provides

an important test of our codes. In Table 5.1, five significant figures are listed to compare

MBPT2 and CCD. The MBPT2 results were checked against the analytical results to ensure

133

that they could be reproduced numerically. Next, the CCD results can be checked against the

MBPT2 results. While these are two very different methods, they should give results that

are reasonably close to one another, especially at small values of the interaction strength

g. At g = 0, the particles are no longer interacting, so a first simple check of a code is

Table 5.1: Coupled cluster and MBPT2 results for the simple pairing model with eight single-
particle levels and four spin-1/2 fermions for different values of the interaction strength g.

g
-1.0
-0.5
0.0
0.5
1.0

Eref ∆EM BP T 2 ∆ECCD
-0.21895*
3
-0.06306
2.5
2
0
-0.08336
1.5
1
-0.36956

-0.46667
-0.08874
0
-0.06239
-0.21905

that the correlation energy drops to 0. Also note that the g = −1.0 case diverges without
implementing iterative mixing. Iterative mixing is defined by

t(i) = αt

(i)

no mixing + (1 − α)t(i−1),

(5.1)

where t(i−i) is the t-amplitude from the previous iteration, t

(i)
no mixing is the updated am-

plitude. By choosing a mixing parameter α, we create a simple linear combination of the

current iteration and the previous iteration to use in the next iteration. This can help the

CC iterations converge faster, or in some cases prevent oscillating or diverging iterations.

In Fig. 5.1 we can see that CCD compares quite well to the exact calculation of FCI in

this range of interaction strength g. Also plotted are higher orders of many-body pertur-

bation theory, MBPT3 and MBPT4, which are higher order corrections to the many-body

perturbation theory correlation energy. Coupled cluster doubles does not start diverging

until larger values of interaction strength, as this method includes significantly more many-

134

Figure 5.1: Correlation energy for the pairing model with exact diagonalization, CCD and
perturbation theory to third (MBPT3) and fourth order (MBPT4) for a range of interaction
values.

body correlations. We can say CCD looks good “by eye”, but when validating numerical

results, it is necesary to print out several digits and make sure the code is validated to a set

level of precision.

5.1.2 Infinite Neutron Matter

Once the pairing model is numerically handled, it is good to benchmark against a more

realistic system. By writing the computer program that solves these equations in a modular

way, it is not too difficult to add in new physical systems to calculate via their own module

135

1.00.50.00.51.0Interaction strength, g0.50.40.30.20.10.0Correlation energyExactMBPT3MBPT4CCDto plug in. To calculate properties of infinite neutron matter matter, only a few parameters

are needed for the system. The number of neutrons in the box, the density of these neutrons,

and the number of basis states above the Fermi surface. To get an idea of how the system

behaves, it is typical to calculate the ground state energies at different densities, particle

numbers and basis sizes. Here we present some calculations for a system of neutrons using

many-body perturbation theory (MBPT) and coupled cluster doubles (CCD). Table 5.2 lists

a set of numerical values to check the infinite matter basis with the Minnesota potential [53]

described in Chapter 3. While this is not a very realistic nuclear force, it has enough of the

right symmetries and properties for code validation purposes.

Table 5.2: CCD and MBPT2 results for infinite neutron matter with N = 66 neutrons and a
maximum number of single-particle states constrained by Nmax = 36 (36 plane wave energy
shells).

Density ρ fm−3 EM BP T 2 ECCD
6.468
0.04
0.06
7.932
9.136
0.08
10.074
1.0
10.885
1.2
1.4
11.565
12.136
1.6
12.612
1.8
2.0
13.004

6.472
7.919
9.075
9.577
10.430
11.212
11.853
12.377
12.799

5.2 Taming the Two-Body Basis

The first of many computational considerations that will be examined here is how to

get a grasp on the two-body basis. As was explained in Chapter 2, the full many-body

basis built from Slater determinants is a factorially growing problem with respect to the

136

single particle basis. This motivated the need for more efficient methods like CC theory

and in-medium similarity renormalization group (IM-SRG). However, even in the world of

polynomially scaling methods, the size of the problem is still enormous. Let’s first look at

the memory demands of CC with a two-body force. The two-body force has antisymmetrized

matrix elements (cid:104)pq|ˆv|rs(cid:105). If we have a calculation using 103 single particle states (common
in nuclear matter), then we would need to store 1012 matrix elements, which are usually

complex numbers of double precision. A double precision floating point number, commonly

called a double, holds 8 bytes of data. A complex double needs 16 bytes, a double for

the real part and a double for the imaginary part. With this in mind the full two-body

matrix requires 16, 000 gigabytes of memory. Now, this is not impossibly large for modern

supercomputers, but is inaccessible to anyone attempting this calculation on a laptop.

To reduce the storage requirements of the two-body force, we first look to the CC equa-

tions.

In any particular diagram in the CCD equations, full unrestricted single-particle

indices (p, q, r, s) are never used, only terms of indices of a fixed particle (a, b, c, d) or

hole (i, j, k, l) nature. We can organize the matrix in terms of groups of particles and

holes. For example vpp pp represents the two-body piece in terms where all of the sums

are over particle indices vab

cd := (cid:104)ab|ˆv|cd(cid:105) appear. Following this notation, the interaction
can be grouped into 24 = 16 different sectors vpp ph, vpp hh, . . . . Due to symmetries of

the interaction, not every particle-hole sector has to be stored, as these symmetries mean

that some of the information is redundant up to a phase as seen in Table 5.3, where

all of the terms in parentheses are previously listed. Looking at the symmetries of the

two-body operator, these are the only subsections that need to be stored, since by an-

tisymmetry: (cid:104)pq|ˆv|rs(cid:105) = −(cid:104)qp|ˆv|rs(cid:105) = −(cid:104)pq|ˆv|sr(cid:105) = (cid:104)qp|ˆv|sr(cid:105), and by Hermiticity, so
(cid:104)pq|ˆv|rs(cid:105) = (cid:104)rs|ˆv|pq(cid:105)∗.

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Table 5.3: All possible particle-hole sectors are listed in the left column. In the right column
are 6 particle hole sectors which contain all of the information of the whole matrix, plus how
the other 10 can be equivalently expressed.

vpp pp
vpp ph
vpp hh
vph hh
vhh hh
vph ph

Sector Equivalent Sector
vpp pp
vpp ph
vpp hh
vph hh
vhh hh
vph ph
vpp hp −(vpp ph)
(vpp hh)∗
vhh pp
vhh hp −(vph hh)∗
vhp pp −(vpp ph)∗
(vph ph)
vhp hp
(vpp ph)∗
vph pp
vhp ph −(vph ph)
vph hp −(vph ph)
vhp hh −(vph hh)
(vph hh)∗
vhh ph

These two facts let us reproduce any matrix element while only storing six sectors. Since

the number of occupied states (hole states) is typically much less than the unoccupied states

(particle states) nh << np, the largest term, vpp pp comprises the vast majority of memory

requirements. The storage of the large amount of matrix elements receives some relief by

only storing the non-redundant matrix elements, but the size of vpp pp alone grows large

enough to make many calculations impossible.

Fortunately for most two-body interactions, this matrix is incredibly sparse. To write

the two-body interaction as a matrix, all of the two-body states need to be organized. One

multi-index scheme maps two single-particle indices i, j which run from 0 to N to a single

column index can be written as: Column Index = (N + 1) ∗ i + j as shown in Table 5.4.

While this is a simple way to organize the two-body interaction matrix, it has some

138

Table 5.4: A straight forward scheme to organize the two-body basis in columns.

Column Number
Two-Body State

0
|00(cid:105)

1
|01(cid:105)

2
|02(cid:105)

. . . N + 1 N + 2
. . .

|10(cid:105)

|11(cid:105)

. . . N 2
. . .

|N N(cid:105)

serious drawbacks. In this scheme, the matrix appears very sparse, but the non-zero matrix

elements are organized essentially randomly throughout the matrix. There has been much

study of sparse matrices, the underlying physics of the problem serves as a guide to a custom

compressed data structure. To look at how to get some more significant reductions, we look

at the symmetries of the Hamiltonian.

We know that the quantum numbers of the eigenstates of a Hamiltonian describe the

values of the conserved quantities of that Hamiltonian. So it is natural to try and select a

single-particle basis for a calculation that is labeled by relevent quantum numbers for the

problem. This way, even the approximate solutions that are computed are assured to have

the correct symmetries. This is almost always done in CC calculations, although the field of

symmetry broken reference states is very active in many-body theory [66]. The advantage

here is that the bra and ket states of a Hamiltonian matrix element must have the same

conserved quantities. That is, we know a priori that any matrix element for which the

bra and ket states do not have the same conserved quantities, must be zero. This is the

fundamental fact that guides the compressed data structure frequently used in many-body

theory. We can use the symmetries of the single-particle basis that are unbroken by the

Hamiltonian to throw away vast amounts of zeroed matrix elements. Looking back to the

two-body force, we want a way to categorize the matrix elements that we know are going to

be zero by symmetry arguments. This is done by organizing the two-body basis into “blocks”,

where a block is uniquely determined by the conserved quantities of the two body states |pq(cid:105).

139

For example, the proton and neutron plane wave basis described in Chapter 3 has single

particle states with the quantum numbers {px, py, pz, sz, tz}. A particular two-body state
|ij(cid:105) would be sorted into a block with ξij = {pxi+pxj , pyi+pyj , pzi+pzj , szi+szj , tzi+tzj},
where ξij is a compact notation for the set of summed quantum numbers. Now we have a

convenient system for finding symmetry exluded matrix elements. For example, if two basis

states |13(cid:105) ,|46(cid:105) such that ξ13 (cid:54)= ξ46, then (cid:104)13|v|46(cid:105) = 0. The goal now is to sort through the
whole two-body basis |pq(cid:105) ,∀p, q, and group each state into a symmetry block. Then only
two-body states in the same block will produce non-zero matrix elements, and we can ignore

all of the rest. In many bases, like the plane wave basis, this yields an enormous reduction in

the number of matrix elements that need to be stored. The reason these are called blocks, is

because they show that the matrix can be organized into a block diagonal structure. Figure

5.2 shows this block diagonal structure, as well as just how sparse vpppp is.

Figure 5.2: The pp-pp sector of a two-body interaction matrix for a simple neutron matter
system with 40 single-particle states above the Fermi level.

Figure 5.2 is a case with 54 single-particle basis states and 14 particles, yielding 14 hole

140

states and 40 particle states. Even in this modest basis size, we can see just how sparse

the matrix is.

In a more realistic calculation with 1000 single-particle states, a factor of

about 1000 can be saved, going from 16Tb of storage needed down to 16Gb, which plenty

of laptops nowadays have. This particular matrix in the figure is the two-body matrix

for a pure neutron matter calculation with the Minnesota potential with 54 states in the

single-particle basis. This was used for visualization, since showing off larger matrices would

contain almost entirely white space in this plot. Along with greatly reducing the memory

needs of the calculation, this also yields enormous speed improvements, since we can now

skip multiplying by zero millions of times.

5.3 Performance Testing Matrix-Matrix Multiplication

Matrix-matrix multiplication, which is frequently used in many-body calculations, is a

nice case study for computational speedup. For three matrices A, B, C with matrix elements

apq, the product of C = A ∗ B is written as:

(cid:88)

r

cpq =

apr ∗ brq ∀p, q.

(5.2)

Counting operations, each element cpq is calculated with about r addition operations and r

multiplication operations. This must be done for all p, q elements in C. For notational con-

sistency, let’s say that the first index p is the row index, and the second index q is the column

index. If A, B, C are all N × N square matrices, this would mean the calculation of C would
require O(N 3) operations. There are mathematical speedups like the Strassen algorithm
[67], which scales as O( N 2.8), and algorithms which can further lower this computational

141

complexity, but let’s first just look at this simple sum in Eqn. (5.2). While mathematical

complexity is very important as better scaling is almost always favorable, they leave out im-

portant details, such as potentially large (or small) coefficients in front of these polynomial

powers. These coefficients can manifest themselves in unexpected ways if you haven’t exam-

ined how the mathematics is actually carried out on a low level. To show this, let’s introduce

two algorithms that are mathematically equivalent. First, when selecting which matrix el-

ement cpq to compute, we will first loop over p, the rows of C, followed by q, the columns.

In the second algorithm, we will loop over the columns first, then the rows. Importantly,

the innermost loop over the summed index is unchanged, so on paper this looks identical.

142

for row p in C do

for col q in C do

c[p][q] = 0.0;

for col r in A do

c[p][q] += a[p][r] * b[r][q]

end

end

end
Algorithm 1: Basic matrix-matrix multiplication, looping over rows then columns

for col q in C do

for row p in C do

c[p][q] = 0.0;

for col r in A do

c[p][q] += a[p][r] * b[r][q]

end

end

end
Algorithm 2: Basic matrix-matrix multiplication, looping over columns then rows
Now, in Figure 5.3 is a timing plot, showing how the cpu timings of these algorithms com-

pare, with an optimized routine (dgemm) included as well. The results are pretty dramatic.

As the size of the matrix gets larger, algorithm 2 becomes substantially faster than algorithm

1. This is largely due to how data are moved from memory to the processor, but the exact

details of this are saved for a later section. The major takeaway is that considerations of how

data are accessed in matrices, or tensors in the CC case, is of considerable importance. The

red dotted line labeled “dgemm” is a BLAS (Basic Linear Algebra Subprograms) routine

143

Figure 5.3: Implementation of the same mathematics can have very different run times.

which computes a general matrix-matrix multiplication. The name comes from the fact that

the matrices are double precision (d) of general structure (ge), meaning not symmetric, and

it is matrix-matrix (mm) muliplication, and not matrix-vector or anything else. Looking at

the dgemm speeds where many layers of optimizations have been made, we can see that even

for small matrices of size 1000 × 1000, full order of magnitude savings can be made. This
implementation of dgemm comes from OpenBLAS, which is an open source package [68].

This is not a one-off example, but just a glimpse at how important computational details are.

Many-body theory is becoming an increasingly interdisciplinary field, as expertise in physics,

applied mathematics, and computational science are all often needed in equal importance to

access interesting questions in nature.

144

5.4 Tensor Contractions as Matrix Multiplication

Let’s look at a particularly expensive CC diagram. We can find very expensive terms for

a CCDT calculation which includes the full normal-ordered three-body force WN . In this
case, there will be terms that look like (cid:104)Φabc
ijk|( ˆWN ˆT3)C|Φ0(cid:105). In the triples equation, we will
(cid:88)
have

(5.3)

1
8

def

wabc
def t

def
ijk .

This triple sum over particle states must be computed for all a, b, c, i, j, k which means that

p) = O(n9). So even for modest single-particle basis sizes, this
the scaling cost is O(n3
will be costly to compute. Similar to how the two-body basis was organized in terms of

hn6

symmetry blocks, we can do the same for the three-body basis. This has two-fold benefits.

The first is an enormous reduction in the memory required to store them, which is shown in

Chapter 6. The second is that this three-body basis creates a mapping from the three single

particle indices into one three-body index {a, b, c} → {A}. This way, we can write the n9
diagram as

wA
BtB
I ,

(5.4)

(cid:88)

B

1
8

where the inner index that is being summed over. This shows that with this index remapping,

we have exactly the definition of matrix-matrix multiplication. This is a big win, because

from the plot 5.3, we can now take advantage of the extremely optimized OpenBLAS dgemm

routine.

One complication is that the symmetry organized three-body basis grouped the matrix

into blocks, and we don’t want to perform matrix-matrix multiplication on the entire matrices

together, keeping track of many zero elements. The solution is that within each symmetry

145

block ξpqr, the column index B of wA

B will still match perfectly with the row index of tB

I , as

in the two-body case described before. This means that we can just loop over the set of all

symmetry blocks {ξpqr}, and do a block-to-block matrix-matrix multiply.

(cid:88)

(cid:88)

{ξpqr}

B

1
8

w

B(ξ)
A(ξ)
B t
I

(5.5)

where the superscript (ξ) is denoting that the we now also have a dependence on the symme-

try blocks. This computational strategy is seen across a variety of many-body calculations.

In CC theory, there are many terms like Eqn. (5.5), and in IM-SRG there are terms like

(cid:88)
uv (cid:104)qr|η|uv(cid:105)(cid:104)uv|Γ|st(cid:105)

1
2

block

=

1
2

(cid:88)

(cid:88)

{ξuv}

U

(cid:104)Q|η|U(cid:105)(ξ) (cid:104)U|Γ|S(cid:105)(ξ) ,

(5.6)

for qr → Q, uv → U and st → S, where the right-hand side has been written in block
matrix notation to show that these terms can also be written as matrix-matrix products.

Any operator which conserves the symmetries of the two-body states can be written as a

block diagonal structure, allowing for the efficient storage of non-zero matrix elements and

the usage of efficient matrix-matrix multiplication. While this work is largely focused on

coupled cluster theory, it is important to stress that these tools have applicability to a large

range of many-body methods.

A simple many-body method to consider is the expression for the correlation energy from

MBPT2. We rewrite

(cid:88)

abij

1
4

(cid:104)ij|ˆv|ab(cid:105)(cid:104)ab|ˆv|ij(cid:105)


ab
ij

,

∆EM BP T 2 =

(5.7)

by defining the matrices ˆA and ˆB with new indices I = (ij) and A = (ab). The individual

146

matrix elements of these matrices are

and

AIA = (cid:104)I|ˆv|A(cid:105),

BAI = (cid:104)A|ˆv|I(cid:105)


A
I

.

We can define the intermediate matrix ˆC as

(cid:88)

Aξ

(ξI )
IJ =

C

(cid:104)I|ˆv|A(cid:105)(ξ) (cid:104)A|ˆv|J(cid:105)(ξ)


A
I

,

(5.8)

which is the matrix product over the blocks of ( ˆA)( ˆB). We have written A with a subscript

ξ as notation to restrict this sum to the symmetry block ξI defined by the left-hand side of

the equation. From there we can rewrite the correlation energy from MBPT2 as

∆EM BP T 2 =

1
4

C

(ξ)
II ,

(5.9)

(cid:88)

(cid:88)

{ξij}

Iξ

which is the trace over the blocks of the matrix product ˆC = ˆA ˆB. Again the (ξij) is denoting

the block defined by the quantum numbers of the two-body states |ij(cid:105), and the sum over Iξ
is denoting the restricted sum over two-body states contained within the symmetry block.

By writing the inner sum over the A index as a series of matrix-matrix products between

the blocks (ξ) of matrices A and B defined above, the entire expression is almost computed.

Only the sum over the hole indices is left, which is taken care of by the trace.

Figure 5.4 shows the difference between the brute force summation over single-particle

states of Eq. (5.7) and the block matrix set up, that is Eq. (5.8). In these calculations we

147

Figure 5.4: MBPT2 contribution to the correlation for pure neutron matter with N = 14
neutrons and periodic boundary conditions. Up to approximately 1600 single-particle states
have been included in the sums over intermediate states in Eqs. (5.7) and (5.8)
.

have only considered pure neutron matter with N = 14 neutrons and a density n = 0.08

fm−3 and plane wave single-particle states with periodic boundary conditions, allowing for

up to 1600 single-particle basis states. The Minnesota interaction model [53] has been used

in these calculations. With 40 single-particle shells for example, we have in total 2713

single-particle states. Using the block matrix algorithm the final calculation time is 2.4 s

(this is the average time from ten numerical experiments). The total time using the brute

force summation over single-particle indices is 100.6 s (again the average of ten numerical

experiments), resulting in a considerable speed up. It is useful to dissect the final time in

148

02004006008001000120014001600Number of Single Particle States10-1100101log[Time to Calculate E_MBPT2 (sec)]MBPT2 Timing Tests   Time to Calculate (log scale) vs Basis SizeBlock MatricesFull Structureterms of different operations. For the block matrix algorithm most of the time is spent

setting up the matrix elements for the two-body channels and to load the matrix elements.

The final matrix-matrix multiplication takes only 1% of the total time. For the brute force

algorithm, the multiplication and summation over the various single-particle states takes

almost half of the total time. This is how code optimization typically progresses, take a

section which is the current computational bottleneck and tackle that. At which point the

next most expensive subroutine become prominent and must be tackled until the code runs

sufficiently fast for the task at hand.

This performance speed up is very nice, but unfortunately it is not always so easy. There

are many terms in the CCD equations where the tensor contractions do not have their indicies

aligned as matrix-matrix products. For example,

(cid:88)

ld

− ˆP (ab|ij)

(cid:104)lb|X|id(cid:105)(cid:104)ad|t|lj(cid:105) ,

(5.10)

(5.11)

where

(cid:104)lb|X|id(cid:105) = (cid:104)lb|ˆv|id(cid:105) −

(cid:88)
em (cid:104)lm|v|ed(cid:105)(cid:104)eb|t|im(cid:105) .

1
2

We can see that the contracted indices do not match up bra to ket in either of these cases,

so some additional work must be done.

Looking at Eqn. (5.10), we can write the sum as a matrix product if we permute the

indices by

(cid:88)

ld

− ˆP (ab|ij)

(cid:104)a¯j|t|l ¯d(cid:105)(cid:104)l ¯d|X|i¯b(cid:105) ,

(5.12)

149

where the bar over the index represents that it has been permuted from bra to ket or from

ket to bra. Normally, this would be a relatively straightforward transpose type operation

where the elements of the tensor are reshuffled, but due to the symmetry block structure

of the tensors it gets a bit more complicated. This can be seen most strikingly with the

t-amplitudes which are always in the format of two-particle two-hole excitations tpp hh, but

when permuted get shuffled into a form that looks like tph hp. This structure does not exist

for the t-amplitudes, which is why we must be careful to put the bars over the indices to

indicate this non-standard placement.

To maintain the block diagonal structure of this permuted tensor, we must rewrite what

the symmetry blocks represent. A conservation law for (cid:104)ab|t|ij(cid:105) that looked like ka + kb =
ki + kj will now look like ka − kj = ki − kb for (cid:104)a¯j|t|i¯b(cid:105). This is still the same conservation
law, just shuffled around. Since the momenta are subtracted, this is functionally a time

reversed state which is why the bar symbol for the anti-particle is used here.

Now that Eqn. (5.12) has been permuted, it would also help to write the intermediate

in terms of the permute indices as well,

(cid:104)l ¯d|X|i¯b(cid:105) = (cid:104)l ¯d|ˆv|i¯b(cid:105) −

(cid:88)
em (cid:104)l ¯d|v|e ¯m(cid:105)(cid:104)e ¯m|t|i¯b(cid:105) .

1
2

Plugging these in together we get

− ˆP (ab|ij)

(cid:104)a¯j|t|l ¯d(cid:105)

(cid:88)

l ¯d

(cid:32)
(cid:104)l ¯d|ˆv|i¯b(cid:105) −

(cid:33)
(cid:88)
e ¯m (cid:104)l ¯d|v|e ¯m(cid:105)(cid:104)e ¯m|t|i¯b(cid:105)

,

1
2

(5.13)

(5.14)

where now we see that we have rewritten the tensor contractions as matrix products: one

over l ¯d and one over e ¯m. The resulting t-amplitude will then be in the format of (cid:104)a¯j|t|i¯a(cid:105),

150

so an additional permutation must be done to get it back into the correct format.

It is

interesting now to think if all of this was worth it. After all, the main point of the matrix-

matrix product was to take advantage of efficient data movement, but this now looks like

a lot of wasted inefficient data movement. While the t-amplitudes do need to be permuted

and un-permuted every iteration, the two-body interaction elements do not. The permuted

two-body matrix elements only need to be calculated once at the beginning of the code, and

can then be used for every loop of the CC iterations.

5.5 Parallel Computing

The next step to increase the performance of the code is parallelization.

If the block

matrix strategy is still not yet fast enough to target a physical system of interest, then

the next step is to look towards a faster computer. While Moore’s law has continued the

increased transitor density, this has not lead to a one-to-one increase in processor speed. To

continue the exponential growth of computational power, modern computers have begun to

increase the number of processing cores available per computer. This trend has been going

strong for a couple of decades now, with supercomputers capable of performing hundreds

of quadrillions of floating point operations per second (hundreds of petaFLOPs), and the

next generation of supercomputers is predicted to break into the exaFLOP era of computing.

However, accessing this level of performance brings many challenges, as the parallel paradigm

requires rethinking algorithms and data structures at a fundamental level. Programs which

are written and optimized for a single thread of execution, which is called a serial program,

often have to be completely overhauled to run in parallel.

The first step to take advantage of parallel computing is to identify regions of the program

151

where a many computations are performed that are independent of each other. This usually

corresponds to regions of the code for which many iterations of a loop need to be done, and

the execution of the loop entries can be done in any order. For example let’s examine an

algorithm to compute the correlation energy of many-body perturbation theory at second

order (MBPT2), that is, Eqn. (5.7). An algorithm which implements this expression in a

straight forward way can be seen in Algorithm 3. The only necessary data structures are

a list of hole states, a list of particle states, the two-body interaction, and an array of the

single-particle energies. The energy can the be calculated with a simple set of nested loops.

energy = 0.0;

omp parallelization directive goes here for i ∈ hole states do

for j ∈ hole states do

for a ∈ particle states do

for b ∈ particle states do

numerator = twoBodyInteraction(i,j,a,b);

numerator = numerator*numerator;

denominator = spEnergy[i] + spEnergy[j] - spEnergy[a] - spEnergy[b];

energy += 0.25*numerator/denominator;

end

end

end

end

Algorithm 3: Basic algorithm for calculating the many-body perturbation theory energy

at second order.

If the number of single-particle states is large, then this type of pattern is perfect for

parallelization, since all of the computations are independent of one another. Parallelizing a

152

loop like this is as simple as including a compiler directive from the Open Multi-Processing

(OpenMP) [69] application programming interface (API) before the first loop. OpenMP

is typically used to parallelize code where a single computational node has multiple cores,

allocating one thread of execution for each computation core by default. OpenMP’s compiler

directives create multiple threads of execution to distribute the work from loops to the

available processors, cutting down to overall time to compute. The only thing to take care

of here, is that each thread needs to have its own copy of the “energy” variable so that they

are not trying to overwrite each other. Then, each thread can combine its partial sum into

a total energy upon exiting the loop using the “reduce” directive.

Unfortunately, this seems like we have taken a step backwards to take a step forwards,

since we have only parallelized the brute force version of this calculation. This was to illus-

trate cases where parallelization is very easy, and much can be gained for a small amount

of effort. This is a case where it is significantly easier to parallelize the simple implemen-

tation of the code, but the advantage we gain by compressing the matrices into blocks is

too good to give up. An algorithm for the more optimized block matrix implementation

for MBPT2 is in algorithm 4, where the loops are now over the block diagonal structure.

153

energy = 0.0;

for block ∈ symmetryBlock.blocks do

for row ∈ symmetryBlock(block).rows do
for col ∈ symmetryBlock(block).cols do

i = symmetryBlock(block,row,col).holeIndex1;

j = symmetryBlock(block,row,col).holeIndex2;

a = symmetryBlock(block,row,col).particleIndex1;

b = symmetryBlock(block,row,col).particleIndex2;

numerator = twoBodyBlockInteraction(block,row,col);

numerator = numerator*numerator;

denominator = spEnergy[i] + spEnergy[j] - spEnergy[a] - spEnergy[b];

energy += 0.25*numerator/denominator;

end

end

end

Algorithm 4: Block diagonal algorithm for calculating the many-body perturbation theory

energy at second order.

One consequence of looping over only the non-zero matrix elements is that a new data

structure, here called “symmetryBlocks” needs to keep track of how many total blocks there

are, how many rows and columns are within each block, and the single-particle indices

that generated each matrix element. While this is a worthwhile trade-off, it does make

the parallelization more difficult. While putting the usual parallelization directive at the

top most loop will again split the amount of work up and distribute it to the different

threads. However, the basic directives will distribute a roughly equal number of blocks to

154

each thread, and the blocks vary in size considerably. This leads to what is called “load

imbalance”, where some processors have a much larger amount of work to be done than

others. Once the processors with the least amount of work finish, they just sit idle while

the other processors keep going. Thus the time it takes to complete the parallel section is

completely bottlenecked by which thread was given the most amount of work. In Fig. 5.5, a

cartoon of the problem is shown with an example of four threads of execution. Even if the

Figure 5.5: A cartoon of how the interaction matrix might be split into work loads for
different threads of execution for the naive storage and the block storage schemes.

parallelization scheme is terribly load balanced the block matrix implementation will still

be significantly faster, but optimizing the parallelization of the blocks can be tricky since

the sizes of the blocks varies considerably as seen in Fig. 5.2. This load imbalance becomes

worse and worse the larger the matrices become, and the more threads that are used for

parallelization.

155

5.6 Distributed Memory Parallelization

As the system increases in size, even the block diagonal compression of the full interac-

tion becomes insufficient, as the memory required to store the blocks becomes to large to

contain in RAM. So the speed of the computation becomes an irrelevant question, as the

computation is impossible. This memory wall can be circumvented with distributed mem-

ory parallelization, where multiple computational nodes are linked together via a network

and work together. This allows the combined RAM of multiple nodes to be leveraged to

solve larger problems. In practice, this is done with the Message Passing Interface (MPI)

API, which launches a copy of a program on each node, and allows the communication of

the copies amongst each other. For this application, the OpenMPI [70] implementation was

used. While the increased memory of many compute nodes helps with the memory problem,

it does not circumvent the load balancing problem. Whereas OpenMP allows each thread of

execution to view the whole matrix in memory, the load imbalancing was more a matter of

which thread is responsible for which computations. With distributed memory parallelism

each copy of the program, or “rank” as they are called in MPI, can only access the matrix

elements on other ranks via costly communication accross the network. Now Fig. 5.5 can

represent the difficulty in distributing the interaction matrix across multiple compute nodes.

If we distribute the blocks of the interaction matrix such that each MPI rank has an equal

number of blocks (with no consideration of block size), then the ranks with the smallest

number of matrix elements finish much faster than the ranks with more work to do. This

disparity can be seen in Fig. 5.6, where the time between the fastest rank and the slowest

rank becomes increasingly larger for increasing number of MPI ranks.

To correct for this, before the matrix elements are computed, the block sizes are computed

156

Figure 5.6: Performance of MBPT2 calculations with increasing number of MPI ranks. The
speed of the calculation is measured in s−1, the black data are the inverse time required
to finish the calculation on the fastest rank, and the red data are the speeds of the slowest
rank.

and sorted so that they can be distributed such that each rank is responsible for roughly the

same number of non-zero matrix elements, rather than the same number of blocks. This has

the added benefit that the calculation is load balanced for the computation of the matrix

elements, such that the time it takes to load the matrices is roughly equal among the ranks

as well. In the case of calculations for infinite matter, the two-body basis forms many more

small blocks than large blocks. The blocks can be passed to the ranks by a bin-packing

algorithm, which passes the largest blocks to the ranks starting from the largest block. Once

the largest blocks have been distributed, the small blocks can passed in to keep all of the

ranks roughly even with respect to number of non-zero matrix elements.

157

5.7 Final Parallel Algorithm

We finally have all of the ingredients to describe the distributed memory algorithm for

performing coupled cluster calculations implemented for this work. The full algorithm is

shown via a series of cartoons and histograms below to illustrate the work flow. These

figures and this distribution scheme were a joint effort by the author, Stephanie Lauber

and Peter Ahrens. The first step is to pre-compute the size of the two-body basis blocks,

and assign the blocks to MPI ranks in a way to minimize load imbalance. This is shown

in Fig. 5.7, where the three columns represent three MPI ranks, and the blocks colored in

blue represent which blocks each rank is responsible for. The stacked layers of the matrices

represents that this is happening for multiple sectors of the interaction matrix, although

due to its size, most of this discussion is focusing on the vpp pp sector. On the right side

of the figure is a histogram of the time required to load the matrices as a function of block

size for an example caculation neutron matter with the Minnesota potential using the CCD

approximation, although the results are largely general to any infinite matter calculation.

The highest peak is at the first bin, which shows that loading the blocks with 50,000 or

less non-zero matrix elements constitutes the most total compute time at this stage in the

calculation. The distribution has a fairly long jagged tail with the final bin being a single

block with about 2.5 million non-zero matrix elements. The next step is to perform the

tensor contractions, starting with the terms which require some permutation to be aligned

as a matrix-matrix product. In CCD, one such term is

(cid:88)

dl

1
2

(cid:104)kl|v|cd(cid:105)(cid:104)db|t|lj(cid:105)

(5.15)

158

Figure 5.7: The blocks are distributed to the ranks to try and keep the number of non-zero
matrix elements equal among ranks. The histogram shows that the time it takes to load
these blocks is dominated by an enormous amount of small blocks, which is ideal for load
balancing.

where the inner indices are not aligned, and so a permutation step is required to write this

as

(cid:88)

dl

1
2

(cid:104)χ(kc)|v|χ(dl)(cid:105)(cid:104)χ(dl)|t|χ(jb)(cid:105) ,

(5.16)

where the χ represents the permutation operation which remaps the data to align the tensor

contraction as a matrix-matrix multiply. In Fig. 5.8, the tall skinny blocks represent the

t-amplitudes, since they are always of the form tpp hh, the particle dimension is always

considerably larger than the hole dimension. Secondly, tensor elements are now colored

in yellow to represent the fact that for a given rank the permutation operation requires t-

amplitudes which are on other ranks. Here, the overall time to compute these diagrams is

smaller than the loading step, but it is interesting to see from the histogram that it is now

the large blocks which are taking the longest. The actual matrix-matrix product is not very

costly here, most of the time in this step is spent receiving and transmitting data across

the network for the permutation operation. Furthermore, the communication pattern can

be entirely pre-computed before the first iteration. This allows for efficient communication

across the network, as each rank can group all of the elements that need to be sent into a

159

Figure 5.8: The t-amplitudes are permuted as needed for the tensor contractions which are
not aligned as a matrix-matrix product. In the histogram, the larger blocks have begun to
take more of the total time relative to the previous step.

separate buffer from each other rank that needs the data.

The next step is to compute the tensor contractions which do not require any permutation

to be expressed as a matrix-matrix product. This step includes the most costly term, that

of

(cid:88)

cd

1
2

(cid:104)kl|v|cd(cid:105)(cid:104)cd|t|ij(cid:105)

(5.17)

which needs the vpp pp sector of the two-body interaction. The histogram in Fig. 5.9 shows

that the work is much more even across block sizes than the load step. This is because

loading the matrices scales as N 2 for a matrix of row size N , and a matrix-matrix product

scales as N 3. So the large dense matrices take proportionally longer than adding together

many small matrix products.

The last step, shown in Fig. 5.10, is to update the t-amplitudes using the partial sums

from the various diagrams. Once each rank has updated the t-amplitudes, it computes its

partial sum of the CCD correlation energy, and a global reduce between all of the ranks

sums the final energy for that step. If the energy is still changing rapidly (above some set

160

Figure 5.9: The largest tensor contractions are now performed, which are already aligned as
matrix-matrix products. The O(N 3) scaling of the matrix-matrix product causes the larger
matrices to contribute significantly to the total processing time in this step.

tolerance) compared to the previous iteration, these new amplitudes are used for the next

iteration, which begins at step 2 of the algorithm. Step 1 does not need to be recomputed

each step, since the interaction matrices only need to be loaded once at the beginning of the

calculation.

Figure 5.10: The t-amplitudes are summed together and the correlation energy is calculated.
If the energy has not converged to the set tolerance, another iteration of the CC equations
are performed, using these new t-amplitudes in step 2.

Once the energy has converged, the CC correlation energy for the ground state of the

system has been found. The results for some selected physical systems and more concrete

161

performance tests can be seen in Chapter 6.

162

Chapter 6

Results

One quantity of interest to many physical scientists is the equation of state for different

forms of matter. The equation of state describes some state variables of matter under certain

conditions. For example, Boyle’s law which describes the relationship between the pressure

and volume of an ideal gas. If the gas is in a plunger and you push down on it, the equation

of state describes how much pressure pushes back on the plunger. The equation of state of

nuclear matter is of great importance in understanding the interior of neutron stars.

This matter has enormous gravitational pressure on it, and understanding the equation

of state of such dense matter, how hard the matter pushes back, would allow the mass-radius

relationship of neutron stars to be calculated. This is observationally very difficult due to

how small and dim neutron stars are, so a first principles calculation of the equation of state

is of great interest to the nuclear astrophysics community. One way to simulate the interior

of a neutron star is to calculate the energy of a large slab of neutrons with the many-body

Schr¨odinger equation. As outlined in previous chapters, by choosing a single particle basis of

plane waves in a box with periodic boundary conditions, we can systematically increase the

number of states, the number of neutrons, and the size of the box to better approximate this

extreme environment. Due to the computational demand of adding additional single-particle

states and number of neutrons, a polynomially scaling many-body method like coupled

cluster (CC) theory is a great choice.

To find the equation of state it is standard to calculate the energy of the system at a

163

range of densities. This energy-density relationship provides a similar insight into nuclear

matter, as the pressure-volume relationship of Boyle’s law provides for an ideal gas. A

detailed calculation would use a state-of-the-art chiral effective field theory (χ-EFT) [6,

7] derived neutron-neutron interaction with three-body forces. As a proof-of-concept, the

results in the chapter will present calculations of neutron matter using the simple nuclear

force model called the Minnesota potential, and show capabilities of the CC code on the

most computationally demanding term of full three-body force calculation.[53] These are the

essential ingredients for a future project which is being planned to leverage the computational

power of a supercomputing center for such a highly accurate ab initio calculation of the

nuclear equation of state.

6.1 Neutron Matter

Fig. 6.1a is a plot of the equation of state of neutron matter with the Minnesota potential.

We see that as the density increases the energy per particle of the system monotonically

increases. To get an idea for how much of this calculation is beyond mean field contributions,

we look at just the CCD correlation energy in Fig. 6.1b. With this system, due to the

short range nature of the nuclear force, the many-body correlations become more and more

important as the density increases.

The next plot of interest increases not only the size of the basis, but also the number of

particles in the box. The limit where the number of nucleons A goes to infinity, as the volume

V of the system goes to infinity and N/V = const is called the thermodynamic limit. Figure

6.2a shows the convergence of the system towards the thermodynamic limit as function of

the number of particles for the CCD approximation with the Minnesota interaction model

164

(a) The equation of state for pure neutron matter
with the Minnesota potential.

(b) Correlation enenrgy per particle as a function
of density.

Figure 6.1: Two different energy per particle plots at low densities of neutron matter with
the Minnesota potential [53] computed in the CCD approximation with 54 neutrons and
an Nmax = 100 truncation (100 plane-wave energy shells), corresponding to 10754 single
particle states.

[53] with Nmax = 20.

Notice that A = 54 is lower than A = 14 and A = 186 is above. This shows that the

convergence towards the thermodynamic limit is not monotonic. Before worrying about this

limit however it is necessary to check, for an individual calculation at fixed particle number,

that the calculation has converged within the basis.

Figure 6.2b shows the convergence in terms of different model space sizes with a fixed

number of neutrons N = 114. The EoS lines appear to get closer together in Figure 6.2b, and

Nmax = 25 seems to be relatively converged. To get a more quantitative look at convergence,

it is better to look at the relative error among model spaces of a particular density on the plot.

These types of calculations are important to get a grasp on basis truncation errors. There

is active work in fitting the energy curves and extrapolating to the infinite basis limit, and

while this is a powerful technique, it is necessary to actually calculate quantities with large

basis sets to validate the extrapolation. It is also important to ask about the universality

165

(a) Equations of state for different numbers of par-
ticles with Nmax = 20 (874 single particle states)

(b) Equations of state for different model space
sizes with A = 114.

Figure 6.2: Energy per particle of pure neutron matter computed in the CCD approximation
with the Minnesota interaction model [53].

of basis convergence.

In the case of neutron matter, Figure 6.3 shows how for different

numbers of particles, the convergence of the calculation can be dramatically different. For

14 neutrons, one-thousand basis states quickly converges. However for larger calculations,

like 114 neutrons, the calculation is only stable to 3-4 digits at 3, 500 basis states.

The final piece to mention on the topic of basis set convergence is about extrapolating

to the thermodynamic limit of infinite matter. Since this matter is meant to simulate an

effectively infinite expanse of neutrons or electrons, it is important to also increase the number

of particles in the system. The thermodynamic limit of bulk matter is when N → ∞, V → ∞
and N/V ∝ const. Here it is helpful to reconsider the “box” the calculation is being done in.
The periodic boundary conditions (PBC) φ(xi) = φ(xi + L) are arbitrarily chosen boundary

conditions that constrain the wavefunctions. Any number of other boundary conditions could

have been chosen, like anti-periodic boundary conditions φ(xi) = − φ(xi + L). Studies have
shown that the difference between these two choices gives an idea of how much the correlation

energy is affected by this basis truncation [71, 72, 23].

166

0.000.050.100.150.200.250.300.35ρ[fm−3]46810121416E/A[MeV]Nmax=20A=14A=54A=186A=3580.000.050.100.150.200.250.300.35ρ[fm−3]46810121416E/A[MeV]A=114Nmax=10Nmax=15Nmax=20Nmax=25Figure 6.3: The relative error shows how much the CCD correlation energy is changing
between subsequent calculations at different model spaces sizes ranging from Nmax = 10 to
70 for neutron matter with the Minnesota potential at density 0.2 fm−3.

One solution to this problem is by integrating over solutions between periodic and anti-

periodic conditions, known as twist-averaging [73]. This is an attempt at allowing more

freedom in the basis functions at the boundary. The single-particle states are multiplied by

a phase for each direction, characterized by a twist-angle, θi,

φ(cid:126)k((cid:126)x + (cid:126)L) → ei(cid:126)θφ(cid:126)k((cid:126)x) .

(6.1)

For periodic boundary conditions (PBC) θi = 0 and θi = π for anti-periodic boundary

conditions (APBC)

(cid:126)θ
(cid:126)k → (cid:126)k +
L
π2
π
(cid:126)k · (cid:126)θ +
L2 .
L


(cid:126)k → 
(cid:126)k +

167

(6.2)

(6.3)

These twist phases effectively change the momentum of the basis states. This yields new

single-particle energies. This correction disappears as L → ∞, which is desired, since all
boundary conditions should become irrelevent in that case. Since the single particle energies

are changing, this changes the shell structure of the basis. Depending on the twist chosen,

certain particle states can jump to holes or holes to particles. It is therefore necessary to fill

hole states separately for each (cid:126)θ since the CC framework developed so far is only effective

for a closed-shell reference state. Integration over a quantitiy is approximated by a weighted

sum, such as Gauss-Legendre quadrature, over the quantity for each set of twist angles.

The algorithm is described in Algorithm 1. By using twist-averaged boundary conditions,

Build mesh points and weights for each direction;
Etwist = 0;
for (θxi, wxi) ∈ {θx, wx} do

for (θyi, wyi) ∈ {θy, wy} do

for (θzi, wzi) ∈ {θz, wz} do

Build Basis States with ki → ki +
Order States by Energy and Fill Holes;
Get Result E (T,HF,CCD);
Etwist = Etwist + 1

π3 wxwywzE;

θi
L ;

end

end

end

Algorithm 5: Twist-Averaged Boundary Condition Algorithm

the extrapolation towards the thermodynamic limit is significantly smoother. However, this

comes at a price, since a full CCD calculation is done at each of these steps. If, for example,

10 twist angles (called TABC10) in each direction are used, this requires 1000 full CCD

calculations. For a computationally cheaper glimpse into the effects of twist-averaging, it is

easy to calculate the kinetic energy per particle and the Hartree-Fock energy per particle,

which avoids the full CCD calculation. It is clear in Figure 6.4a how much more stable the

168

energy calculations are with respect to particle number. These calculations can be compared

to the exact values for infinite matter, which are calculated by integrating the relevant values

up to the Fermi surface. The kinetic energy is given by

Tinf =

32k2
f
10m

,

while the potential energy to first order (corresponding to the Hartree-Fock contribution)

reads

HFinf =

1

(2π)6

L3
2ρ

d(cid:126)k1

d(cid:126)k2 (cid:104)(cid:126)k1(cid:126)k2|ˆv|(cid:126)k1(cid:126)k2(cid:105) .

(cid:90) kf

0

(cid:90) kf

0

(a) Kinetic energy.

(b) Hartree-Fock energy.

Figure 6.4: Finite-size effects in different energies of pure neutron matter computed with the
Minnesota interaction model [53] as a function of the number of particles for both periodic
boundary conditions (PBC) and twist-averaged boundary conditions (TABC5).

Similarly, Fig. 6.4b displays the corresponding Hartree-Fock energy (the reference energy

as defined in Chapter 2) obtained with the Minnesota interaction using both periodic and

twist-averaged boundary conditions. The results show again a weaker dependence on finite

size effects. These are some of the tools needed to push towards the realistic thermodynamic

limit calculations that are necessary. Of course these calculations will require a more sophis-

169

101102103A10-510-410-310-210-1|1−TN/Tinf|ρ=0.16fm−3Tkin(PBC)Tkin(TABC5)101102103A10-510-410-310-210-1|1−TN/Tinf|ρ=0.16fm−3Tkin(PBC)Tkin(TABC5)ticated nuclear potential, so it is instructive to examine another infinite matter system to

check for similarities and discrepancies.

To conclude this section, a comparison of pure neutron matter calculations for several

different many-body methods is presented in Fig. 6.5. Configuration interaction Monte

Carlo (CIMC) and the algebraic diagrammatic construction (ADC) of the self-consistent

Green’s function scheme [74, 75, 76], were not detailed in this text, but are other many-

body methods of interest to many researchers [77]. We see that all of these methods add

configurations that contribute correlations much beyond the reference energy. Additionally,

they all have the same qualitative features. This is as expected, since they are all solving

the same system, any differences between the methods are due to differences in many-body

correlations they add. The collaborative work of [77] provides a detailed comparison of the

many-body methods for the Minnesota potential.

(a) The equation of state for pure neutron matter.

(b) The correlation energy per particle for each
method.

Figure 6.5: Energy per particle for pure neutron matter with the Minnesota potential [53].
Here the calculations have been performed with IM-SRG(2), CCD, CIMC [77], and the
ADC(3) Self-Consistent Green’s Function scheme [77].

170

0.050.10.150.2ρ[fm−3]6.07.08.09.010.011.012.013.014.0E/A[MeV]ReferenceenergyMBPT(2)CIMCADC(3)CCDIMSRG(2)0.050.10.150.2ρ[fm−3]−1.4−1.2−1.0−0.8−0.6Ecorr/A[MeV]MBPT(2)CIMCADC(3)CCDIMSRG(2)6.2 Homogeneous Electron Gas

In Figure 6.6, a similar procedure using CCD to calculate the ground state energy of the

homogeneous electron gas (HEG) at a range of densities has been calculated. While nuclear

matter calculations are computed with respect to the particle density ρ, HEG calculations

are usually phrased in terms of the Wigner-Seitz radius (rs), so the energy per particle vs.

Wigner-Seitz Radius forms an equation of state for the HEG. The Wigner-Seitz radius is

defined as

4
3

πr3

s =

1
ρ

,

rs =

4πρ

(cid:18) 3

(cid:19)1/3

.

(6.4)

The plots in this chapter will give rs in units of the Bohr radius, rb =


mcα , where m is the

electron mass, c is the speed of light, and α is the fine structure constant. Unlike the nuclear

force, the Coulomb force between electrons is well known, and so the electron gas has been

studied much more extensively. [23, 78].

Increasing in the independent variable in Figure 6.6, rs, corresponds to decreasing the

density of the system. The plot shows that as the particles are squeezed tighter together the

repulsive force increases the energy of the system, similar to the case with the Minnesota

potential.

What might be less intuitive is how the many-body contributions look. Subtracting out

the reference energy and plotting just the CCD correlation energy, as seen in Figure 6.7,

shows that as the electrons spread apart (rs increasing), the many-body correlations from

CCD monotonically increase. However, this is a bit misleading, since the absolute magnitude

of the energy is very large as seen for the smallest rs regime of the EoS. Figure 6.8 shows that

at small rs (high densities), the reference energy is very nearly 100% of the total, whereas

at rs = 1.0, many-body correlations make up about 5% of the total energy, which is a very

171

Figure 6.6: The CCD energy per particle for the homogeneous electron gas for a range of
Wigner-Seitz Radii with A = 14 electrons

significant contribution in the high accuracy field of quantum chemistry. This means that

at high densities, the reference (Hartree-Fock) energy contributes the vast majority of the

total energy and the state is well approximated by a single Slater determinant. Conversely,

as the electrons spread out, the many-body correlations become increasingly important.

Again, it is necessary to check the convergence of the calculations with respect to the

single particle basis size. Figure 6.9 shows the relative error of the CCD correlation energy

for the HEG with rs = 1.0 and A = 14 electrons. The results are quite striking when

compared with Figure 6.3, which shows that at 2500 basis states the relative error for A =

14 was down to 10−9, whereas here, the relative error is still at 10−3! This is much closer

to the A = 114 particle case, showing that for the same number of particles, electron gas

calculations need a much larger basis to converge.

This convergence is significantly slower, meaning that the HEG needs much larger basis

172

Figure 6.7: Contributions to the energy from purely CCD many-body correlations.

Figure 6.8: Fractional contribution to the energy from the Hartree-Fock reference state.

173

Figure 6.9: The relative error shows how much the CCD correlation energy is changing
between model spaces sizes ranging from Nmax = 10 to 60 for the electron gas at rs = 0.5.

sets for the same level of precision when compared to neutron matter. This is again due to the

very long range tail of the Coulomb potential coupling electrons across large distances and

thus calculations for the HEG need a larger “box” to perform the calculation in. However, it

is hard to make a direct comparison, when the densities and Wigner-Seitz radii have not been

tuned to be equivalent. To gain some insight to how significant this is, Figure 6.10 shows

the same plot, but at a much higher density of rs = 0.1. This shows that the convergence

trend is not dependent on densities for the electron gas, and that it has more to do with the

nature of the Coulomb force than particle number.

6.3 Computational Results

To get an idea of just how slow the basis convergence is for the HEG or for neutron matter

with a large number of particles, a calculation using an extremely large basis set was ran on

174

Figure 6.10: The relative error shows how much the CCD correlation energy is changing
between model spaces sizes ranging from Nmax = 10 to 60 for the electron gas at rs = 0.1.

2048 of the XE compute nodes on the Blue Water supercomputer. Each of these nodes has

32 cores, totalling 65,536 cores for this calculation. Figure 6.11 shows that it is not until

around 25,000 basis states that the calculation approaches the ∼ 10−9 level of accuracy of
the much smaller basis used for the Minnesota potential calculation. In this calculation, the

tolerance for the CCD iterative solver was set to ∼ 10−9, so the drop at the final data point
is just a random fluctuation as it is beyond the convergence tolerance.

However, with large calculations like this, choosing the proper basis size for the calculation

at hand can be done via an interpolation, which is generally a more accurate method of

prediction than extrapolation.

Even with just a two-body force, this calculation required 54, 000 Gigabytes of mem-

ory to store the interaction tensor. This proved to be an excellent case to validate the

distributed memory implementation described in Chapter 5. Figure 6.12 shows the time

175

Figure 6.11: The relative error of the CCD correlation energy is changing between model
spaces sizes ranging from Nmax = 100 to 200 for the neutron matter with the Minnesota
potential at A = 54 and ρ = 0.08.

required to compute each data point from the above plot, as well as the breakdown of the

major computational kernels. First, it is worth noting that despite the extreme memory

requirements even the largest calculation here took less than 2 hours due to the high level

of parallelism that can be exploited. Understandably, the tensor contractions are the most

expensive component of these calculations, since even though the computations are dense in

floating point operations, it is also this stage of the calculations that has the most commu-

nication overhead across the network. The line labeled setup is the nearly serial bottleneck

at the beginning of the code, and this part of the code has since been parallelized, but it is

difficult to run another timing test of this scale. Lastly, the load step is calculating all of the

Minnesota potential matrix elements required for the interaction tensors. In a more realistic

calculations, this would be the file I/O step, which could hopefully employ a similar level of

parallelism.

176

Figure 6.12: The time required for the large basis set Minnesota potential calculations.

To get an idea of how well the code is parallelized, it is common to look at strong and weak

scaling. Ideal strong scaling is when doubling the amount of processors doubles how fast the

code runs. This can be seen in many cases that are called “embarassingly parallel”, where

the calculation can be perfectly divided amoung compute cores, without any communication

between cores needed. However, many calculations have parts of the code that run in serial,

or communication overhead which causes the speedup to not follow the ideal case. Figure

6.13 is a strong scaling plot for the distributed memory implementation described in the

computational methods chapter, with a line plotted to show what ideal strong scaling would

look like. The code scales quite well up to about 100 cores for this calculation, but ceases to

gain much speedup from increasing the cores beyond that, diverging more and more rapidly

from the strong scaling line. While ideal strong scaling would be nice, it is not much of a

surprise that this is not the case here. Any calculation which is even possible on a single core

is a case where 1, 000 cores is entirely unnecessary. The parts of the code that are highly

177

Figure 6.13: Strong scaling of distributed memory code, dark green line shows ideal case.

parallel are being computed almost instantly, leaving just the serial parts of the code which

now take up 99% of the compute time.

However, in many-body physics, it is not often the case that parallelism is used to solve

the same problem faster, but to solve larger and larger problems to increase the accuracy of

calculations. This is where weak scaling is a more useful metric. Weak scaling is the idea

that, given 1000 times more cores, can a problem 1000 times larger be tackled? If so, the

ideal case would be that a problem size 1000 times larger would take the same amount of

time if given 1000 times more cores. However, it is sometimes hard to easily quantify the

problem size, since to compare apples to apples it would need to be measured in total floating

point operations (FLOPs). In the case of CC calculations however, the limiting factor is the

memory for the interaction tensors. How many gigabytes of memory are needed to store the

matrix is typically what dictates how many nodes are allocated, and thus how many cores

are used. To plot the weak scaling of the code, Figure 6.14 shows “problem size” vs. time to

178

100101102103Number of Processors100101102103Speedup (Relative to Single Processor)Strong Scaling (54 Particles, 44 Energy Shells)complete calculation. A calculation is chosen to run on one core, which requires some amount

of gigabytes to store the interaction matrix. This matrix size is now the unit which the other

quantites are measured against. A problem which has an interaction matrix roughly 1, 000

times larger and is then run on 1, 000 cores. This procedure is done for many points, trying

to increase the matrix size and number of processors proportionally. The green line shows

the ideal weak scaling case, which the code is again diverging from. However, by chosing

problem size equal to the matrix size we should not expect ideal weak scaling since for a

square matrix of row size N , the matrix size scales as N 2 while matrix-matrix multiplication

scales as N 3. From the plot we can see that solving a system which needs 1000 times more

memory from some base case only takes 2.5 times as long as this case is given 1000 times

more processors. This is encouraging, as it shows that given the computational resources, the

distributed memory algorithm is capable of solving proportionally larger and larger problems

without too much additional overhead.

While extreme basis sets for CCD with two-body forces is nice, the real motivation here is

to handle even modest basis sets for CCDT or three-body force calculations. To get an idea

of how large the (cid:104)abc|w|def(cid:105) tensor is, Figure 6.15 shows the naive N 6 amount of matrix
elements as well as the number of non-zero elements that the compressed block-diagonal

stucture has.

While these are large numbers, a more pragmatic calculation of the size of the three-body

tensor in gigabytes is in Figure 6.16.

From this plot, we can see that even “small” basis sets over 700 single-particle states

would require a billion gigabytes of memory naively. Even with the 104 compression factor

of the block-diagonal tensor, this calculation will require 10 to 100 terabytes of memory.

While this is a staggering number, it is not out of reach of modern supercomputers. Since

179

Figure 6.14: Weak scaling of distributed memory code, the dark green line shows the ideal
case.

the basis for the electron gas and nuclear matter is so similar, this feature is universal across

these calculations. Coupled cluster theory frequently runs into these memory issues, which

could be circumvented by not storing the interactions, but by computing them on-the-fly

[79]. This method has had success in quantum chemistry, but the nuclear potential has

proven to be too costly to employ this method. However, as the computational power of

these machines grows, it is not unthinkable that this could be done in the future.

These calculations would need to be highly optimized at the node level, exploiting as much

parallelism as possible for many-core and GPU architectures. Figure 6.17 shows the on-node

timing tests for computing the (cid:104)abc|w|def(cid:105)(cid:104)def|t|ijk(cid:105) tensor contraction which, scaling at
N 9, is the most expensive component of a full CCDT calculation with three-body forces.

Understanding this term and developing and optimizing it will be extremely important for

future calculations. The primary challenge is performing hundreds of thousands of matrix-

180

020040060080010001200Problem Size (Keeping Number of Processors Proportionally Constant)0.00.51.01.52.02.53.03.5Time (Relative to Single Processor Case)Weak Scaling (54 Particles, Up to 102 Energy Shells)Figure 6.15: Number of tensor elements required for the 3-body force in the infinite matter
basis with and without block-diagonal compression.

matrix multiplications across all of the symmetry blocks. In the plot, three methods were

tested: OpenMP [69] parallelization over the blocks, serial batching of cuBLAS on the

blocks, and serial batching of multithreaded OpenBLAS zgemm (complex double matrix-

matrix multiplication) calls. The differen BLAS operation calls typically have a leading

character that determines the data type, d for double, z for complex double. OpenBLAS

[68] was used for the CPU implementation of BLAS, and cuBLAS was used for the GPU

implementation, where the “cu” in cuBLAS is a reference to CUDA, a programming model for

writing software for Nvidia GPUs [80]. From this plot, we can see that the multi-threaded

zgemm calls are the fastest, while the OMP parallelized loops are the slowest, not much

faster than cuBLAS. The many-core calculations were ran on a node which has two 2.4Gz

14-core Intel Xeon processors and the GPU calculations where ran on an NVIDIA Tesla K80.

By raw performance, the GPU calculations should run faster, but there are difficulties in

181

Figure 6.16: Size of tensor in gigabytes required for the 3-body force in the infinite matter
basis with and without block-diagonal compression.

getting the calculations to run efficiently. The large disparity between the thousands of tiny

matrix-matrix multiplies versus the relatively few large matrix-matrix multiplies means that

a hybrid scheme will likely be necessary for doing batched calls for the small matrices and

regular gemm calls for the large matrices. However, since this calculation is just doing tens

of thousands of cuBLAS calls, the benefit of the GPU speed is overcome by the enormous

amount of call overhead.

182

Figure 6.17: On node timing tests for the tensor contraction of three-body force diagrams.

183

Chapter 7

Conclusions and Perspectives

The future of ab initio many-body physics is bright, especially in nuclear theory where

methods like coupled cluster (CC) theory and in-medium similarity renormalization (IM-

SRG) are still relatively young to the field. Accurate calculations with predictive power

are necessary for answering questions where experimental data are lacking. In systems like

neutron stars, theory and computation could be our only tools for answering questions about

their internal structure. In this thesis, we reviewed and implemented the formalism for several

many-body methods, with a focus on coupled cluster theory which is capable of computing

properties of very large systems while maintaining a link to all of the fundamental degrees

of freedom.

The computer program that implements these many-body methods is designed with

accuracy as the first goal. Analytical techniques were used to validate the accuracy of the

program before any other considerations were made. The code is designed in a modular

way which allows any physical system to be included without modifying any of the existing

infrastructure. This allows properties of the pairing model, the homogeneous electron gas

and infinite nuclear matter to be computed with minimal additional effort, and allows the

addition of other systems in the future. Importantly, the program implements distributed

memory algorithms and data structures which allow the code to run at high-performance

computing centers. Investigating the strong and weak scaling showed that the program can

perform increasingly large calculations as long as a proportional increase in computational

184

resources is provided. This enables extremely large basis sets to be used for physical systems

to reach levels of precision that would otherwise be impossible.

Looking towards the future, with these data structures and algorithms implemented, the

program can be extended to tackle many interesting topics in many-body theory and com-

putational physics. Coupled cluster theory with doubles and triples excitations (CCDT)

calculations of the homogeneous electron gas with twist-averaged boundary conditions is a

likely first target. The addition of triples correlations will likely provide a significant correc-

tion to the correlation energy of the system. While approximate triples contributions have

been included, it is currently unknown how important the role of the full triples correlations

are in this system. Next, the inclusion of full three-body forces in calculations of nuclear

matter with chiral effective field theory Hamiltonians has up until now been avoided due to

the extreme memory requirements of the full three-body forces and the computational effort

required for the tensor contractions. Tackling this calculation will be a serious undertaking

in high performance computing, but with the next generation of high-performance comput-

ing facilities, we may be able to handle the increased dimensionality. The data structures

and algorithms to handle the large number of matrix elements and computationally expen-

sive tensor contractions are largely in place. Further work on the GPU implementation is

on-going, which would allow the program to deploy on some of the modern supercomputers.

The next step is to include all of the less computationally heavy, but necessary three-body

diagrams. With these in place, the program will be in position to examine several important

many-body questions:

1. What is the role of the full three-body interaction in infinite matter using chiral effective

field theory? How much the normal ordered 0-,1-, and 2-body terms miss? Can the

inclusion of the full three-body interaction lead to the accurate prediction of the nuclear

185

saturation density?

2. Can we quantify the errors of approximated triples by implementing full triples? Are

there patterns in these errors to predict the errors of approximated triples in regions

where full triples are too expensive?

3. What is the role of full triples in the homogeneous electron gas?

4. How does the nuclear equation of state differ at various neutron to proton fractions?

What about at β-stable equilibrium?

Along with CC calculations, the distributed memory data structures and algorithms

implemented can be quickly ported into a new many-body method, namely IM-SRG with the

Magnus expansion. A distributed memory implementation of IM-SRG does not currently

exist, but due to the modular nature of the program and the generic nature of the data

structures, adding IM-SRG functionality is a logical next step forward. This would alleviate

many of the memory constraints of the method, allowing the calculation of many new physical

systems under a new theoretical perspective.

186

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