TRANSPORT PROPERTIES OF RANDOM SCHR ¨ODINGER OPERATORS ON

CORRELATED ENVIRONMENTS.

By

Rodrigo Bezerra de Matos

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics – Doctor of Philosophy

2020

ABSTRACT

TRANSPORT PROPERTIES OF RANDOM SCHR ¨ODINGER OPERATORS ON

CORRELATED ENVIRONMENTS.

By

Rodrigo Bezerra de Matos

This Ph.D. thesis presents recent developments in the theory of random Schr¨odinger op-

erators. Differently from what is often studied in the subject, our main results consider

potentials which are not independent at distinct sites but, rather, display some form of long

range correlation. These are natural objects to investigate if one wishes to understand the

long term behavior of a single particle which evolves in a disordered environment but also

interacts with different members of this environment (other particles, spins, etc).

In chapter 2 it is shown that, within the Hartree-Fock approximation for the disordered

Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit localiza-

tion, suitably defined as exponential decay of eigenfunction correlators. Our result holds in

any dimension in the regime of large disorder and at any disorder in the one dimensional case.

As a consequence of our methods, we are able to show H¨older continuity of the integrated

density of states with respect to energy, disorder and interaction using known techniques.

This is based on joint work with Jeffrey Schenker [46]

Chapter 3 is based on joint work with Jeffrey Schenker and Rajinder Mavi. There we

present simple, physically motivated, examples where small geometric changes on a two-
dimensional graph G, combined with high disorder, have a significant impact on the spectral
and dynamical properties of the random Schr¨odinger operator −AG +Vω obtained by adding
a random potential to the graph’s adjacency operator. Differently from the standard An-

derson model, the random potential will be constant along vertical line, hence the models

exhibit long range correlations. Moreover, one of the models presented here is a natural

example where the transient and recurrent components of the absolutely continuous spec-

trum, introduced by Avron and Simon in [9], coexist and allow us to capture a sharp phase

transition present in the system.

Copyright by
RODRIGO BEZERRA DE MATOS
2020

To Marsya, Sandra and Cory.

v

ACKNOWLEDGEMENTS

I would like to thank Jeff Schenker for many things, in particular for supervising the projects

which compose this thesis. The importance of his guidance during my Ph.D. goes well

beyond research, travel support and recommendation letters. Jeff is an extremely generous

and open-minded individual from whom I learned a lot. I am thankful to Russell Schwab,

firstly for being my initial contact at Michigan State University and, secondly, for supervising

my readings and research projects in partial differential equations. Russell has been very

patient with me and encouraged me during the last five years.

I am also thankful to Ilya Kachkovskiy for the reading groups and seminars organized

by him and for the interesting discussions we have had in the last three years, from which

I have certainly benefited a lot. I thank Jun Kitagawa and Willie Wong for the excellent

graduate courses they have taught and for always being available and helpful. I would like

to thank Tsveta Sendova, Rachael Lund and Andy Krause for supporting my teaching and

making my experience as an instructor much easier.

I have been lucky to make a number of friends at MSU, specially Abhishek Mallick,

Leonardo Abbrescia, Christos Grego, Ioannis Zachos, Reshma Menon, Wenchuan Tian, Zak

Tilocco and Andr´es Galindo. I have also met amazing individuals while traveling for confer-

ences. By naming Wencai Liu I thank them all.

I would like to express my gratitude to Fabio Montenegro, for his guidance during my

undergraduate education and encouragement to study abroad. I have no words to describe

how much I learned from him and his attitude towards mathematics. I thank Rafael Alves

for ten years of friendship, starting from the first semester of college. Being able to often

exchange ideas with him during this period was very important to me.

This journey would not have happened without love and support from my parents, Sandra

and Cory, and brothers Bruno and Pedro. By naming them I thank my whole family for

everything they did to me.

vi

I am specially grateful to Marsya, who made my life brighter from the first day we met.

Her love, patience and support during the final years of this program were crucial to me.

Her intelligence, dedication and grace are a constant source of inspiration.

vii

TABLE OF CONTENTS

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

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 A short introduction to Anderson Localization . . . . . . . . . . . . . . . . .
1.2 Dynamical localization at large disorder
. . . . . . . . . . . . . . . . . . . .
1.3 Green’s function decay: A step towards dynamical

localization.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 First proof theorem 1.2: the self-avoiding walk expansion . . . . . . . . . . .
1.5 Second proof of theorem 1.2: the ’depleted’ resolvent identity . . . . . . . . .
1.5.1 Further Aspects of A-priori bounds . . . . . . . . . . . . . . . . . . .
1.6 From Green’s function decay to dynamical localization . . . . . . . . . . . .
1.6.1
Independence as a key tool . . . . . . . . . . . . . . . . . . . . . . . .
Instructive consequences of dynamical localization . . . . . . . . . . . . . . .
1.7.1 From dynamical localization to finite averaged moments
. . . . . . .
1.7.2 From dynamical localization to pure point spectrum . . . . . . . . . .
Localization centers and Eigenfunction decay . . . . . . . . . . . . .
1.7.3
1.8 Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

CHAPTER 2

2.2 Definitions and Statement of the Main Result

LOCALIZATION AND IDS REGULARITY IN THE HUBBARD
MODEL WITH HARTREE-FOCK THEORY . . . . . . . . . . . . .
2.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Discussion of the results and main obstacles . . . . . . . . . . . . . .
2.1.2 Hartree-Fock theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Background on Localization for interacting systems . . . . . . . . . .
. . . . . . . . . . . . . . . . .
2.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Definition of the operators . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Outline of the Proof of theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . .
2.5 Existence of the effective potential . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Regularity of the effective potential
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
2.7 Decay estimates for the effective potential
Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Proof of lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 The Hartree approximation for the Hubbard model
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
2.9.1 Existence of the Effective potential
2.9.2 Regularity of the effective potential
. . . . . . . . . . . . . . . . . . .
2.9.3 Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 One dimensional Aspects:proof of theorem 2.1 . . . . . . . . . . . . . . . . .
2.11 One dimensional aspects: strategy of the proof of theorem 2.1 . . . . . . . .

2.7.1

x

1
1
5

6
6
8
9
10
11
11
11
12
13
14

16
16
17
18
18
19
19
20
22
24
26
28
32
33
40
44
48
49
49
50
51
51

viii

2.11.1 Main ideas in the i.i.d case . . . . . . . . . . . . . . . . . . . . . . . .
2.11.2 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.3 Proof of Lemma 2.21 . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.4 Proof of lemma 2.22 . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.5 Proof of (2.11.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 H¨older Continuity for the integrated density of states at weak interaction . .

CHAPTER 3 SPECTRAL AND DYNAMICAL CONTRAST ON HIGHLY COR-

3.1

RELATED ANDERSON-TYPE MODELS.

. . . . . . . . . . . . . .
Introduction and Main results . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Background on Correlated Models . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Main Results
3.1.3 Dynamical Contrast between HSym and HDiag
. . . . . . . . . . . .
. . . . . . . . . . . . . .
3.1.4

Spectral contrast between HSym and HDiag

3.1.5 Phase transition within σ(cid:0)HSym
(cid:0)HSym

(cid:1) . . . . . . . . . . . . . . . . . . . .
(cid:1)

3.1.6 Definition of the Models . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Lower Bound on the Averaged Moments for HSym: Proof of Theorem 3.1 . .
3.3 The absolute continuity of µδ(0,0)
. . . . . . . . . . . . . . . . . . .
3.4 Floquet Theory for HSym . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Localization in the horizontal direction . . . . . . . . . . . . . . . . .
3.4.2 Transient and recurrent components: proof of theorem 3.7 . . . . . .
3.5 Absence of Diffusion for GDiag,γ: Proof of theorem 3.2 . . . . . . . . . . . .
3.6 Upper bounds for quantum dynamics revisited . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .

3.6.1 Paking Dimension: Proof of Corollary 3.3

3.7 A Version of Boole’s Equality for level sets of Heglotz functions: proof of

proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 A priori bounds on the Green’s function . . . . . . . . . . . . . . . . . . . .

52
53
57
59
61
62

68
68
68
69
70
71
72
72
73
76

77
79
81
84
85
87

88
90

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

ix

LIST OF FIGURES

Figure 2.1: The Hubbard Model

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

22

Figure 3.1: 1/2 spin and single-particle model . . . . . . . . . . . . . . . . . . . . . .

68

Figure 3.2: Highly correlated Anderson-type models

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

69

x

CHAPTER 1

INTRODUCTION

1.1 A short introduction to Anderson Localization

The effects of disorder on transport properties of quantum systems have drawn a signif-

icant amount of attention in the mathematics community since their introduction in 1958

in the celebrated paper [8] by the Physicist Philip Anderson, who later in 1977 became a

Nobel prize winner. The efforts to encode Anderson’s claim that randomness may prevent

diffusion into a rigorous mathematical statement and to obtain a proof for it gave rise to a

beautiful theory, incomplete up to these days. Absence of diffusion in the disordered context

is now generically referred to as Anderson Localization which we shall briefly review in this

chapter, which aims at introducing the main concepts and results relevant for the remainder

of the thesis. For a more complete historical picture we refer to the survey [58] and the book

[6]. The proofs for well-known results given in this introduction are deeply influenced by the

presentation in these two sources.

The Anderson model is the family of operators
Hω = −∆ + λVω

acting on (cid:96)2(cid:16)Zd(cid:17)
• ∆ is the discrete Laplacian acting on ϕ ∈ (cid:96)2(cid:16)Zd(cid:17)

through

, the space of functions f : Zd → C which are square summable, where

(cid:88)

|m−n|1=1

(∆ϕ) (n) =

(ϕ(m) − ϕ(n)) , n ∈ Zd,

|n|1 = |n1| + ...|nd|.

.

• The on site potential Vω acts as a multiplication operator on (cid:96)2(cid:16)Zd(cid:17)

via

(Vωϕ) (n) = ω(n)ϕ(n) , n ∈ Zd.

1

• ω = {ω(n)}

n∈Zd is a list of independent, identically distributed random variables.

• λ > 0 denotes the disorder strength.

For technical reasons one often includes the following regularity condition on the random

variables

P (ω(0) ∈ I) =

(cid:90)

I

ρ(v) dv.

(1.1.1)

A typical assumption on the density ρ is requiring it to be bounded and of compact sup-
port, which makes the random variable ω(n) bounded for every n ∈ Zd. Sometimes it is

more convenient to work with distributions with unbounded support such as the Gaussian
distribution given by ρ(v) = 1√
π

e−v2

1

(cid:96)2(cid:16)Zd(cid:17)

and the Cauchy distribution given by ρ(v) =
The lattice structure of Zd allow us to make use of the canonical basis {δn}

π(1+v2)
ninZd of
defined by δn(m) = δmn where δmn is the Kronecker delta. The probability of

.

finding a particle at position n and time t, given that it started at position 0 is given by

|(cid:104)δn, e−itH δ0(cid:105)|2.

Dynamical Localization is naturally defined as averaged decay of the matrix elements

|(cid:104)δn, e−itH δ0(cid:105)|.

which can be explicit in a bound as

E(cid:16)|(cid:104)δn, e−itH δ0(cid:105)|(cid:17) ≤ r(n).

(1.1.2)

where

(cid:88)

n∈Zd

r(n) < ∞.

In the best scenario, the function r(n) was shown to have exponential decay. In dimension
d ≥ 2, dynamical localization has been achieved at large disorder, meaning that λ is taken

sufficiently large or at weak disorder and extreme energies, see [6, Theorem 10.4] for a precise

2

statement. We shall outline a proof of localization at large disorder in the following sections

of this introduction.

If one is only concerned whether diffusion takes place or not in a system, a useful tool is

given by the averaged q-moments of the position operator defined as

M q

T (H) =

2
T

−2t
T E(cid:104)0|eitHω|X|qe−itHω|0(cid:105) dt.

e

(cid:90) ∞

0

(cid:88)

Where the position operator X is formally defined as (Xϕ)(n) = |n|ϕ(n). For instance,

whenever

|n|qr2(n) < C < ∞

dynamical localization in the above sense implies the bound

n∈Zd

T (H) ≤ C < ∞
M q

The above inequality is a convenient signature of Anderson localization, whereas its coun-

terpart

T (H) ≥ CT α
M q

for α > 0 indicates diffusion.

Finally, recall that associated to the action of H in the Hilbert space (cid:72) = (cid:96)2(cid:16)Zd(cid:17)

is
the decomposition (cid:72) = (cid:72)pp ⊕ (cid:72)sc ⊕ (cid:72)ac of (cid:72) into the pure point, singular continuous

and absolutely continuous subspaces and the corresponding splitting of its spectrum into
components σ(H) = σac(H)∪ σsc(H)∪ σpp(H). The so called RAGE theorem (after Ruelle,
Amrein, Georgescu and Enss.[6, Theorem 2.6]) provides dynamical characterizations for

the spectral components. In particular, one of its consequences is the fact that dynamical

localization in the form of (1.1.2) implies that H has only pure point spectrum. In other

words, dynamical localization implies that the spectral projection of H into the continuous
component (cid:72)c = (cid:72)sc ⊕ (cid:72)ac vanishes. This means that the spectrum σ(H) is the closure of

the set of eigenvalues for H. If the associated eigenfunctions decay exponentially, the operator

3

H is said to exhibit Anderson Localization. A sufficient condition is the exponential decay

of the eigenfunction correlators

E (|(cid:104)n|g(Hω)|0(cid:105)|) ≤ Ce−µ|n|

sup
|g|≤1

(1.1.3)

where the above supremum is taken over all Borel measurable functions g : R → C and
(cid:104)n|g(Hω)|0(cid:105) = (cid:104)δn, g(Hω)δ0(cid:105) (in this thesis we may alternate between these two notations
for the inner product). For proofs for some of the above facts and more precise statements

In the series of works by Jakˇsi´c and Molchanov [38], [37] and [39] the authors consider a

with randomness being introduced only
= Zd through independent, identically distributed random

we refer to the next few sections of this introduction.

(cid:16)Zd × Z+(cid:17)

Schr¨odinger operator H = −∆+Vω on (cid:96)2(cid:16)Zd × Z+(cid:17)
 0 if n2 (cid:54)= 0.

along the surface ∂
variables {ω(m)}

m∈Zd. More precisely, let

Vω(n1, n2) =

ω(n1) if n2 = 0.

In [37] (see also remark (3) in [39]), for d = 1 a sharp dichotomy is shown as

σpp(H) = σ(H) \ σ(−∆) , σac(H) = σ(−∆)

(1.1.4)

holds almost surely.

An important fact, which will allow us to obtain an analogue of (1.1.4) in the correlated

context later in this thesis, is that the absolutely continuous subspace can be further de-
composed into its transient and recurrent subspaces (cid:72)ac = (cid:72)tac ⊕ (cid:72)rac, an idea which was

introduced by Avron and Simon in [9]. As explained there, one of the motivations is that in

case µψ = χC dx, where C is a Cantor-like set of positive Lebesgue measure, the measure µψ

resembles a singular measure, despite its absolute continuity. This will be a typical situation

in which ψ belongs to the recurrent subspace (cid:72)rac. The subspace (cid:72)tac is the closure of the
set of all ψ ∈ (cid:72)ac such that, for all N ∈ N

.

(1.1.5)

(cid:12)(cid:12)(cid:12)(cid:104)ψ|e−itH|ψ(cid:105)(cid:12)(cid:12)(cid:12) = O

(cid:16)

t−N(cid:17)

4

Recall that, by the Riemann-Lebesgue lemma, if ψ ∈ (cid:72)ac,

(cid:12)(cid:12)(cid:12)(cid:104)ψ|e−itH|ψ(cid:105)(cid:12)(cid:12)(cid:12) = 0

lim
t→∞

hence, considering the spectral measure µψ associated to ψ in (cid:72)tac, we see that its density

with respect to Lebesgue measure

dµψ
dE must have additional regularity properties. In fact,
according to [9, Proposition 3.1] if ψ is a transient vector then µψ = f (x) dx where f is a

smooth function. The converse statement is also true under the additional assumption of

f being compactly supported. We will show in theorem 3.7 below that the transient and

recurrent spectrum can naturally arise and coexist in a situation of physical relevance.

1.2 Dynamical localization at large disorder

We let H = −∆ + λVω for the remainder of this chapter, unless otherwise specified. In
terms of background for this thesis, the following result is of great relevance. Strictly speak-

ing, the version presented here was proven first by Aizenman in [1] although localization at

large disorder was shown earlier by Aizenman and Molchanov [5]. The technical assumptions

on the random potential Vω will be clear when we outline a proof of this result within a few

paragraphs.

Theorem 1.1. There is a λ > 0 such that for all λ > λ0

(cid:18)

E

sup

t

(cid:19)
|(cid:104)δn, e−itH δ0(cid:105)|

≤ Ce−ν|n|.

for all n ∈ Zd and some C < ∞ and ν > 0.

As mentioned in the above section, theorem 1.1 means that, on average, the probability

of finding the wavepacket at a site n at any time is exponentially small. This is the strongest

form of single particle localization, known as exponential dynamical localization. We shall

present a version of this theorem for the interacting case in chapter 2. A related statement

can be achieved via multiscale analysis although with a subexponential rate, see [28]. Some

general consequences of dynamical localization will be detailed in section 1.7.

5

1.3 Green’s function decay: A step towards dynamical

localization.

An object which is often helpful to the analysis of random operators is the Green’s
function. It is defined at z ∈ C by G(m, n; z) = (cid:104)δm, (H − z)−1δn(cid:105) when this expression
makes sense. In particular, since the operators object of our study are self-adjoint, this will
be the case for all z ∈ C \ R. The first step to show dynamical localization as in theorem

1.1 is obtaining exponential decay for the above quantity as follows.
Theorem 1.2. Given s ∈ (0, 1) and λ sufficiently large there exist positive constants CAnd
and µAnd such that for all z ∈ C+

E (|G(m, n; z)|s) ≤ CAnde−µAnd|m−n|

(1.3.1)

We shall outline two different proofs for the above fact, from which the relationship

between the aforementioned constants and the parameters λ and s will be more clear.

1.4 First proof theorem 1.2: the self-avoiding walk expansion

The proposition below is the self-avoiding walk expansion of the Green’s function, also

called Feenberg’s loop-erased expansion (since it seems to date back to Eugene Feenberg in

the 1940s) or the locator expansion (as in Anderson’s seminal work [8]). It will be useful

not only for this chapter but also on chapter 3, when applied to a graph which is different
from Zd. For this reason we keep the general formulation as in [6], i.e, replacing Zd by an
arbitrary vertex set G of a graph with finite degree, denoted henceforth by degG.

Proposition 1.3. The Green’s function of a self-adjoint operator H = H0 + V on (cid:96)2 (G)
admits the expansion

(−1)|τ|H0(τ (0), τ (1))...H0(τ (|τ| − 1), τ (|τ|)) ×

Gk(z).

On any graph, the above expansion converges whenever Imz > (cid:107)T(cid:107). On finite graphs it is
convergent for all z ∈ C+.

SAW(cid:88)

τ :m(cid:55)→n

G(m, n; z) =

|τ|(cid:89)

k=0

6

Here Gk(z) = (cid:104)δτ (k), (Hτ,k − z)−1δτ (k)(cid:105) where Hτ,k denotes the restriction of H to the

set (cid:96)2(cid:0)G \ ∪j<kτ (j)(cid:1) (in case k = 0 we set Hτ,k = H). The summation ranges over all self-

avoiding walks τ from m to n with a finite number of steps. It is then clear from Feenberg’s
expansion that in order to control arbitrary matrix elements of (H − z)−1 can firstly focus
on analysing (meaning obtaining bounds on) the diagonal ones. For that end, fix n ∈ Zd
and let Pn be the projection onto the subspace spanned by δn. Denote by ˆH the operator

obtained from H by setting the potential at site n equal to zero, in other words

ˆH = H − λV (n)Pn.

(1.4.1)

The first resolvent identity gives

(H − z)−1 − ( ˆH − z)−1 = −λV (n)(H − z)−1Pn( ˆH − z)−1.

(1.4.2)

In particular,

G(m, n; z) − ˆG(m, n; z) = (cid:104)δm, ((H − z)−1 − ( ˆH − z)−1)δn(cid:105)

= −λV (n)(cid:104)δm, (H − z)−1Pn( ˆH − z)−1δn(cid:105)
= −λV (n)G(m, n; z) ˆG(n, n; z).

Solving for G(m, n; z) we reach

G(m, n; z) =

ˆG(m, n; z)

1 + λV (n) ˆG(n, n; z)

.

The above formula is the basic ingredient for the following a-priori bound.

Proposition 1.4. For all λ > 0, s ∈ (0, 1) and z ∈ C+ we have

E (|G(n, n; z)|s) ≤ CAP(s)

λs

.

where CAP(s) = 2s(cid:107)ρ(cid:107)s∞
1−s

.

Proof. Letting m = n in (1.4.3) we find that

G(n, n; z) =

1

λω(n) − Σ(z)

7

(1.4.3)

(1.4.4)

(1.4.5)

where Σ(z) = −(cid:16) ˆG(n, n; z)

(cid:17)−1

is a complex number independent of ω(n). Let v = ω(n).

Under the assumption that ρ ∈ L∞ (R) we have that for any interval I ⊂ R

P (v ∈ I) ≤ (cid:107)ρ(cid:107)∞|I|.

(1.4.6)

Combining the above observations with the layer cake representation we discover that

(cid:90) ∞

−∞

1

0

0

P({|v − ξ| ≤ t−1/s})dt
min{1, 2(cid:107)ρ(cid:107)∞t−1/s}dt

(cid:90) ∞
(cid:90) ∞
|v − ξ|s ρ(v)dv =
≤
= 2s(cid:107)ρ(cid:107)s∞ + 2(cid:107)ρ(cid:107)∞
2s(cid:107)ρ(cid:107)s∞
= 2s(cid:107)ρ(cid:107)s∞ + 2(cid:107)ρ(cid:107)∞ s
1 − s
= 2s(cid:107)ρ(cid:107)s∞ + 2(cid:107)ρ(cid:107)∞ s
1 − s

(cid:90) ∞

t−1/sdt

s−1
s

((2(cid:107)ρ(cid:107)∞)s)
(2(cid:107)ρ(cid:107)∞)s−1

2s(cid:107)ρ(cid:107)s∞
1 − s

.

=

finishing the proof of the a-priori bound.

It readily follows from Feenberg’s expansion that whenever λ > CAP degG

E (|G(m, n; z)|s) ≤

∞(cid:88)

degkG

k=|m−n|

(cid:18) CAP

λs

(cid:19)k+1
(cid:18)degG CAP

λs

(cid:19)|m−n|

finishing the proof of theorem 1.2.

=

CAP
λs

1

1 − degG CAP

λs

1.5 Second proof of theorem 1.2: the ’depleted’ resolvent identity

Let us now provide another proof of theorem 1.2, based on the so called ’depleted’ resol-

vent identity which, for m (cid:54)= n, reads

GΛ(m, n; z) = −GΛ(m, m; z)

H0(m, n(cid:48))GΛ\{m}(n(cid:48), n; z).

(cid:88)

n(cid:48)(cid:54)=m

8

where (cid:104)m|(HΛ − z)−1|n(cid:105) and HΛ denotes the restriction of H to a box of size |Λ| centered
at the origin. Using the a-priori bound and that GΛ\{m}(n(cid:48), n; z) is independent of m we

reach

E (|GΛ(m, n; z)|s) ≤ CAP
λs

(cid:88)

n(cid:48)

|H0(m, n(cid:48))|sE(cid:16)|GΛ\{m}(n(cid:48), n; z)|s(cid:17)

Exponential decay is now achieved whenever

with

λs > CAP(cid:107)H0(cid:107)∞,∞

(cid:107)H0(cid:107)∞,∞ := sup

m

(cid:88)

n

|H0(m, n)|.

.

(1.5.1)

(1.5.2)

(1.5.3)

through a suitable iteration of (1.5.1). A Couple of Remarks are in order:

• Note that both proofs make use of independence but that the existence of a conditional

density is enough.

• The second of the above proofs also allows H0 to be of the general form

|H0(m, n)|(cid:16)

(cid:17)
eν|m−n| − 1

sup
m

(cid:88)

n

< η < ∞.

(1.5.4)

In case the matrix elements of H0 only exhibit polynomial decay, the argument given

above yields polynomial decay of the Green’s function.

• In [47], a refined estimate for the large disorder threshold is presented. Remarkably, it

coincides with Anderson’s original prediction of the large disorder regime.

Further comments and details may be found on [6, Section 10.1].

1.5.1 Further Aspects of A-priori bounds

The estimates for the Green’s function given in this section heavily rely on the rank-one

formula (1.4.3). An alternative, and more general, approach, originally due to [2], which

is useful in the continuum and also for the operators studied on chapter 3 will be given in

section 3.8 of that chapter.

9

1.6 From Green’s function decay to dynamical localization

The next result we shall review explains why, in the context of the Anderson model, it

is enough to shown Green’s function decay in order to obtain dynamical localization. A

proof will not be given here, but related ideas are presented in chapter 3, section 3.4.1. The

original argument is due to Aizenman in [1] with a streamlined version being presented in [3,

Appendix A]. We will say that a random operator Hω has uniform Green’s function decay
on an energy domain I ⊂ R when given some s ∈ (0, 1) there exist C > 0 and µ > 0 such

that

E(cid:16)|GΛ(m, n, E)|s(cid:17) ≤ Ce−µ|m−n|

(1.6.1)

holds for all Λ ⊂ Zd and every E ∈ I.

Theorem 1.5. Let Hω = H0 +Vω be a random operator with the hopping term H0 satisfying

(1.5.4). Assume that

• E(|Vω(n)|δ) < ∞ holds for some δ > 0.

• For every n0 ∈ Zd the conditional distribution of Vω(n0) = v at specified values of
{Vω(n)}n(cid:54)=n0

has a density ρn0(v) and

sup
ω∈Ω

sup
n0∈Zd

sup
v∈R

ρn0(v) < ∞.

Suppose further that for some s ∈ (0, 1) the uniform Green’s function decay (1.6.1) is satisfied
in some interval I ⊂ R. Then, there exist C(cid:48) > 0 and µ(cid:48) > 0 such that
≤ C(cid:48)e−µ(cid:48)|m−n|

(cid:33)
(cid:104)m|f (Hω)PI (Hω)|n(cid:105)

(1.6.2)

(cid:32)

E

sup
|f|≤1

where the supremum is taken over all Borel measurable functions f : R → C bounded by one.

10

1.6.1

Independence as a key tool

When establishing localization by the above arguments, independence of the random vari-
ables {ω(n)}

n∈Zd is important in many steps. Strict independence can, at most, be re-
placed by an assumption of a weaker form of independence which cannot be removed. This

is not simply a technical issue related to fractional moments. For instance, the robust

multiscale analysis technique is usually formulated under the assumption of independence

at distance, meaning that there exists a ρ > 0 such that events based on boxes ΛL1
are independent of events based on boxes ΛL2
ΛL(m) = {m(cid:48) ∈ Zd : |m − m(cid:48)|∞ < L

(m), ΛL2
2 }, see [41] for a survey in the subject.

(n) if dist

ΛL1

(m)

(n)

> ρ. Here

(cid:16)

(cid:17)

1.7 Instructive consequences of dynamical localization

Let us point out some consequences of theorem 1.1. More precisely, the facts presented

in this section will follow from dynamical localization as in (1.6.2).

1.7.1 From dynamical localization to finite averaged moments

Let

M q

T (H) =

2
T

(cid:90) ∞

0

T E(cid:16)(cid:104)0|eitHω|X|qe−itHω|0(cid:105)(cid:17)

−2t

e

dt.

be the averaged q-moments of the position operator X, formally defined as (Xϕ)(n) =
|n|ϕ(n). The following proposition holds true.

Proposition 1.6. Assume that dynamical localization in the sense of (1.6.2) holds. Then

for some finite constant C.

T (H) ≤ C
M q

11

Proof. By definition and from (1.6.2), we have that

M q

T (H) =

(cid:90) ∞
≤ C(cid:48) (cid:88)

2
T

0

e

−2t

T (cid:88)
n∈Zd
|n|qe−µ(cid:48)|n|

|n|q|E(cid:16)(cid:104)δn, e−itH δ0(cid:105)|2(cid:17)

dt.

n∈Zd

where the above quantity is finite.

1.7.2 From dynamical localization to pure point spectrum

For the convenience of the reader, we shall recall the definition of the spectral types below

in section 1.8. The next consequence of dynamical localization we would like to point out is

the following.

Proposition 1.7. Suppose that dynamical localization as described by (1.6.2) holds on an
interval I ⊂ R. Then, almost surely, Hω has only pure point spectrum in I.

Proof. By the RAGE theorem (see [6, Theorem 2.6]), given ϕ ∈ (cid:96)2(cid:16)Zd(cid:17)

and denoting by

P c
I the projection onto the continuous component of H within I we have that

(cid:107)P c

I ϕ(cid:107)2 = lim
Let ϕ = δn for some n ∈ Zd. Then,

L→∞ lim
T→∞

1
T

(cid:90) T

0

(cid:107)1

Λc
L

e−itH PI (H)ϕ(cid:107)2 dt.

(1.7.1)

(cid:107)1

Λc
L

e−itH PI (H)ϕ(cid:107)2 ≤ (cid:107)1
≤ (cid:107)1

Λc
L

e−itH PI (H)Pn(cid:107)2(cid:107)δn(cid:107)2
e−itH PI (H)Pn(cid:107)

(cid:12)(cid:12)(cid:104)δm, e−itH PI (H)δn(cid:105)(cid:12)(cid:12).

Λc
L

≤ (cid:88)

|m|>L

12

assumption (1.6.2)

(cid:88)

Taking expectations, using Fatou’s lemma and Fubini’s theorem, we conclude that under the

(cid:107)P c

I ϕ(cid:107)2 ≤ C(cid:48)

e−µ(cid:48)|m−n| = 0.

The general result follows from density of finitely supported functions on (cid:96)2(cid:16)Zd(cid:17)

lim
L→∞

|m|>L

(1.7.2)

.

1.7.3

Localization centers and Eigenfunction decay

Lastly, let us discuss the phenomenon on engenfunction decay in the regime of dynamical

localization.

Definition 1.8. Given a Borel set I ⊂ R, the eigenfunction correlator on I is defined by

Q(m, n; I) = sup
|f|≤1

(cid:104)m|f (Hω)PI (Hω)|n(cid:105) .

(1.7.3)

with the supremum being taken over Borel measurable functions.

If H has only pure point spectrum in I, we have

Q(m, n; I) =

|ϕE,m(m)||ϕE,m(n)|

(1.7.4)

(cid:88)

E∈I

where ϕE,m is the normalized eigenfunction within the cyclic subspace (cid:72)H,δm associated to

the eigenvalue E. Recall that, in general,

(cid:72)H,v = span{(H − z)−1v : z ∈ C \ R}.

(1.7.5)

In case H is a bounded operator, the above set coincides with the closure of the set spanned

by vectors of the form q(H)v where q is a polynomial, see [19, lemma 2.4.3]. The expression

(1.7.4) is obtained expanding (1.7.3) into the eigenfunction basis.

Definition 1.9. Given ϕ ∈ (cid:96)2(cid:16)Zd(cid:17)

with (cid:107)ϕ(cid:107)2 = 1, a point n0 ∈ Zd is said to be a

localization center for ϕ if

:=(cid:80)

where C−1

d

|ϕ(n0)|2 ≥

Cd

(1 + |n0|)d+1

(1.7.6)

n∈Zd

1

(1+|n0|)d+1 .

13

It is immediate to check that any such ϕ admits a localization center.

Proposition 1.10. Assume that for some ν > 0.

(cid:88)

m

Cd

(1 + |m|)d+1

n

(cid:88)

eν|m−n|E(cid:16)

(cid:17)

Q2(m, n; I)

< ∞.

(1.7.7)

Then, almost surely, for any simple eigenvalue E ∈ I of H there is a localization center

nE(ω) such that the normalized eigenfunction ϕE,ω satisfies

|ϕE,ω(n)| ≤ C(I, ω)(1 + |n|)d+1e

− ν|n−nE|

2

.

(1.7.8)

for all n ∈ Zd. Moreover, ω (cid:55)→ C2(I, ω) belongs to L1 (Ω; P).

Proof. Let

(cid:32)(cid:88)

m

(cid:88)

n

Cd

(1 + |m|)d+1

C(I, ω) :=

(cid:33)1/2

eν|m−n|Q2(m, n; I)

.

(1.7.9)

By assumption, C2(I, ω) belongs to L1 (Ω; P) hence it is a finite quantity almost surely. By

proposition 1.7, the spectrum of H is, almost surely, pure point in I. Moreover, by (1.7.4),

any simple eigenfunction of H satisfies

|ϕE,m(m)|2|ϕE,m(n)|2 ≤ C2(I, ω)C−1

d (1 + |n|)d+1e−ν|n−nE|.

(1.7.10)

recalling the definition of localization center and letting m = nE we reach

|ϕE,m(n)|2 ≤ C2(I, ω)C−2

d (1 + |n|)2d+2e−ν|n−nE|.

(1.7.11)

which yields the desired result after taking square roots.

1.8 Spectral Measures

Finally, we conclude this introduction with a few notions on spectral theory. Let (cid:72) be a
Hilbert space and H a self-adjoint operator on (cid:72). Given ψ ∈ (cid:72) , the spectral measure µψ
is the unique finite Borel measure on R satisfying, for all z ∈ C+,

(cid:90)

1
t − z

dµψ(t).

R

(cid:104)ψ, (H − z)−1ψ(cid:105) =

14

The existence and uniqueness of µψ follows from the representation theorem for Herglotz

functions (see [6, Appendix B]) since

F (z) = (cid:104)ψ, (H − z)−1ψ(cid:105)

(1.8.1)

defines a map F : C+ → C+ whenever ψ (cid:54)= 0. By the Radon-Nikodym theorem, we can

decompose µψ (with respect to Lebesgue measure) according to its pure point, singular

continuous and absolutely continuous components.

µψ = µpp

ψ + µsc

ψ + µac
ψ

Associated to the above decomposition is the decomposition of the Hilbert space

where

and

(cid:72) = (cid:72)pp ⊕ (cid:72)ac ⊕ (cid:72)sc

(cid:72)pp = {ψ |µψ = µpp

ψ } , (cid:72)sc = {ψ |µψ = µsc
ψ }

(cid:72)ac = {ψ |µψ = µac
ψ }.

The spectrum of H, denoted by σ(H), is defined as the set of z ∈ C such that H − z
does not have a bounded inverse and the spectral components σpp(H), σsc(H) and σac(H)

are defined by restricting the operator to the subspaces (cid:72)pp,(cid:72)sc and (cid:72)ac.

15

CHAPTER 2

LOCALIZATION AND IDS REGULARITY IN THE HUBBARD MODEL

WITH HARTREE-FOCK THEORY

2.1 Chapter Outline

Our main goal in this chapter is to study Anderson localization in the context of in-

finitely many particles. We shall formulate our results for the disordered Hubbard model

within Hartree-Fock theory. However, as the techniques involved are quite flexible, we expect

that similar statements can be made in a more general framework, under appropriate modi-

fications of the decorrelation estimates in section 2.7.1. The (deterministic) Hubbard model

under Generalized Hartree-Fock Theory has been discussed (at zero and positive tempera-

ture) by Lieb, Bach and Solovej in [11] however, to the best of our knowledge, the localization

properties of the disordered version of this model remained unexplored, even in the context

of restricted Hartree-Fock theory, up to the present work. The main difficulty lies in the

addition of a self-consistent effective field, which will be random and non-local by nature,

to a random Schr¨odinger operator. The conclusion of this chapter can be summarized as

follows: under technical assumptions, the results on (single-particle) Anderson localization
obtained in the non-interacting setting in the regimes of large disorder (in dimension d ≥ 2)

and at any disorder (in dimension d = 1), remain valid under the presence of sufficiently

weak interactions. More specifically, in the regime of strong disorder this is accomplished in

any dimension by theorem 2.2 below. Theorem 2.1 contains the improvement in dimension

one, where any disorder strength leads to localization, provided the interaction strength is

taken sufficiently small. Our methods contain various bounds in the fluctuations of the ef-

fective interaction which are interesting on their own right and potentially useful in different

contexts. To exemplify this, we prove H¨older regularity of the integrated density of states

(IDS) with respect to various parameters by adapting arguments of [29], which is the content

16

of theorem 2.3.

2.1.1 Discussion of the results and main obstacles

Mathematically, our setting can be understood as an Anderson-type model Hω = H0 + λUω

where the values of the random potential U at different sites are correlated in a highly non-

local and self-consistent fashion. The correlations are governed by a nonlinear function of

Hω, as explained in section 2.3. In comparison to the recent result on Hartree-Fock theory

for lattice fermions of [23], achieved via multiscale analysis, we use the fractional moment

method to establish exponential decay of the eigenfunction correlators at large disorder in any

dimension but also at any disorder in dimension one. In particular, in the above regimes, we

show exponential decay (on expectation) of the matrix elements of the Hamiltonian evolution
at all times, which means that, on average, |(cid:104)m|e−itH|n(cid:105)| decays exponentially on |m − n|

for all t > 0.

The result of complete localization in dimension one in such interacting context is new

and deserves attention on its own. There, the main technical difficulty also lies in the non-

local correlations of the potential, which means that standard tools such as Furstenberg’s

theorem and Kotani theory are not available. Moreover, a large deviation theory for the

Green’s function is a further obstacle to establishing dynamical localization even if one

obtains uniform positivity of the Lyapunov exponent. We overcome these challenges using

ideas of [6, Chapter 12], where arguments reminiscent of the proof of the main result in [43]

are presented. We then obtain positivity of the Lyapunov exponent at any disorder using

uniform positivity for the Lyapunov exponent of the Anderson model, combined with an

explicit bound on how this quantity depends on the interaction strength, see theorem 2.19.

When it comes to establishing a large deviation theorem, our modification of the argument

in [6, Theorem 12.8] relies on quantifying the decorrelations on the effective potential, which

is presented on lemma 2.22 in the form of a strong mixing statement. It is worth clarifying

that, since our proof is based on fractional moments, we have not established localization

17

in one dimension for rough potentials as in [34]. Moreover, the gap assumption in [23] is

replaced by working at positive temperature thus our results do not apply to Hartree-Fock

ground states.

2.1.2 Hartree-Fock theory

Hartree-Fock theory has been widely applied in computational physics and chemistry. It also

has a rich mathematical literature which goes well beyond the scope of random operators,

see for instance [31],[32],[11],[12],[33] and references therein.

2.1.3 Background on Localization for interacting systems

The main results of this chapter lie in between the vast literature on (non-interacting) single

particle localization and the recent efforts to study many particle systems, as in the case

of an arbitrary, but finite, number of particles in the series of works by Chulaevsky-Suhov

[15],[16],[17] and Aizenman-Warzel [7]. In comparison to the later, we only seek for a single-

particle localization result but allow for infinitely many interactions, which occur in the form

of a mean field. In comparison to the recent developments on spin chains, as the study of

the XY spin chain in [30] and the droplet spectrum of the XXZ quantum spin chain in [25]

and [13], the notions of localization for a single-particle effective Hamiltonian are more clear

and can be displayed from pure point spectrum to exponential decay of eigenfunctions and

exponential decay of eigenfunction correlators. The later is agreed to be the strongest form

of single particle localization and it is what we end up accomplishing. If fact, as we have

explained in the introduction of this thesis, dynamical localization in the form of theorems

2.1 and 2.2 implies pure point spectrum via the RAGE theorem and exponential decay of

eigenfunctions, see also [58, Proposition 5.3] and [6, Theorem 7.2 and Theorem 7.4].

18

2.2 Definitions and Statement of the Main Result

2.2.1 Notation
In what follows, Zd will be equipped with the norm |n| = |n1| + ... +|nd| for n = (n1, ..., nd).
n∈Λ |ϕ(n)|2 < ∞} and, for
. Throughout this chapter, η will

Given a subset Λ ⊂ Zd, we define (cid:96)2(Λ) := {ϕ : Λ → C|(cid:80)
ϕ ∈ (cid:96)2(Λ), we let (cid:107)ϕ(cid:107)(cid:96)2(Λ) := (cid:0)(cid:80)

n∈Λ |ϕ(n)|2 < ∞}(cid:1)1/2

be a positive constant and Fβ,κ will denote the Fermi-Dirac function at inverse temperature

β > 0 and chemical potential κ, meaning that

Fβ,κ(z) =

1

1 + eβ(z−κ)

.

(2.2.1)

We shall omit the dependence of F on the above parameters whenever it is clear from the

context. For many of our bounds, the specific form of (2.2.1) is not important and F could
denote an arbitrarily chosen function which is analytic in the strip (cid:83) = {z ∈ C :
|Imz| < η}
and continuous up to the boundary of (cid:83), in which case we let (cid:107)F(cid:107)∞ := supz∈(cid:83) |F (z)|.
For the function Fβ,κ in (2.2.1) one can take η = π
2β . However, to obtain robust results
which incorporate delicate fluctuations, further properties of the Fermi-Dirac function are
necessary. Namely, in section 2.7.1 we use the the fact that tF (t) is bounded as t → ∞ and
that t(1− F (t)) is bounded t → −∞. These properties will also play a role in the decoupling

estimates needed in the proof of theorem 2.1 but could be relaxed if one is only interested

in the large disorder proof of theorem 2.2 for a specific distribution with heavy tails (for

instance, the Cauchy distribution).

Our main goal is to study localization properties of non-local perturbations of the Ander-
son model HAnd := −∆ + λVω which naturally arise in the context of Hartree-Fock theory
for the Hubbard model. The random potential Vω is the multiplication operator on (cid:96)2(Zd)

defined as

for all n ∈ Zd and {ωn}

n∈Zd are independent, identically distributed random variables

(Vωϕ) (n) = ωnϕ(n)

(2.2.2)

19

on which we impose technical assumptions described in the next paragraph. The hopping
operator ∆ is the discrete Laplacian on Zd, defined via

(∆ϕ) (n) =

(ϕ(m) − ϕ(n)) .

(2.2.3)

(cid:88)

|m−n|=1

The proofs of localization via fractional moments usually do not require the hopping to

be dictated by the Laplacian and, indeed, we will replace ∆ by a more general operator

H0 whose matrix elements decay sufficiently fast away from the diagonal. It is technically

useful to formulate some of our results in finite volume, i.e, we will work with restrictions

of the operators to (cid:96)2(Λ) but the estimates obtained will be volume independent, meaning
that all the constants involved are independent of Λ ⊂ Zd. We will use 1Λ to denote the
characteristic function of Λ as well as the natural projection PΛ : (cid:96)2(Zd) → (cid:96)2(Λ). With
these preliminaries we are ready to define the Schr¨odinger operators studied in this chapter.

2.2.2 Definition of the operators

Let HAnd = H0 + λVω be the Anderson model on (cid:96)2(cid:16)Zd(cid:17)
(cid:17)
eν|m−n| − 1

|H0(m, n)|(cid:16)

(A1)

ζ(ν) := sup
m

(cid:88)

n∈Zd

where:

η
2

,

for some ν > 0 fixed.

<

(cid:90)

(A2) Vω is defined as in (2.2.2) and the random variables {ω(n)}

n∈Zd are independent,

identically distributed with a density ρ:

P (ω(0) ∈ I) =

ρ(x) dx,

for I ⊂ R a Borel set .

(A3) We also assume that supp ρ = R with

I

≥ e−c1(ρ)|x−x(cid:48)|(1+c2(ρ) max{|x|,|x(cid:48)|})

ρ(x)
ρ(x(cid:48))

(2.2.4)

for some c1(ρ) > 0, c2(ρ) ≥ 0 and any x, x(cid:48) ∈ R.

20

Before stating the remaining assumptions on ρ, we need to introduce some notation. Assume

that ρ satisfies (2.2.4). Let

ρ(x) =

where

h(x) =

(cid:82) ∞
−∞ ρ(α)h(x − α) dα

ρ(x)

e−cρ|x|

if c2(ρ) = 0.

e−cρ|x|2

if c2(ρ) > 0.

(2.2.5)

(2.2.6)



(A4) The function ρ is bounded for some cρ > 0.

Remark 1. The technical assumptions (A3)−(A4) will be needed for the large disorder result
of theorem 2.2 below. They include, for instance, the Cauchy distribution, the Gaussian,

and the exponential distribution ρ(v) = m

2 e−m|v|. Further details an related comments will

be provided on the appendix of this thesis, section A.1.

The above requirements will suffice to show localization at large disorder on theorem

2.2 below. To show complete localization in dimension one, theorem 2.1 will also require a

moment condition on ρ, which is the following.

(A5) For some ε > 0 and some cρ > 0,(cid:82) ∞

−∞ |x|ερ(x) dx < ∞.

Remark 2. The assumption (A5) covers, for example, the Gaussian and the exponential

distributions but it does not cover the Cauchy or other distribution with heavy tails.

It

will be necessary for the one dimensional result of theorem 2.1 below. More specifically, this

requirement will imply a moment condition which will be needed to relate the Green’s function

to the Lyapunov exponent, see sections 2.11 and 2.11.5.

Remark 3. The specific bound on ζ(ν) is necessary to ensure that the Combes-Thomas
bound |G(m, n; E + iη)| ≤ 2
(cid:104)m|(H − z)−1|n(cid:105), whenever this quantity is defined.

η e−ν|m−n| holds [6, Theorem 10.5], where G(m, n; z) denotes

21

0

HHub =

 H↑(ω)

Define the operator HHub, acting on (cid:96)2(cid:16)Zd(cid:17) ⊕ (cid:96)2(cid:16)Zd(cid:17)
 H0 + λVω + gV↑(ω)
 :=
where the operators H↑(ω) and H↓(ω) act on (cid:96)2(cid:16)Zd(cid:17)
 V↑(ω)(n)
 =
 (cid:104)n|F (H↓)|n(cid:105)
 .

are defined via

0

H↓(ω)

V↓(ω)(n)

(cid:104)n|F (H↑)|n(cid:105)

0

H0 + λVω + gV↓(ω)

and the so-called effective potentials

by

0

 (2.2.7)

(2.2.8)

Note that the above equations only define H↑(ω) and H↓(ω) implicitly. Existence and
uniqueness of V↑ and V↓ will be shown in section 2.9.1 via a fixed point argument. The model

(2.2.7) is usually referred to as the Hartree approximation, due to the absence of exchange

terms. In section 2.3 below we will show that the terminology Hartree-Fock approximation

is justified when g < 0, which represents a repulsive interaction.

The Hubbard model is schematically represented in the following picture. The black

(horizontal) edges represent hopping between sites and the red (vertical) edges represent the
effective interaction between the two layers, which are identical copies of Zd.

Figure 2.1: The Hubbard Model

n↑

n↓

(0,0)

2.2.3 Main Theorems

Fix an interval I ⊂ R and define the eigenfunction correlator through

(cid:0)|(cid:104)m|ϕ(H↑)|n(cid:105)| + |(cid:104)m|ϕ(H↓)|n(cid:105)|(cid:1) .

QI (m, n) := sup
|ϕ|≤1

(2.2.9)

22

The operators H↑ and H↓ are defined as in (2.2.7) and the supremum is taken over Borel
measurable functions bounded by one and supported on the interval I. In case I = R we

simply write Q(m, n). Our first result is the following:

Theorem 2.1. In dimension d = 1, let H0 = −∆ and assume that the conditions (A1)−(A5)
hold. For any λ > 0 and any closed interval I ⊂ R , there is a constant g1 > 0 such that
whenever |g| < g1 we have

E (QI (m, n)) ≤ Ce−µ1|m−n|.

(2.2.10)

for any m, n ∈ Zd and positive constants µ1 = µ1(λ, ν, η, I), C(η, g, λ,(cid:107)F(cid:107)∞, I).
Theorem 2.2. Suppose that the conditions (A1) − (A4) hold. For any dimension d ≥ 1,
there exists a constant gd = g(d, η,(cid:107)F(cid:107)∞, ν) such that, whenever |g| < gd, there is a positive
constant λ0(g) for which

E (Q(m, n)) ≤ Ce−µd|m−n|.

(2.2.11)
holds for λ > λ0(g), any m, n ∈ Zd and some positive constants µd = µ(d, λ, g, ν, η),
C(η, ν, d, g, λ,(cid:107)F(cid:107)∞).
(cid:16)

Remark 4. It will follow from the proof that the constant gd in theorem 2.2 can be taken

1−e−ν(cid:17)d
(cid:107)F(cid:107)∞ .
(cid:16)
(cid:107)F(cid:107)∞ and the upper bound obtained in corollary 2.20, which also

1−e−ν(cid:17)

factor proportional to
depends on the lower bound for the Lyapunov exponent of the Anderson model on (cid:96)2 (Z).

Remark 5. The constant g1 in theorem 2.1 can be taken equal to be the minimum among a

proportional to

η

η

Before proceeding to the next result, recall the definition of the integrated density of

states for an ergodic operator H :

NH (E) = lim|Λ|→∞

TrP(−∞,E)(1ΛH1Λ)

|Λ|

.

(2.2.12)

Recall also that H is said to be ergodic (on Zd) when for every ω ∈ Ω and j ∈ Zd

HTj ω = UjHωU∗
j .

23

(2.2.13)

In the above equality, T : Ω → Ω denotes the measure preserving transformation given by
Tjω(n) = ω(n − j) and the unitary maps Uj are defined via

(cid:0)Ujϕ(cid:1) (n) = ϕ(n − j).

(2.2.14)

In fact, one can relax the above conditions and allow the maps Uj to be of more general form,

the interested reader may consult [6, Definition 3.4]. In what follows, we denote by N0(E)

the corresponding quantity for the free operator H0 defined above, which is assumed to be

ergodic for the result of theorem 2.3, where we shall be concerned with the small disorder
regime and aim for bounds which do not depend upon λ as λ → 0.

Theorem 2.3. Assume that (A1) − (A2) hold with x2ρ(x) bounded and that g2 < λ and
g(cid:48)2 < λ(cid:48). Fix a interval I where E (cid:55)→ N0(E) is α0-H¨older continuous and a bounded interval
J ⊂ R. The integrated density of states Nλ,g(E) of HHub is H¨older continuous with respect
to E and with respect to the pair (λ, g). More precisely:

• For E, E(cid:48) ∈ I

|Nλ,g(E) − Nλ,g(E(cid:48))| ≤ C(α, I, g)|E − E(cid:48)|α

(2.2.15)

for α ∈ [0,

α0
2+α0

] and C(α, I, g) independent of λ.

• If λ, λ(cid:48) ∈ J, we have that, for any E ∈ I, α ∈ [0,

|Nλ,g(E) − Nλ(cid:48),g(cid:48)(E)| ≤ C(α0.d, I)

α0
2+α0

] and β ∈ [0,

(cid:16)|λ − λ(cid:48)|β + |g − g(cid:48)|β(cid:17)

2α

α+3d+4],

.

(2.2.16)

2.3 Motivation

We shall explain the motivation for the above choice of the effective potential. We are

only going to outline the derivation of the self-consistent equations later in section A.2 as

this is a standard topic, see, for instance, [44, Chapter 3].

24

Let Λ ⊂ Zd be a finite set. Following the notation of [11], we use Γ to denote a one
particle density matrix, i.e, a 2 × 2 matrix whose entries are operators on (cid:96)2 (Λ) and which
satisfies 0 ≤ Γ ≤ 1. We then write

 Γ↑ Γ↑↓

Γ↓↑ Γ↑



Γ :=

†
where Γ↓↑ = Γ
↑↓.

As in [11, Equation 3a.8], the pressure functional (cid:80)(Γ) is defined as

−(cid:80)(Γ) = (cid:69)(Γ) − β−1(cid:83)(Γ).

The energy functional is

(cid:69)(Γ) = Tr (H0 − κ + λVω) Γ + g

(cid:104)n|Γ↑|n(cid:105) (cid:104)n|Γ↓|n(cid:105) ,

(cid:88)
 H0 − κ + λVω

n

0

(2.3.1)

(2.3.2)

.

where we have identified H0 − κ + λVω with

The entropy is given by

0

H0 − κ + λVω

(cid:83)(Γ) = −Tr (Γ log Γ + (1 − Γ) log(1 − Γ)) .

(2.3.3)

Generally, the choice of energy functional (2.3.2) is referred to as Hartree approximation

as exchange terms are neglected. However, in the case of a repulsive interaction among the

particles, it is easy to prove that such exchange terms do not affect the choice of minimizer for
−(cid:80)(Γ) and the process may be referred to as the Hartree-Fock approximation. Indeed, the
Hartree-Fock energy for the repulsive interaction would incorporate the term −g|(cid:104)n|Γ↑↓|n(cid:105)|2,
which is non-negative when g < 0, inside the summation. Thus, for repusive interactions,

off-diagonal terms can be disregarded for minimization purposes, see the analogue discussion
in [11, Section 4a]. The minimizer Γ of −(cid:80)(Γ) exists since Λ is a finite set. Moreover, it

satisfies

(cid:104)n|Γ↑|n(cid:105) = (cid:104)n|

1

β(H0−κ+λVω+Diag

1 + e

(cid:16)

(cid:17)|n(cid:105) .

Γ↓)

(2.3.4)

25

(cid:17)|n(cid:105) .
Thus, the effective Hamiltonian on (cid:96)2 (Λ) ⊕ (cid:96)2 (Λ) is determined by

β(H0−κ+λVω+Diag

(cid:104)n|Γ↓|n(cid:105) = (cid:104)n|

1 + e

1

(cid:16)

Γ↑)

 H0 + λω(n) + gV Λ↑ (n)

0

HΛ

ω :=

H0 + λω(n) + gV Λ↓ (n)



0

|n(cid:105)

|n(cid:105) .

(2.3.5)

(2.3.6)

(2.3.7)

V Λ↑ (ω)(n) := (cid:104)n|
V Λ↓ (ω)(n) := (cid:104)n|

1

1

β(H0−κ+λω+gV↓)

1 + e

β(H0−κ+λω+gV↑)

1 + e

It will follow from arguments given below that if ΛR is an increasing sequence with

∪R∈NΛR = Zd then, for fixed m ∈ Zd,

lim
R→∞ V

ΛR
eff (m) = Veff (m)

(2.3.8)

and this fact ensures that, for localization purposes in the Hubbard model, it suffices to

study HHub and its finite volume restrictions.

2.4 Outline of the Proof of theorem 2.2

We now want to outline the proof of the theorem 2.2 in the related model where HHub

is replaced by the operator

acting on (cid:96)2(cid:16)Zd(cid:17)

with

H = H0 + λω(n) + gVeff (n)

Veff (n) = (cid:104)n|

1

1 + eβ(H0+λω+gVeff )

|n(cid:105) .

In this case, the eigenfunction correlator is defined as

QI (m, n) := sup
|ϕ|≤1

|(cid:104)m|ϕ(H)|n(cid:105)|.

(2.4.1)

(2.4.2)

(2.4.3)

where the supremum is taken over functions ϕ which are Borel measurable and supported

on I.

26

The above operator exhibits the main mathematical features of the Hubbard model,

namely: the effective potential is defined self-consistently as a non-local and non-linear

function of H. Thus, it is natural to first illustrate our methods here. For now let’s assume

the existence and uniqueness of Veff are proven, as well as its regularity with respect to
{ω(n)}

n∈Zd. Combined with estimates on the derivatives of Veff , the above facts form a
significant portion of the proof which is developed in sections 2.5 and 2.6. The, somewhat

straightforward, extension of the proof to HHub will be explained in section 2.9. A feature

which theorem 2.1 and theorem 2.2 have in common is that the eigenfunction correlator

decay will be achieved via analysis of the Green’s function of HΛ = 1ΛH1Λ, which is H
restricted to a finite set Λ ⊂ Zd. Let

GΛ(m, n, z) = (cid:104)m|(HΛ − z)−1|n(cid:105) .

(2.4.4)

Using the basics of the fractional moment method, which dates back to [5] and [1], we aim
at showing that, for some s ∈ (0, 1),

E(cid:16)(cid:12)(cid:12)(cid:12)GΛ(m, n; z)

(cid:12)(cid:12)(cid:12)s(cid:17) ≤ Ce−µd|m−n|

(2.4.5)

holds uniformly in z ∈ C+, with positive constants C = C and µ depending on the param-
eters (d, s, g, λ, ν, η,(cid:107)F(cid:107)∞) but independent of the volume |Λ|. In this context, the Green’s
function decay expressed by equation (2.4.5) implies
−µ(cid:48)
E (Q(m, n)) ≤ C(cid:48)e

d|m−n|

(2.4.6)

for some exponent µ(cid:48)

d = µ(cid:48)(d, s, g, λ, ν, η,(cid:107)F(cid:107)∞) > 0 and C(cid:48) = C(cid:48)(η, ν, d, g, λ, s,(cid:107)F(cid:107)∞).

This is well known and explained in the introduction of this thesis.

Another aspect which is shared by the proofs of theorems 2.1 and 2.2 is that the starting

point to obtain (2.4.5) will be the following a-priori bound.
Lemma 2.4. Given a finite set Λ ⊂ Zd, there exist a constant CAP depending on the
parameters (η, ν, d, g, λ, s,(cid:107)F(cid:107)∞), but independent of Λ, such that

(2.4.7)

E(cid:16)(cid:12)(cid:12)(cid:12)GΛ(m, n; z)

(cid:12)(cid:12)(cid:12)s(cid:17) ≤ CAP

27

holds for any m, n ∈ Λ.

The proof of lemma 2.4 will follow from lemma 2.5 below. Let

Uω(n) = ω(n) +

g
λ

Veff (n, ω).

(2.4.8)

be the “full” potential at site n. From now on, to keep the notation simple, we drop the
dependence on ω in the new variables {U (n)}n∈Λ. Note that U (n) and U (m) are correlated
for all values of m and n. The strategy is to show that, for g sufficiently small, they still

behave as if they were independent in the following sense:

Lemma 2.5. Fix Λ ⊂ Zd finite and n0 ∈ Λ. The conditional distribution of U (n0) =
u at specified values of {U (n)}n∈Λ\{n0} has density ρΛ
. Moreover, under assumptions
(A1) − (A4) we have that

n0

sup
u∈R
If, additionally, assumption (A5) holds then ρΛ
n0

sup
n0∈Λ

sup
Λ

(u) < ∞.

ρΛ
n0
(u) ∈ L1 (R,|x|εdx).

(2.4.9)

The proof of the above result is detailed in section 2.8; it requires exponential decay of

∂ω(m) | and | ∂2Veff (n)
| ∂Veff (n)

∂ω(m)ω(l)| with respect to |m− n| and |m− n| +|l− n|, respectively. The need

for this decay is the main reason to require β > 0 or, in other words, to require analiticity

of F in a strip. The intuitive explanation for lemma 2.5 is that the random variables U (n)
and U (n0) decorrelate in a strong fashion as |n − n0| becomes large. As explained in the
introduction of this manuscript, lemma 2.5 implies (2.4.5) for any 0 < s < 1 as long as λ is

taken sufficiently large, see also [6, Theorem 10.2].

The proof of theorem 2.1 will require additional efforts involving tools which are specific

to one dimension, which we shall comment on below.

2.5 Existence of the effective potential

To justify the definition of the effective potential in (2.4.1), let Φ(V ) : (cid:96)∞(Zd) → (cid:96)∞(Zd)
be given by Φ(V )(n) := (cid:104)n|F (T + λVω + gV )|n(cid:105) . Recall that F is analytic, bounded in the

28

strip S = {|Imz| < η} and continuous up to the boundary of S. Our goal is to check that Φ

is a contraction in (cid:96)∞(cid:16)Zd(cid:17)

, meaning that

(cid:107)Φ(V ) − Φ(W )(cid:107)

(cid:96)∞(Zd)

< c(cid:107)V − W(cid:107)

(cid:96)∞(Zd)

(2.5.1)

holds for some c < 1 and all V, W ∈ (cid:96)∞(Zd). Let R(z, H) = (H − z)−1 and T be a self-

adjoint operator. Using the analiticity of F we have the following representation for F (T )

[4, Equation (D.2)]

(cid:90) ∞
−∞ (R(t + iη, T ) − R(t − iη, T )) f (t) dt

1
2πi

F (T ) =

(2.5.2)
for all V ∈ (cid:96)∞(Zd), where f = F+ + F− − D ∗ F for F±(u) = F (u ± iη) and D(u) =
η2+u2(cid:17) is the Poisson kernel. As explained in [4], the above representation is a consequence

η

(cid:16)

π
of the identities

D+ + D− = δ + D ∗ D

(with δ denoting the Dirac delta) and

(D+ + D−) ∗ F = D ∗ (F+ + F−).

(2.5.3)

(2.5.4)

It follows immediately that (cid:107)f(cid:107)∞ ≤ 3(cid:107)F(cid:107)∞. This is a prelude for the following fixed point
argument, where the operator T will be assumed to satisfy

<

η
2

.

(2.5.5)

Proposition 2.6. The following contraction estimates hold true.

(cid:17)
eν|m−n| − 1

m

sup
n

(cid:88)

|T (m, n)|(cid:16)
• For any self-adjoint operator T on (cid:96)2(cid:16)Zd(cid:17)
(cid:12)(cid:12)(cid:12)(cid:104)m|(F (T + V ) − F (T + W ))|n(cid:105)(cid:12)(cid:12)(cid:12) ≤ 72
(cid:16)

V, W , we have, for any ν(cid:48) ∈ (0, ν), that

satisfying (2.5.5) and bounded potentials

√

2e−ν(cid:48)|m−n|

1 − eν(cid:48)−ν(cid:17)d

η

(cid:107)F(cid:107)∞(cid:107)V − W(cid:107)∞.

(2.5.6)

29

• For any self-adjoint operator T on (cid:96)2(cid:16)Zd(cid:17) ⊕ (cid:96)2(cid:16)Zd(cid:17)
potentials V, W on (cid:96)2(cid:16)Zd(cid:17) ⊕ (cid:96)2(cid:16)Zd(cid:17)
(cid:12)(cid:12)(cid:12)(cid:104)m|(F (T + V ) − F (T + W ))|n(cid:105)(cid:12)(cid:12)(cid:12) ≤ 144
(cid:16)
1 − eν(cid:48)−ν(cid:17)d

2e−ν(cid:48)|m−n|

√

η

satisfying (2.5.5) and bounded

we have, for any ν(cid:48) ∈ (0, ν), that

(cid:107)F(cid:107)∞(cid:107)V − W(cid:107)∞

(2.5.7)

• For any m, n, j ∈ Zd, the matrix elements (cid:104)m|F (T + gV )|n(cid:105) are differentiable with

respect to V (j) and

(cid:12)(cid:12)(cid:12) ∂ (cid:104)m|F (T + gV )|n(cid:105)

∂V (j)

(cid:12)(cid:12)(cid:12) ≤ |g|72

√
2e−ν(|m−j|+|n−j|)

η

(cid:107)F(cid:107)∞(cid:107)V (cid:107)∞.

(2.5.8)

Proof. The resolvent identity gives

(cid:104)m|

1

T + V − t − iη

−

= (cid:104)m|(

1

T + V − t − iη

1

T + W − t − iη
−

1

|n(cid:105) + (cid:104)m|

1

T + W − t + iη

−

)(W − V )

1

T + W − t − iη

|n(cid:105)

1

T + V − t + iη
|n(cid:105)

(cid:18)

− (cid:104)m|

1

T + W − t + iη

(W − V )

1

T + V − t + iη

|n(cid:105) .

(cid:19)

T + V − t + iη
−

T + W − t − iη

1

Taking absolute values in the first term on the right-hand side we obtain

(cid:19)

|n(cid:105)(cid:12)(cid:12)(cid:12)

(cid:18)
(cid:12)(cid:12)(cid:12)(cid:104)m|
≤ (cid:88)
(cid:88)

l∈Zd
≤ 24

l

× (cid:104)m|

1

−

1

T + V − t − iη
|GV (m, l; t + iη) − GV (m, l; t − iη)||(W − V )(l)|GW (l, n; t + iη)|

T + W − t − iη

T + V − t + iη

(W − V )

1

|(V − W )(l)|e−ν(|l−n|+|l−m|) (cid:104)l|

1

(T + V − E)2 + η2/2

|l(cid:105)1/2

1

(T + V − t)2 + η2/2

|m(cid:105)1/2 .

30

In the last step we made use of the Combes-Thomas bound |GW (m, n; t + iη)| ≤ 2

η e−ν|m−n|

as well as lemma 3 in [4, appendix D] to estimate the difference between the Green functions

as

|GV (m, l; t + iη) − GV (m, l; t − iη)|
≤ 12ηe−ν|m−l| (cid:104)m|

1

(T + V − t)2 + η2/2

|m(cid:105)1/2 (cid:104)l|

1

(T + V − E)2 + η2/2

|l(cid:105)1/2 .

Integrating over t we conclude, using Cauchy-Schwarz and the spectral measure representa-

tion, that

(cid:82) ∞

−∞

(cid:12)(cid:12)(cid:12)(cid:104)m|(cid:16)

(cid:17)

T +W−E−iη|n(cid:105)(cid:12)(cid:12)(cid:12) dt

1

(cid:80)
T +V −t−iη −
1
√
≤ 24
η

2π

(W − V )

T +V −t+iη
l |(V − W )(l)|e−ν(|l−n|+|l−m|).

1

The above implies that

(cid:90) ∞

1
−∞
√
2π
≤ 12
2
η
√
12
η

=

2

1

−

T + V − t − iη

(cid:18)
(cid:12)(cid:12)(cid:12)(cid:104)m|
(cid:107)V − W(cid:107)∞e−ν(cid:48)|m−n| (cid:88)
(cid:16)
(cid:107)V − W(cid:107)∞e−ν(cid:48)|m−n|

l∈Zd

1

T + V − t + iη
e(ν(cid:48)−ν)|l−n|

1 − eν(cid:48)−ν(cid:17)d

1

.

(cid:19)

(W − V )

1

T + W − t − iη

(2.5.9)

(2.5.10)

|n(cid:105)(cid:12)(cid:12)(cid:12) dt

As a similar bound holds for the term

(cid:90) ∞

−∞

(cid:18)
(cid:12)(cid:12)(cid:12)(cid:104)m|

1
2π

1

H0 + W − t + iη

−

1

H0 + W − t − iη

(cid:19)

(V − W )

1

H0 + V − t + iη

|n(cid:105)(cid:12)(cid:12)(cid:12) dt

we conclude the proof of the inequality (2.5.6) by recalling that (cid:107)f(cid:107)∞ ≤ 3(cid:107)F(cid:107)∞. The
inequality (2.5.7) follows from the same argument with the only difference that one has to

sum two geometric series, hence the modification of the upper bound by a factor of two. The
e−ν|j−n|e−ν|m−j| is an upper bound
bound (2.5.8) is proven similarly: note that |g||h| 24

2π

√
η

for the left-hand side of equation (2.5.9) with V replaced by gV and W = g(V + hPj), where

31

Pj denotes the projection onto Span{δj}. We also observe that this time there will be no
summation over l, hence the introduction of the ν(cid:48) is unnecessary. We then conclude that

(cid:12)(cid:12)(cid:12) (cid:104)m|F (T + gV + hPj)|n(cid:105) − (cid:104)m|F (T + gV )|n(cid:105)

h

(cid:12)(cid:12)(cid:12) ≤ |g|72

η

√

2

e−ν|j−n|e−ν|m−j|.

(2.5.11)

Letting h → 0 finishes the proof.

Taking m = n we conclude that the contraction mapping condition (2.5.1) holds whenever

η(cid:0)1 − e−ν(cid:1)d

√
2(cid:107)F(cid:107)∞ .
72

|g| <

(2.5.12)

This observation yields the following.

Proposition 2.7. Let gd =

potential Veff ∈ (cid:96)∞(cid:16)Zd(cid:17)

(cid:16)

1−e−ν(cid:17)d
2(cid:107)F(cid:107)∞ . Then, for |g| < gd, there is a unique effective

√

η

72

satisfying

Veff (n) = (cid:104)n|F (H0 + λω + gVeff|n(cid:105) .

(2.5.13)

Moreover, for Λ ⊂ Zd, there is a unique V Λ
HΛ = 1ΛH1Λ.
Remark 6. Replacing H0 by H0− κI we can incorporate a chemical potential in our results.
For simplicity, we shall make no further reference to κ during the proofs and assume it was

eff in (cid:96)2 (Λ) satisfying (2.5.13) with H replaced by

already incorporated to H0.

2.6 Regularity of the effective potential

Our goal in this section is to conclude that, for a fixed finite subset Λ ⊂ Zd with |Λ| = n,
the effective potential Veff is a smooth function of {ω(j)}j∈Λ. This will be of relevance for
several resampling arguments later in the thesis. For that purpose, define a map

ξ : (cid:96)∞(cid:16)Zd(cid:17) × Rn → (cid:96)∞(cid:16)Zd(cid:17)

by

ξ(V, ω)(j) = V (j) − (cid:104)j|F (H0 + λω + gV )|j(cid:105)

32

(2.6.1)

(2.6.2)

Then, Veff is the unique solution of ξ(V, ω) = 0. Thus, its regularity can inferred via the
implicit function theorem once we check that the derivative Dξ(·, ω) is non-singular. Note

that

∂ξ(V, ω)(j)

∂V (l)

= δjl − ∂ (cid:104)j|F (H0 + λω + gV )|j(cid:105)

∂V (l)

.

(2.6.3)

Using proposition 2.6, we have that

(cid:12)(cid:12)(cid:12)∂ (cid:104)j|F (H0 + λω + gV )|j(cid:105)

(cid:12)(cid:12)(cid:12) ≤ |g|72

√

2e−2ν|j−l|

(cid:107)F(cid:107)∞.

∂V (l)
√
η(1−e−2ν )d < 1 we have that the operator Dξ(ω, .) : (cid:96)∞ (Λ) →
In particular, whenever |g| 72
(cid:96)∞ (Λ) is invertible since it has the form I + gM where gM has operator norm less than one.
It is worth observing that the smallness condition on g is independent of Λ ⊂ Zd. It is a

2(cid:107)F(cid:107)∞

η

(2.6.4)

consequence of the implicit function theorem that V is a smooth function of (ω(1), ..., ω(n)).

2.7 Decay estimates for the effective potential

We start this section with the following lemma, which will be useful in order to formulate

the decay of correlations between U (n) and U (m) as |m − n| → ∞.

√
2|g|(cid:107)F(cid:107)∞
η(1−e−ν)d < 1, there exist constants C1(d, λ, g, η,(cid:107)F(cid:107)∞, ν) and
Lemma 2.8. Whenever 72
C2(d, λ, g, η,(cid:107)F(cid:107)∞, ν) such that
eν|n−m|(cid:12)(cid:12)(cid:12)∂Veff (n)
eν(|l−n|+|n−m|+|l−m|)(cid:12)(cid:12)(cid:12) ∂2Veff (n)

eν|n−m|(cid:12)(cid:12)(cid:12) ∂Veff (n)
(cid:12)(cid:12)(cid:12)(cid:111) ≤ C1
(cid:12)(cid:12)(cid:12) ≤ C2.

(cid:110)(cid:88)
(cid:88)

(cid:88)

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

∂ω(m)

∂ω(m)

(2.7.2)

(2.7.1)

max

m

n

∂ω(m)∂ω(l)

l,m,n

Moreover C1 and C2 can be bounded from above by a constant of the form λD

1−gθ with D and

θ independent of g and these constants are explicit in the proof.

Proof. For convenience of notation we let Veff = V in this proof. As in section 2.5 we write

(cid:90) ∞

(cid:18)

−∞

F (H) =

1
2πi

(cid:19)

f (t) dt

1

H − t − iη

−

1

H − t + iη

33

where f is bounded by 3(cid:107)F(cid:107)∞. Thus

V (n, ω) =

(cid:90) ∞
−∞ K(n, t, ω)f (t) dt

1
2πi

(2.7.3)

(cid:96)2(cid:16)Zd(cid:17)

where K(n, t, ω) = G(n, n; t − iη) − G(n, n; t + iη). Denote by Pm the projection mapping

onto (cid:96)2 (Span{δm}). Using difference quotients, it is easy to check

∂

∂ω(m)

1

H − z

+ g

1

H − z

∂V

∂ω(m)

1

H − z

= −λ

1

H − z

Pm

1

H − z

.

(2.7.4)

Taking matrix elements we obtain

∂K(n, t, ω)

∂ω(m)

= −g

(cid:88)

l

˜G(l, n)

∂V (l)
∂ω(m)

+ λr(m, n).

˜G(l, n) := G(l, n; t + iη)G(n, l; t + iη) − G(l, n; t − iη)G(n, l; t − iη).

r(m, n) := G(n, m; t + iη)G(m, n; t + iη) − G(n, m; t − iη)G(m, n; t − iη).

Note

˜G(l, n) = (G(l, n; t + iη) − G(l, n; t − iη)) G(n, l; t + iη)
+ (G(n, l; t + iη) − G(n, l; t − iη)) G(l, n; t − iη).

(2.7.5)

We now make use of [4, Lemma 3]:

|G(l, n; t + iη) − G(l, n; t − iη)| ≤ 12ηe−ν|l−n| (cid:104)n|

|n(cid:105)1/2

(2.7.6)

1

(H−t)2+η2/2
|l(cid:105)1/2 .
1

(H−t)2+η2/2

×(cid:104)l|

This, together with the Combes-Thomas bound |G(l, n, t ± iη)| ≤ 2

η e−ν|l−n| and (2.7.5)

implies

| ˜G(l, n)| ≤ 48e−2ν|l−n| (cid:104)n|

1

(H − t)2 + η2/2

|n(cid:105)1/2 (cid:104)l|

1

(H − t)2 + η2/2

|l(cid:105)1/2 .

|r(m, n)| ≤ 48e−2ν|m−n| (cid:104)m|

1

(H − t)2 + η2/2

|m(cid:105)1/2 (cid:104)n|

1

(H − t)2 + η2/2

|n(cid:105)1/2 .

34

To summarize, we have shown the following inequality

−∞

˜r(m, n) :=

˜K(l, n) :=

(cid:90) ∞
(cid:90) ∞
(cid:32)
(cid:12)(cid:12)(cid:12) ∂V (n)
(cid:12)(cid:12)(cid:12) ≤ 3(cid:107)F(cid:107)∞
|g|(cid:88)
1−e−2ν(cid:17)d < 1 we have that

∂ω(m)

−∞

2π

l

√

2|g|

72

√
| ˜G(l, n)| dt ≤ 48
2π
η
√
|r(m, n)| dt ≤ 48

2π

η

e−2ν|l−n|

e−2ν|m−n|.

(cid:12)(cid:12)(cid:12) ∂V (l)

∂ω(m)

(cid:33)
(cid:12)(cid:12)(cid:12) + λ˜r(m, n)

.

˜K(l, n)

|g|(cid:107) ˜K(cid:107)∞,∞ < 1

Thus

Whenever

(cid:16)

η

where

(cid:107) ˜K(cid:107)∞,∞ = sup
Considering the weight W (n) := eν|m−n| we let

l

θ :=

3(cid:107)F(cid:107)∞

2π

sup
n

By the triangle inequality,

(cid:88)

m

˜K(l, m).

(cid:88)

l

W (n)
W (l)

˜K(n, l).

eν|n−l| ˜K(n, l)

(cid:88)

θ ≤ 3(cid:107)F(cid:107)∞
sup
2π
n
√
2(cid:107)F(cid:107)∞
≤ 72
η (1 − e−ν)d

l

.

hence, whenever

√
72

2|g|

η(1−e−ν)d < 1, we have that

|g|θ < 1.

35

(2.7.7)

(2.7.8)

(2.7.9)

(2.7.10)

Moreover, with the choice

we have

(cid:88)

n

D1 :=

W (n)˜r(m, n)

√
D1 ≤ 72

2(cid:107)F(cid:107)∞
η (1 − e−ν)d

.

(2.7.11)

(2.7.12)

After conditions (2.7.7), (2.7.10) and (2.7.12) have been verified, the general result [6,

Theorem 9.2] applies, yielding

eν|n−m|(cid:12)(cid:12)(cid:12) ∂V (n)

∂ω(m)

(cid:88)

m

Differentiating (2.7.4) with respect to ω(l),

(cid:12)(cid:12)(cid:12) <
(cid:18) ∂

λD1
1 − gθ

:= C1(d,(cid:107)F(cid:107)∞, λ, g, η, ν).

(2.7.13)

∂2

∂ω(m)∂ω(l)

H − z

1

(cid:18) ∂

∂ω(m)
1

+ g

= −λ

1

H − z
∂V

+ g

(cid:18) ∂
(cid:19)

∂ω(l)

∂ω(l)

H − z

Pm

(cid:19) ∂V

∂ω(m)
1

H − z

1

(cid:19)

+ g

H − z
1

H − z

− λ

∂ω(l)

1

H − z
1

H − z

1

H − z
∂2V

(cid:18) ∂

1

(cid:19)

H − z
1

∂ω(m)∂ω(l)

Pm

∂ω(l)

H − z

Repeating the previous argument and using the established decay of ∂V (n)

∂ω(m) we reach

(2.7.2), finishing the proof.

Given a finite set Λ ⊂ Zd, let us define (cid:84) : R|Λ| → R|Λ| by

((cid:84)ω) (n) = ω(n) +

g
λ

Veff (n).

(2.7.14)

Let U (n) := ((cid:84)ω) (n) be the new coordinates in the probability space. The bound (2.7.1)
implies that, for |g| sufficiently small, (cid:84) is a differentiable perturbation of the identity by an
operator with norm less than one hence (cid:84)−1 is well defined.

Fix n0 ∈ Λ and denote by Uα = U + (α − U (n0)) δn0 the new potential obtained from

U by changing its value at n0 to α ∈ R. Let ωα(n) =(cid:0)(cid:84)−1Uα

(cid:1) (n). The variables ωα(n)

correspond to the change in ω(n) when a resampling argument is applied to the new proba-

bility space at the point n0. Intuitively, the exponential decay shown above guarantees that

this change is not too large if n and n0 are far away. This is the content of the result below.

36

Lemma 2.9. Given α ∈ R and |g| < λC−1

(cid:88)

n(cid:54)=n0

eν|n−n0|(cid:12)(cid:12)ωα(n) − ω(n)(cid:12)(cid:12) ≤ C1|g|

1 , we have

λ

(cid:16)|α − U (n0)| + 2

(cid:16)

1 − |g|

λ C1

(cid:17)

.

|g|(cid:107)F(cid:107)∞

λ

(cid:17)

where C1 is the upper bound in equation (2.7.1).
Proof. Using the given definitions and the mean value inequality we obtain, for n (cid:54)= n0,

λ

|V (n, ω) − V (n, ωα)|

|ω(n) − ωα(n)| ≤ |g|
(cid:12)(cid:12)(cid:12) ∂Veff (n, ˆωα)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ωα(l) − ω(l)
(cid:12)(cid:12)(cid:12)
(cid:88)
≤ |g|
(cid:12)(cid:12)(cid:12)(cid:18)
(cid:12)(cid:12)(cid:12)∂Veff (n, ˆωα)
≤ |g|
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ωα(l) − ω(l)
(cid:12)(cid:12)(cid:12) ∂Veff (n, ˆωα)
(cid:12)(cid:12)(cid:12).
(cid:88)
|g|
λ

|α − U (n0)| + 2

∂ω(n0)

l∈Zd

∂ω(l)

∂ω(l)

λ

λ

l(cid:54)=n0

(cid:19)

+

|g|(cid:107)F(cid:107)∞

λ

Where ˆωα denotes some configuration in the probability space with ˆωα(l) in the interval
connecting ω(l) to ωα(l). Let W (n) = eν|n−n0|. According to (2.7.1),

(cid:88)

l

sup
n

W (n)
W (l)

(cid:12)(cid:12)(cid:12) ∂Veff (n, ˆωα)

∂ω(l)

(cid:12)(cid:12)(cid:12) ≤ sup

n

(cid:88)

l

eν|n−l|(cid:12)(cid:12)(cid:12)∂Veff (n, ˆωα)

∂ω(l)

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

≤ C1.

Once again, the conditions of [6, Theorem 9.2] are satisfied for |g| < λC−1
|g|(cid:107)F(cid:107)∞

eν|n−n0|(cid:12)(cid:12)ωα(n) − ω(n)(cid:12)(cid:12) ≤ C1|g|

(cid:16)|α − U (n0)| + 2

(cid:88)

λ

1

(cid:17)

.

(cid:16)

(cid:17)

λ

therefore

1 − |g|

λ C1

n(cid:54)=n0

Since another application of the mean value theorem gives, after a possible correction on

ˆωα that

(cid:12)(cid:12)(cid:12) ∂Veff (n, ω)

∂ω(m)

− ∂Veff (n, ωα)

∂ω(m)

(cid:12)(cid:12)(cid:12) ≤ (cid:88)

l∈Zd

(cid:12)(cid:12)(cid:12) ∂2Veff (n, ˆωα)

∂ω(l)∂ω(m)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω(l) − ωα(l)

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

37

we obtain that, for any ν(cid:48) ∈ (0, ν), the difference

above by

C2|g|
λ

C1

(cid:16)|α − U (n0)| + 2
(cid:16)
(cid:17)

(cid:17)
|g|(cid:107)F(cid:107)∞
(1 − eν(cid:48)−ν)d

1 − |g|

λ

λ C1

∂ω(m)

(cid:12)(cid:12)(cid:12) is bounded from

∂ω(m) − ∂Veff (n,ωα)

(cid:12)(cid:12)(cid:12) ∂Veff (n,ω)
 e−ν(cid:48)(|m−n|+|n−n0|+|m−n0|).

+ 2(cid:107)F(cid:107)∞

where C2 is the constant in (2.7.2).

In particular, letting ν(cid:48) = ν/2 and with the choices

we have(cid:88)

(m,n)∈Λ×Λ

|(A − B)m,n| ≤ C2|g|2

A =

g
λ

(cid:18) ∂Veff (ni, ωα)

∂ω(nj)

(cid:19)
C1
(cid:19)
(cid:18) ∂Veff (ni,ωα)

λ2

Lemma 2.10. Let A = g
λ
|g|

λ C1 < 1 we have (cid:88)

(m,n)∈Λ×Λ

We summarize the above observation as a lemma.

∂ω(nj)

(cid:19)
(cid:18) ∂Veff (ni, ω)
(cid:17)

|g|(cid:107)F(cid:107)∞
(1 − e−ν/2)3d

λ

, B =

g
λ

|Λ|×|Λ|

(cid:16)|α − U (n0)| + 2
(cid:16)
(cid:17)

1 − |g|

λ C1

|Λ|×|Λ|

 .

2(cid:107)F(cid:107)∞

(1 − e−ν/2)2d

+

(2.7.15)

(cid:18) ∂Veff (ni,ω)

∂ω(nj )

(cid:19)

. Whenever

|Λ|×|Λ|

∂ω(nj )

|Λ|×|Λ|

and B = g
λ

|(A − B)m,n| ≤ |g|2 (C3|α − U (n0)| + C4) .

(2.7.16)

Moreover, the constant C3 can be chosen independent of λ and C4 is proportional to 1
λ.

Finally, we analyze how the effective potential varies with respect to disorder and inter-

action. This will be relevant to the Integrated Density of States regularity with respect to

disorder and interaction strengths developed in chapter four.

Lemma 2.11. For a fixed ω ∈ Ω
|Vλ,g(n) − Vλ(cid:48),g(cid:48)(n)| ≤ C5(d,(cid:107)F(cid:107)∞, g, η, ν, ω)
1 − gC6(d,(cid:107)F(cid:107)∞, g, η, ν)

|λ − λ(cid:48)| + C7(d,(cid:107)F(cid:107)∞, g, η, ν)|g − g(cid:48)|.

(2.7.17)

38

Note when λ (cid:54)= λ(cid:48) the bound depends on ω through the constant C5.

Proof. Let Rλ,g(z) =

H0+λω+gVλ,g−z and Rλ(cid:48),g(cid:48)(z) =

1

1

H0+λ(cid:48)ω+g(cid:48)Vλ(cid:48),g(cid:48)−z

for z = t + iη.

Similarly as in the above proofs, it is immediate to check that

Rλ,g(z) − Rλ(cid:48),g(cid:48)(z) =(λ(cid:48) − λ)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z) + (g(cid:48) − g)Rλ,g(z)Vλ(cid:48),g(cid:48)Rλ(cid:48),g(cid:48)(z)

(cid:16)

(cid:17)

+ gRλ,g(z)

Vλ,g − Vλ(cid:48),g(cid:48)

Rλ(cid:48),g(cid:48)(z).

Replacing z by ¯z and subtracting the resulting equations:

(cid:17)
(cid:0)Rλ,g(z) − Rλ,g(¯z)(cid:1) −(cid:16)
(cid:0)Rλ,g(z) − Rλ,g(¯z)(cid:1)(cid:16)
Rλ(cid:48),g(cid:48)(z) − Rλ(cid:48),g(cid:48)(¯z)
(cid:16)
(cid:17)(cid:16)
(λ(cid:48) − λ)Vω + (g(cid:48) − g)Vλ(cid:48),g(cid:48)
(cid:17)
+ g(cid:0)Rλ,g(z) − Rλ,g(¯z)(cid:1)(cid:16)
(λ(cid:48) − λ)Vω + (g(cid:48) − g)Vλ(cid:48),g(cid:48)
Rλ(cid:48),g(cid:48)(z) − Rλ(cid:48),g(cid:48)(¯z)
(cid:16)
(cid:17)(cid:16)
(cid:17)
Vλ,g − Vλ(cid:48),g(cid:48)
Rλ(cid:48),g(cid:48)(z)
Rλ(cid:48),g(cid:48)(z) − Rλ(cid:48),g(cid:48)(¯z)

Vλ,g − Vλ(cid:48),g(cid:48)

Rλ(cid:48),g(cid:48)(z)

+ gRλ,g(z)

+ Rλ,g(¯z)

(cid:17)

=

.

(cid:17)

Taking matrix elements, multiplying by f (t), integrating with respect to t and taking absolute

values we can read from the representation (2.7.3) that, denoting

Kλ,g(n, l) = |Gλ,g(n, l; z) − Gλ,g(n, l; ¯z)|,

|ω(l)|

(cid:90) ∞
(cid:16)
(cid:90) ∞
(cid:16)|Gλ,g(n, l)|Kλ(cid:48),g(cid:48)(l, n) + Gλ(cid:48),g(cid:48)(n, l)|Kλ,g(l, n)

Kλ,g(n, l)Kλ(cid:48),g(cid:48)(l, n) + |Gλ,g(n, l)| ˜Kλ(cid:48),g(cid:48)(l, n)
(cid:17)

−∞

dt.

−∞

(2.7.18)

(cid:17)

dt

|Vλ,g(n) − Vλ(cid:48),g(cid:48)(n)| ≤
3(cid:107)F(cid:107)∞

|λ − λ(cid:48)|(cid:88)
|g − g(cid:48)|(cid:88)
2π
3(cid:107)F(cid:107)2∞
(cid:90) ∞
(cid:88)

l∈Zd

l∈Zd

2π

+

+ 2g

−∞

l∈Zd

|Gλ,g(n, l)||Vλ,g(l) − Vλ(cid:48),g(cid:48)(l)|Kλ(cid:48),g(cid:48)(l, n) dt.

Using equation (2.7.6) together with [6, Theorem 9.2] we conclude the proof.

39

2.7.1

Improvements

We will now improve upon the previous bounds. More specifically, we need robust estimates

which also reflect the decay of the derivatives of Veff (n) when the local potential ω(n) is large.

The content of this section will be important for a general fluctuation analysis on section 2.8

and for complete localization in the one dimensional setting. Before stating the main result

of the section we start with the following deterministic estimate, which incorporates ideas

from [4, Lemma 3].

Lemma 2.12.

|G(m, l; t + iη)| ≤

√

2(cid:104)l|

1

(H − t)2 + η2|l(cid:105)1/2 e−ν|m−l|

(2.7.19)

Proof. To keep the notation simple, we set t = 0 without loss of generality. Let Hf =
ef He−f where f (n) = ν min{|n − l|, R} for a fixed l ∈ Zd and R > 0. By choosing R
sufficiently large, we may assume that |m − l| < R. We then have
eν|m−l|G(m, l; iη) = (cid:104)m|(Hf − iη)−1|l(cid:105) .

||(Hf − iη)−1(H2 + η2)1/2|| ≤

√

2.

(2.7.20)

We claim that

Indeed,

||(Hf − iη)−1(H2 +

η2
2

)1/2||2 = ||(H2 +
= ||(H2 +

η2
2
η2
2

)1/2(H∗

)1/2

f + iη)−1(Hf − iη)−1(H2 +
η2
2

(H2 +

η2
2
)1/2||

1

f + iη)

)1/2||

where by [4, Eq D.9] (with f replaced by −f ) we have

(Hf − iη)(H∗
(cid:19)
(cid:18)

H2 +

η2
2

(Hf − iη)(H∗

f + iη) ≥ 1

2

showing the claim in (2.7.20). The inequality (2.7.19) will now follow from

|(cid:104)m|(Hf − iη)−1|l(cid:105)| ≤ (cid:107)(Hf − iη)−1(H2 +

)1/2(cid:107)|(H2 +

η2
2

)−1/2δl|

η2
2

√

≤

2(cid:104)l|(H2 +

)−1|l(cid:105)1/2 .

η2
2

40

Lemma 2.13. There exists C7(λ, η, d, g,(cid:107)F(cid:107)∞, ν) > 0 such that, for m (cid:54)= n,

max{|ω(n)|,|ω(m)|}(cid:12)(cid:12)(cid:12) ∂V (n)

∂ω(m)

(cid:12)(cid:12)(cid:12) ≤ C7e−2ν|m−n|

and, for n (cid:54)= n0,

(cid:12)(cid:12)ω(n)(ωα(n) − ω(n))(cid:12)(cid:12) ≤ C7|g|

λ − |g|C1

(cid:32)

|α − U (n0)| + 2

|g|(cid:107)F(cid:107)∞

λ

+

1

(1 − e−ν)d

(2.7.21)

(cid:33)

e−ν|n−n0|.

(2.7.22)

Moreover, whenever

|g|
λ C1 < 1, C7 can be chosen to be uniformly bounded as a function of

the parameters λ and g.

Proof. Recall that U (n) = ω(n) + g

λ Veff (n) denotes the “full” potential at site n. We divide

the proof into two cases.

• Case one: U (n) ≥ 0.

Let us start by noting that lemma 2.7.19 implies that for n, l ∈ Zd

√
|G(n, l; t + iη)G(l, n; t + iη)| dt ≤ 2

From the previous section we already know that

∂V (n)
∂ω(m)

=

r(n, l)

∂V (l)
∂ω(m)

+ λr(m, n)

f (t) dt

(2.7.23)

where f (t) = F+(t + iη) + F−(t − iη) − D ∗ F (t) and

r(m, n) = G(n, m; t + iη)G(m, n; t + iη) − G(n, m; t − iη)G(m, n; t − iη).

Observe that, for z = t + iη and n (cid:54)= m,

λ|U (n)G(n, m; z)G(m, n; z)|

(cid:88)

l

λ|U (n)|
|λU (n) − z|
|z|

(cid:18)

=

≤

1 +

|λU (n) − z|

(cid:19)(cid:88)

l

41

|H0(n, l)G(l, m; z)G(m, n, z)|

|H0(n, l)G(l, m; z)G(m, n, z)|

(cid:90) ∞

−∞

(cid:32)

(cid:90) ∞

−∞

−g

(cid:88)

l

2π
η

e−2ν|n−l|.
(cid:33)

where we made use of the identity

(λU (n) − z)G(n, m; z) = δmn −(cid:88)

Note that if U (n) ≥ 0 and t = Rez < 0, then

l

|z|

|λU (n) − z| ≤ 1.

(cid:90) ∞

Using the fact that tf (t) goes to zero as t → ∞ we conclude that
|U (n)G(n, m; t+iη)G(m, n; t+iη)||f (t)| dt ≤ C(ν, η,(cid:107)F(cid:107)∞)

−∞

H0(n, l)G(l, m; z).

(2.7.24)

(2.7.25)

e−2ν|m−n|. (2.7.26)

λ

Since a similar equation holds with m replaced by l, we can proceed as in the previous
section and, using the exponential decay of ∂V (n)

∂ω(m), conclude the proof.

• Case two: U (n) < 0.

In this case, the argument given above must be modified to take into account that the

inequality

|z|

|λU (n) − z| ≤ 1.

(2.7.27)

is satisfied when t = Re > 0. In this case, the immediate use of (2.7.23) would result
in a problem as tf (t) is unbounded as t → −∞. This can be addressed by observing

that the Fermi-Dirac function F (z) has the following symmetry

1

1 + eβ(z−µ)

= 1 −

1

1 + eβ(−z+µ)

.

Hence we can make use of the representation (2.7.23) corresponding to

−

1

1 + eβ(−z+µ)

:= F∗

µ(z)

(2.7.28)

(2.7.29)

since, for m (cid:54)= n, the constant term does not affect the calculation of ∂V (n)

∂ω(m). Denoting

by

f∗(t) = F∗

+(t + iη) + F∗−(t − iη) − D ∗ F∗(t)

42

we reach

(cid:32)

(cid:90) ∞

−∞

−g

(cid:88)

l

∂V (n)
∂ω(m)

=

r(n, l)

∂V (l)
∂ω(m)

(cid:33)

+ r(m, n)

f∗(t) dt

(2.7.30)

where now tf∗(t) → 0 as t → −∞. Proceeding as in the first case the proof is finished,

showing (2.7.21). Following the proof of lemma 2.9 and using (2.7.21) we conclude

(2.7.22)

Lemma 2.14. Let Λ1 and Λ2 be subsets of Zd with dist(Λ1, Λ2) ≥ r. Let V c
potential defined by

2 be the effective

2 (n) = (cid:104)n|F (HΛc
V c

2)|n(cid:105) for n ∈ Zd

where HΛc

2 denotes the restriction of H to the complement of Λ2.

Then, for any n ∈ Λ1

(cid:12)(cid:12)(cid:12) ∂V (n)

∂ω(m)

(cid:12)(cid:12)(cid:12) ≤ C(η, d, λ, g,(cid:107)F(cid:107)∞, ν)e−ν(|m−n|+r)

− ∂V c

2 (n)
∂ω(m)

(2.7.31)

Proof. The proof follows the same steps as in the previous results. The only modification
which is required comes when comparing the quantities r(m, n) and rΛc

2(m, n) given by

r(m, n) = G(n, m; t + iη)G(m, n; t + iη) − G(n, m; t − iη)G(m, n; t − iη)

rΛc

2(m, n) = GΛc

2(n, m; t + iη)GΛc

2(m, n; t + iη) − GΛc

2(n, m; t − iη)GΛc

2(m, n; t − iη).

We observe that

G(n, m; z)G(m, n; z) − GΛc
G(m, n; z) − GΛc

G(n, m; z)

(cid:16)

2(n, m; z)GΛc

2(m, n; z) =

2(m, n; z)

+

G(n, m; z) − GΛc

2(n, m; z)

GΛc

2(m, n; z).

(cid:17)

Moreover,

G(m, n; z) − GΛc

2(m, n; z) = −λ

G(m, l; z)ω(l)GΛc

2(l, n; z)

l∈Λ2
G(m, l; z) (Veff (l) − V c

+

g
λ

2 (l)) GΛc

2(l, n; z).

(cid:17)

(cid:16)

(cid:88)
(cid:88)

l

43

Since Veff (l)− V c
is now finished using arguments identical to the proof of lemma 2.8 and the improvement on

2 (l) can be expressed in terms of Green’s functions of H and HΛc

2 the proof

lemma 2.13.

2.8 Proof of lemma 2.5

We now verify the existence of the density ρΛ
n0

. Fix Λ ⊂ Zd finite. Recall that we have

defined

U (n, ω) = ω(n) +

g
λ

Veff (n, ω).

(2.8.1)

Until the end of this section we suppress the ω dependence on U (n) and Veff . Note that, for
m, n ∈ Λ,

We have denoted the above change of variables by (cid:84) : R|Λ| → R|Λ|, which reads

∂U (m)
∂ω(n)

= δmn +

g
λ

∂Veff (m)
∂ω(n)

.

(2.8.2)

(cid:84)(ω(n1), ..., ω(n|Λ|)) = (U (n1), ..., U (n|Λ|)).

(2.8.3)

It is now simple compute the joint distribution of the variables {U (n)}

n∈Zd. Using the
n∈Zd have a common density ρ we conclude that for

−1U (nk)

ρ

(cid:84)

dU (n1)...dU (n|Λ|)

∂Veff (ni, (cid:84)−1U )

∂U (nj)

ρ

U (nk) − g
λ

Veff (nk, (cid:84)

(cid:17)
−1U )

dU (nk).

(cid:17)

=

fact that the random variables {ω(n)}
all Borel sets I1, ..., IN in R:

(cid:17) |Λ|(cid:89)

P(cid:16)
U (n1) ∈ I1, ... , U (n|Λ|) ∈ I|Λ|
(cid:90)
(cid:84)−1(cid:16)
(cid:90)
(cid:90)

I1×...×I|Λ|

I1×...×I|Λ|

|Λ|(cid:89)

k=1

| det J

(cid:12)(cid:12) det

(cid:84)−1|
(cid:16)

I +

=

=

k=1

(cid:16)

g
λ

I1×...×I|Λ|

ρ(ω(nk)) dω(nk)

(cid:16)

(cid:17)
(cid:17)(cid:12)(cid:12) |Λ|(cid:89)

k=1

44

Therefore the joint distribution of {U (nk)}|Λ|

k=1 is given by the measure

(cid:16)

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

∂Veff (ni, (cid:84)−1U )

∂U (nj)

g
λ

I +

(cid:16)

(cid:17)(cid:12)(cid:12)(cid:12) |Λ|(cid:89)

k=1

(cid:17)
−1U )

ρ

U (nk) − g
λ

Veff (nk, (cid:84)

dU (n1)...dU (n|Λ|).

(2.8.4)
It follows that for each n0 ∈ Λ the conditional distribution of U (n0) at specified values of
{U (n)}n(cid:54)=n0
(cid:82) ∞

(cid:17)(cid:12)(cid:12)(cid:81)|Λ|
k=1 ρ(cid:0)U (nk) − g
(cid:17)(cid:12)(cid:12)(cid:81)|Λ|
k=1 ρ(cid:0)U α(nk) − g

λ Veff (nk, (cid:84)−1U )(cid:1)
λ Veff (nk, (cid:84)−1U α)(cid:1) dα

∂Veff (ni,(cid:84)−1U )
∂Veff (ni,(cid:84)−1U α)

(cid:16)
(cid:12)(cid:12) det
(cid:16)
(cid:12)(cid:12) det

has a density given by

I + g
λ

I + g
λ

∂U (nj )

ρΛ
n0

−∞

=

∂U (nj )

Where U α(n) := U (n) + (α − U (n0)) δn=n0. This strategy naturally leads to the analysis of

ratios of determinants. A sufficient condition for finding an upper bound to the right-hand

(2.8.5)

side of (2.8.5) is to obtain positive constants C = Cfluct(U (n0)) and D = D(α) which are
independent of |Λ| and such that the following estimates hold true

(cid:90) ∞
−∞ D(α)ρ

where

dπα :=

(cid:16)
(cid:12)(cid:12) det
(cid:16)
(cid:12)(cid:12) det
(cid:16)

α − g
λ

(cid:17)(cid:12)(cid:12)
(cid:17)(cid:12)(cid:12) ≥ D(α).

∂U (nj )

∂U (nj )

I + g
λ

I + g
λ

Veff (n0, (cid:84)

dπα ≥ Cfluct(U (n0))

∂Veff (ni,(cid:84)−1U α)
∂Veff (ni,(cid:84)−1U )
(cid:17)
−1U α)
λ Veff (n, (cid:84)−1U α)(cid:1)
ρ(cid:0)U α(n) − g
(cid:89)
ρ(cid:0)U (n) − g
λ Veff (n, (cid:84)−1U )(cid:1) dα.
λ Veff (n0, (cid:84)−1U )(cid:1)
(u) ≤ ρ(cid:0)u − g

.

n∈|Λ|\{n0}

ρΛ
n0

Cfluct(u)

(2.8.6)

(2.8.7)

(2.8.8)

(2.8.9)

The bounds (2.8.7) and (2.8.6) readily imply that letting U (n0) = u

Lemma 2.5 will follow from a precise control of the right-hand side of equation (2.8.9).

We now execute the strategy which was outlined above. The ratio of determinants can
be controlled through the following bound, where (cid:107)M(cid:107)1 denotes the trace norm of a matrix
M .

45

Lemma 2.15. Let A, B be square matrices with I + B invertible. Then,

(cid:12)(cid:12)(cid:12)(cid:12) det (I + A)

det (I + B)

(cid:12)(cid:12)(cid:12)(cid:12) ≤ e(cid:107)(A−B)(I+B)−1(cid:107)1.

Proof. We make use of the elementary identities

det(I + A)
det(I + B)

= det(I + A)(I + B)−1

and

(I + A)(1 + B)−1 = I + (A − B)(I + B)−1.

The proof is now finished with the inequality

|det(1 + M )| ≤ e(cid:107)M(cid:107)1

which holds in the general setting of trace class operators, see [49, Lemma 3.3]

The triangle inequality for the trace norm implies the following.

(2.8.10)

(2.8.11)

(2.8.12)

Corollary 2.16. Under the above conditions

(cid:12)(cid:12)(cid:12)det(I + B)
(cid:18) ∂V (ni,ω)
(cid:19)

det(I + A)

(cid:12)(cid:12)(cid:12) ≥ e
−(cid:80)

(A−B)(I+B)−1(cid:17)
m,n |(cid:16)
(cid:18) ∂V (ni,ωα)

(cid:19)

|

(2.8.13)

mn

Letting A = g
λ

∂U (nj )
see that, for |g| < λC−1
1 , (1 + B)−1 has uniformly bounded operator norm. Using lemma
2.10 and corollary 2.16 we conclude that (2.8.6) holds with D(α) = e−|g|2C3(|α−U (n0)|+C4).

∂U (nj )

|Λ|×|Λ|

|Λ|×|Λ|

and using lemma 2.8 we

, B = g
λ

We now check that equation (2.8.7) holds when ρ satisfies the fluctuation bound (2.2.4).

We divide the proof in two cases:

• Suppose that c2(ρ) > 0.
Let cρ = max{c1(ρ), c2(ρ)}. The left-hand side of (2.8.7) is bounded from below by
(cid:90) ∞
−∞ D(α)ρ

e−cρ|ω(n)−ωα(n)|(1+|ω(n)|+|ωα(n)|) dα

(cid:17) (cid:89)

α − g
λ

−1U α)

Veff (n0, (cid:84)

(cid:16)

n∈Λ\{n0}

46

(cid:16)

α − g
λ

Veff (n0, (cid:84)

(cid:17)
−1U α)

eS(n) dα

which equals

for

S(n) =

(cid:90) ∞
−∞ D(α)ρ
(cid:88)

n∈Λ\{n0}

−cρ|ω(n) − ωα(n)|(1 + |ω(n)| + |ωα(n)| ).

(2.8.14)

Due to the triangle inequality and lemmas 2.9 and 2.13, we conclude that there is a
positive constant C = C(d,(cid:107)F(cid:107)∞, g, η, ν) with limg→0 C(d,(cid:107)F(cid:107)∞, g, η, ν) < ∞ such
that for n (cid:54)= n0

− cρ|ω(n) − ωα(n)| (1 + |ω(n)| + |ωα(n)|) ≥
− cρ|ω(n) − ωα(n)| (1 + 2|ω(n)| + |ωα(n) − ω(n)|)

C2|α − U (n0)|2 + 2C|α − U (n0)|(cid:17)

≥ −|g|cρe−ν|n−n0|(cid:16)

.

Therefore,

− cρ

− |g|

(cid:88)

n∈Λ\{n0}
2cρ

(1 − e−ν)d

|ω(n) − ωα(n)| (1 + |ω(n)| + |ωα(n)|) ≥

C2|α − U (n0)|2 + 2C|α − U (n0)|(cid:17)

(cid:16)

).

Thus, by choosing |g| sufficiently small so that |g|

(1−e−ν)d C2 < cρ and using the
assumption (A4) we obtain that the integral below is finite and bounded from below

4cρ

by a positive constant independent of Λ and n0.

(cid:90) ∞
−∞ D(α)

ρ(cid:0)α − g
ρ(cid:0)U (n0) − g

λ Veff (n0, (cid:84)−1U α)(cid:1)
λ Veff (n0, (cid:84)−1U )(cid:1) eS(n) dα.

where S(n) was defined in (2.8.14). This, together with (2.8.9), verifies lemma 2.5
(u) ∈

when c2(ρ) > 0. If ρ satisfies the assumption (A5) the above argument yields ρΛ
n0
L1 (R,|u|εdu).

47

• Assume that c2(ρ) = 0:

Similarly to the above argument, the left-hand side of (2.8.7) is bounded from below

by

Where, from (2.2.4)

(cid:90) ∞
−∞ D(α)

−|g|

λ Veff (n0, (cid:84)−1U α)(cid:1)
ρ(cid:0)α − g
ρ(cid:0)U (n0) − g
λ Veff (n0, (cid:84)−1U )(cid:1) e
λ Veff (n0, (cid:84)−1U α)(cid:1)
ρ(cid:0)α − g
ρ(cid:0)U (n0) − g
λ Veff (n0, (cid:84)−1U )(cid:1) ≥ e
λ Veff (n0, (cid:84)−1U α)(cid:1)
ρ(cid:0)α − g
λ Veff (n0, (cid:84)−1U )(cid:1) (cid:89)
ρ(cid:0)U (n0) − g

(cid:18)
−c1(ρ)

(cid:90) ∞
−∞ D(α)

0 <

Again, choosing |g| sufficiently small we conclude that

n∈Λ\{n0}

c1(ρ)

(1−e−ν)d

C|α−U (n0)|

.

(2.8.15)

(cid:19)

.

|g|
λ

(2.8.16)

|α−U (n0)|+2

e−c1(ρ)|ω(n)−ωα(n)| dα < ∞.

finishing the proof.

2.9 The Hartree approximation for the Hubbard model

Let us now explain how the techniques from the previous sections can adapted to the



0

0

H0 + λVω + gV↓(ω)

0

0

H↓(ω)

 :=

 H↑(ω)

Hubbard model. Recall that HHub is defined as

 H0 + λVω + gV↑(ω)
acting on (cid:96)2(cid:16)Zd(cid:17) ⊕ (cid:96)2(cid:16)Zd(cid:17)
is the standard Anderson model acting on (cid:96)2(cid:16)Zd(cid:17)
 V↑(ω)(n)
 =

 (cid:104)n|F (H↓)|n(cid:105)
 .

(cid:104)n|F (H↑)|n(cid:105)

V↓(ω)(n)

. The operators H0 and Vω are defined as before, i.e; H0 + λVω

. The effective potentials are given by

(2.9.1)

Mathematically, the treatment of the above model is very similar to the one explained

above, therefore most details are skipped and we just indicate the required modifications.

48

2.9.1 Existence of the Effective potential

Let Φ(X, Y ) : (cid:96)∞(cid:16)Zd(cid:17) ⊕ (cid:96)∞(cid:16)Zd(cid:17) → (cid:96)∞(cid:16)Zd(cid:17) ⊕ (cid:96)∞(cid:16)Zd(cid:17)

be given by

Φ(X, Y )(m, n) := ((cid:104)n|F (H0 + Vω + gY )|n(cid:105) , (cid:104)m|F (H0 + Vω + gX)|m(cid:105)) .

using proposition 2.6, we immediately reach

|g|

√
72

(cid:96)∞(cid:16)Zd(cid:17)⊕(cid:96)∞(cid:16)Zd(cid:17) ≤
(cid:96)∞(cid:16)Zd(cid:17)(cid:33)
(cid:32)
(cid:107)Φ(X1, Y1) − Φ(X2, Y2)(cid:107)
(cid:16)
1 − eν(cid:48)−ν(cid:17)d
(cid:107)X1 − X2(cid:107)
(cid:16)
1−eν(cid:48)−ν(cid:17)d(cid:107)F(cid:107)∞ < 1 we conclude Φ has a unique fixed point

(cid:96)∞(cid:16)Zd(cid:17) + (cid:107)Y1 − Y2(cid:107)

(cid:107)F(cid:107)∞

2

√

72

2

η

η

Veff =(cid:0)V↑, V↓(cid:1)

.

(2.9.2)

Therefore, if |g|

belonging to (cid:96)∞(cid:16)Zd(cid:17) ⊕ (cid:96)∞(cid:16)Zd(cid:17)

.

2.9.2 Regularity of the effective potential
Fix Λ ⊂ Zd finite and define functions ξ : ((cid:96)∞ (Λ) ⊕ (cid:96)∞ (Λ)) × Rn → (cid:96)∞ (Λ) ⊕ (cid:96)∞ (Λ)

through

ξ↑(V, ω)(j) = V ↑(j) − (cid:104)j|F (H0 + λω + gV↓)|j(cid:105) .
ξ↓(V, ω)(j) = V ↓(j) − (cid:104)j|F (H0 + λω + gV↑)|j(cid:105) .

(2.9.3)

(2.9.4)

Our goal is to conclude V ↑,V ↓ are smooth functions of an arbitrary, but finite, list {ω(j)}j∈Λ.

Again, this can be done via implicit function theorem once we check that the derivative

∂ξ(ω, V )(j)

∂V (l)

= δjl − ∂ (cid:104)j|F (H0 + λω + gV )|j(cid:105)

∂V (l)

.

is non-singular. Using lemma 2.6, we have that for (cid:93) ∈ {↑,↓}

(cid:12)(cid:12)(cid:12) ∂ (cid:104)j|F (H0 + λω + gV(cid:93))|j(cid:105)

∂V (l)

(cid:12)(cid:12)(cid:12) ≤ |g|72

√

2e−2ν|j−l|

η

(cid:107)F(cid:107)∞.

(2.9.5)

(2.9.6)

49

In particular, whenever |g| 144

√
η(1−e−2ν )d < 1 we have that the operator Dξ(·, ω) : (cid:96)∞ (Λ)⊕
(cid:96)∞ (Λ) → (cid:96)∞ (Λ) ⊕ (cid:96)∞ (Λ) has an inverse. From the implicit function theorem it follows
that V is a smooth function of (ω(1), ..., ω(n)) for n = |Λ|.

2(cid:107)F(cid:107)∞

2.9.3 Decay estimates

The decay rate in the case of the Hubbard model is dictated by

∂ω(m)

(cid:12)(cid:12)(cid:12) ∂V↑(n)
(cid:12)(cid:12)(cid:12) ∂V↓(n)

∂ω(m)

(cid:12)(cid:12)(cid:12) ≤ 3|g|(cid:107)F(cid:107)∞(cid:88)
(cid:12)(cid:12)(cid:12) ≤ 3|g|(cid:107)F(cid:107)∞(cid:88)

l

l

∂ω(m)

(cid:12)(cid:12)(cid:12) ∂V↓(l)
(cid:12)(cid:12)(cid:12) ∂V↑(l)

∂ω(m)

(cid:12)(cid:12)(cid:12) + ˜r↓(n).
(cid:12)(cid:12)(cid:12) + ˜r↑(n).

˜K↓(l, m)

˜K↑(l, m)

where, for (cid:93) ∈ {↑,↓}

(2.9.7)

(2.9.8)

˜G(cid:93)(l, m) := G(cid:93)(l, n; t + iη)G(cid:93)(n, l; t + iη) − G(cid:93)(l, n; t − iη)G(cid:93)(n, l; t − iη).

r(cid:93)(m, n) := G(cid:93)(n, m; t + iη)G(cid:93)(m, n; t + iη) − G(cid:93)(n, m; t − iη)G(cid:93)(m, n; t − iη).

˜K(cid:93)(l, m) :=

˜r(cid:93)(n) :=

−∞

| ˜G(cid:93)(l, m)| dt.

(cid:90) ∞
(cid:90) ∞
(cid:16) ˜K↑(l, m) + ˜K↓(l, m)

|r(cid:93)(n)| dt.

−∞

In particular,

(cid:12)(cid:12)(cid:12)∂V↑(n)

∂ω(m)

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

(cid:12)(cid:12)(cid:12)∂V↓(n)

∂ω(m)

(cid:12)(cid:12)(cid:12) ≤ 3|g|(cid:107)F(cid:107)∞(cid:88)
+(cid:0) ˜r↑(n, m) + ˜r↓(n, m)(cid:1) .

l

(cid:17)(cid:18)(cid:12)(cid:12)(cid:12) ∂V↑(l)

∂ω(m)

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

∂ω(m)

(cid:12)(cid:12)(cid:12)(cid:19)
(cid:12)(cid:12)(cid:12) ∂V↓(l)
M11
. The

M21

The analysis from the previous sections applies and we obtain lemmas 2.8,2.9,2.10 and

2.13 with |.| being replaced by the matrix norm |M| = |M11| +|M21| for M =

effective potential and its derivatives are to be interpreted as follows:

50

Recall in this case H0 = −∆ hence, we define HHub acting on(cid:0)(cid:96)2 (Z) ⊕ (cid:96)2 (Z)(cid:1) by

(cid:76)(z) = −E(cid:0)ln|G+(0, 0; z)|(cid:1)
And(0, 0; z)|(cid:1) .
(cid:76)And(z) = −E(cid:0)ln|G+

 H↑(ω)
 HAnd + gV↑(ω)

 :=

HHub =

H↓(ω)

0

0

0

 .

0

(2.10.1)

(2.10.2)

(2.10.3)

(2.10.4)

V

↑
eff (n)
↓
eff (n)
V

 ,

Veff (n) =

∂Veff (n)
∂ω(m)

=

 ∂V

↑
eff
↓
eff

(n)
∂ω(m)
∂V
(n)
∂ω(m)

 and

∂2Veff (n)
∂ω(m)∂ω(l)

=

 ∂2V

↑
eff
↓
eff

(n)
∂ω(m)ω(l)
∂2V
(n)
∂ω(m)ω(l)

 .

2.10 One dimensional Aspects:proof of theorem 2.1

In this section we will prove theorem 2.1. We let H+ = H[0,∞)∩Z and denote by

G+(m, n; z) the Green’s function of H+. Recall the definition of the Lyapunov exponent:

where, denoting by HAnd the standard Anderson model −∆ + Vω on (cid:96)2 (Z),

 H↑(ω)

0

H↓(ω)

0

HAnd + gV↓(ω)

The effective potentials are defined as (2.2.8). In the theorem below, we will use an abbre-

viation and (cid:76)(z) will refer to the Lyapunov exponent of either H↑ or H↓ whereas (cid:76)And(z)
will denote the Lyapunov exponent of the Anderson model on (cid:96)2 (Z).

2.11 One dimensional aspects: strategy of the proof of theorem

2.1

The argument for proving theorem 2.1 follows closely the approach in the proof of theorem

12.11 in [6], which we now recall.

51

2.11.1 Main ideas in the i.i.d case

In the reference [6, Chapter 12] the decay rate for the Green’s function is described in terms

of the moment generating function, defined by

ϕ(s, z) = lim|n|→∞

ln E(|G(0, n; z)|s)

|n|

.

(2.11.1)

The existence of the above quantity for all z ∈ C+ and s ∈ (0, 1) and its relationship to the

Lyapunov exponent are a consequence of Fekete’s lemma:

Lemma 2.17 (Fekete). Let {an}n∈N be a sequence of real numbers such that, for every pair
(m, n) of natural numbers,

an+m ≤ an + am.

(2.11.2)

Then, α = limn→∞ an

n exists and equals infn∈N an
n .

It is an elementary observation that if, instead, the sequence {an}n∈N satisfies the in-
equality an+m ≤ an + am + C then the above result applies to bn := an + C and that an ana-
logue statement holds for superadditive sequences, i.e, sequences which satisfy (2.11.2) but
with the reversed inequality sign. In the i.i.d. context, the sequence an = ln E (|G(0, n; z)|s)
is shown to be both subbaditive and superadditive, meaning that there exist constants
C−(s, z) and C+(s, z) for which

an + am + C− ≤ an+m ≤ an + am + C+.

(2.11.3)

holds for all m, n ∈ N, see [6, Lemma 12.10]. A consequence of this fact, together with a

precise control of the arising constants, is stated below where we operate under the assump-
tion that the random variables {ω(n)}n∈Z have a density ρ which satisfies ρ ∈ L1 (R;|u|εdu)
with ε ∈ (0, 1).

Lemma 2.18. [6, Theorem 12.8] For any z ∈ C+, there are cs(z), Cs(z) ∈ (0,∞) such that
for all n ∈ Z

c−1
s (z)eϕ(s,z)|n| ≤ E (|GAnd(0, n; z)|s) ≤ Cs(z)eϕ(s,z)|n|.

(2.11.4)

52

Moreover, for any compact set K ⊂ R and S ⊂ [−ε, 1), we have the local uniform bound

sup
s∈S

sup

z∈K+i(0,1]

max{cs(z), Cs(z)} < ∞

(2.11.5)

and the same result holds with z replaced by its boundary value E + i0 for Lebesgue almost

every E.

On the other hand, for fixed z ∈ C+, ϕ(s, z) is shown to be a convex function of s and
∂s = −(cid:76)(z). It is a
non-increasing in [−ε, +∞), with its derivative at s = 0 satisfying ∂ϕ(0,z)
consequence of these facts that for almost every E ∈ R there exists a value s = s(E) ∈ (0, 1)

such that

ϕ(s, E) ≤ − s
2

(cid:76)(E).

(2.11.6)

The above is the content of [6, Equation (12.86)]. Dynamical localization is shown to hold

locally as a consequence of the inequality (2.11.3) along with lemma 2.18, the inequality
(2.11.6) and Kotani theory, which establishes that (cid:76)(E) is positive for almost every E ∈ R.

2.11.2 Modifications

In this section we will outline the proof theorem 2.1 with HHub again replaced by the operator

defined in (2.4.1). For simplicity we set λ = 1 since the disorder strenght does

H on (cid:96)2(cid:16)Zd(cid:17)
(cid:96)2(cid:0)Z+(cid:1) and denote by G+(m, n; z) the Green’s function of H+. Recall the definition of the

not play an important role in theorem 2.1. Let H+ = H[0,∞)∩Z be the restriction of H to

Lyapunov exponent: initially, for z ∈ C+, we let

(cid:76)(z) = −E(cid:0)ln|G+(0, 0; z)|(cid:1) .

(2.11.7)

By Herglotz theory (see, for instance, [6, Appendix B] and references therein) it is seen that,

for Lebesgue almost every E ∈ R, (cid:76)(E) is well defined as limδ→0+ (cid:76)(E + iδ). Finally, recall
the uniform positivity of the Lyapunov exponent for the Anderson model on (cid:96)2 (Z):

ess inf

E∈R (cid:76)And(E) > (cid:76)And

(2.11.8)

53

for some (cid:76)And > 0. The first step towards Green’s function decay (2.4.5) will be showing

uniform positivity of (cid:76)(E), which is accomplished by the following.
Theorem 2.19. There exists a constant CLyap(s, η, g,(cid:107)F(cid:107)∞) > 0 such that

|(cid:76)(z) − (cid:76)And(z)| ≤ CLyap|g|s

(2.11.9)

for all z ∈ C+.

Proof. From the resolvent identity we obtain

|G+(0, 0; z)|
|G+

And(0, 0; z)| ≤ 1 + |g|(cid:107)F(cid:107)∞(cid:88)
|G+(0, 0; z)| ≤ 1 + |g|(cid:107)F(cid:107)∞(cid:88)
(cid:32) |G+(0, 0; z)|
(cid:88)

And(0, 0; z)|

(cid:33)

|G+

n

n

ln

|G+

And(0, 0; z)|

s

n

Using the bound ln(1 + x) ≤ xs

And(n, 0; z)|
|G+(0, n; z)||G+
|G+
And(0, 0; z)|
And(0, n; z)||G+(n, 0; z)|
|G+
|G+(0, 0; z)|

(2.11.10)

(2.11.11)

s for 0 < s < 1 and x > 0 we reach, for 0 < s < 1/2,
≤ |g|s

.

(cid:107)F(cid:107)s∞

(2.11.12)

And(n, 0; z)|s
|G+(0, n; z)|s|G+
And(0, 0; z)|s
|G+
(cid:18)|G+

(cid:19)1/2

(n,0;z)|2s
(0,0;z)|2s

And

|G+

And

:= CLyap(s, η, ν,(cid:107)F(cid:107)∞)|g|s.

(2.11.13)

Taking expectations, using the definition of the Lyapunov exponents and the Cauchy-Schwarz

inequality

(cid:76)And(z) − (cid:76)(z) ≤ |g|s

s (cid:107)F(cid:107)s∞ supn E(cid:0)|G+(0, n; z)|2s(cid:1)1/2(cid:80)

E

n

The fact that CLyap is a finite quantity follows from a couple of remarks. Firstly, by Feen-

berg’s expansion [6, Theorem 6.2] we have the identity

|G+

And(n, 0; z)| = |G+

And(0, 0; z)||G+

And(1, n; z)|

(2.11.14)

where G+

And(1, n; z) denotes the Green’s function of HAnd restricted to (cid:96)2 (Z)∩ [1,∞). From

the a-priori fractional moment bound on lemma 2.4 combined with the Green’s function

decay for dimensional Anderson model

E(cid:16)|G+

And(1, n; z)|2s(cid:17)

< C(s)e−µAnd|n|

(2.11.15)

we conclude that that CLyap < ∞. The estimate for (cid:76)(z) − (cid:76)And(z) is similar.

54

In principle one might worry that the pre-factor CLyap on the above bound will depend

on g. However, it is easy to see from the arguments in the proof of lemma 2.4, that CAP
converges to a finite quantity as g → 0, thus we shall disregard its dependence on g.

(cid:19)1/s

(cid:18)

(cid:76)And
CLyap

Corollary 2.20. Whenever |g| <

holds for some s ∈ (0, 1/2), we have

(cid:76)0 := ess inf

E∈R (cid:76)(E) > 0.

(2.11.16)

We can now proceed to the second step of the proof of theorem 2.1, which consists

of establishing Green’s function decay from corollary 2.20. For that purpose, an important
detail to keep in mind is that, in the correlated context, if we choose an = log E (|G(0, n; z)|s),
the condition (2.11.3) will not be fulfilled for all pairs (m, n) due to the lack of independence

between the potentials. This means that Fekete’s lemma is not applicable. Moreover, its

well-studied modifications (for instance by P. Erd¨os and N. G. de Bruijn [24]) do not seem

to suffice either.

To the best of our knowledge the result given below is new. Its formulation takes into

account the strong decorrelation between the potentials in the Hubbard model and introduces

a notion of approximate subbaditivity.

Lemma 2.21 (Fekete-type lemma for approximately subbaditive sequences). Let δ > 0 be
given and {an}n∈N be a sequence of real numbers such that, for every triplet m, n, r of natural
numbers with r ≥ δ max{log m, log n}, the inequality

an+m+r ≤ an + am + C

holds with a constant C independent of m, n and r. Then,

α = lim
n→∞

an
n

exists and equals infn∈N an+C

n

. Moreover, α ∈ [−∞, 0].

Note that, as a consequence, we have

an ≥ nα − C

55

(2.11.17)

(2.11.18)

(2.11.19)

for all n ∈ N, where C is the same constant as in (2.11.17). The following decoupling

estimate guarantees the applicability of the above lemma with the choice

(cid:16)| ˆG(0, n; z)|s(cid:17)

an = log E

[0,n]

where ˆG(0, n; z) = (cid:104)0|(H[0,n] − z)−1|n(cid:105) is the Green’s function of the operator H restricted
to (cid:96)2 ([0, n] ∩ Z) and E

[0,n] denotes the expectation with respect to U (0), ..., U (n).

Lemma 2.22. [Strong mixing decoupling] There exist constants CDec(s, ν, η, g,(cid:107)F(cid:107)∞) and
δ = δ(η, ν, g,(cid:107)F(cid:107)∞) such that the inequality

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≤ CDecE

(cid:16)| ˆG(0, n; z)|s(cid:17) E

(cid:16)| ˆG(0, m; z)|s(cid:17)

[0,n]

[0,m]

E
[0,n+m+r]

holds whenever r ≥ δ log max{m, n}.

(2.11.20)

A combination of lemmas 2.22, 2.21 and equation (2.11.19) yields the lower bound

C−1
Deceϕ(s,z)n ≤ E

[0,n]

for all n ∈ N.

(2.11.21)

(cid:16)| ˆG(0, n; z)|s(cid:17)

As we shall see in section 2.11.5 below, an application of the lower bound (2.11.21) in

combination with the superadditive version of lemma 2.21 applied to the sequence bn =
−ϕ(s, E)n + log E

is enough to establish an upper bound

(cid:16)| ˆG(0, n; z)|s(cid:17)
E(cid:16)| ˆG(0, n; z)|s(cid:17) ≤ C(s, z)eϕ(s,z)n for all n ∈ N.

[0,n]

(2.11.22)

where the constant C(s, z) is locally uniform in (s, z) ∈ (0, 1) × C+.

After obtaining an analogue of lemma 2.18, the final step will be to relate the moment-

generating function to the Lyapunov exponent through an inequality of the type

ϕ(s, E) ≤ − s
2

(cid:76)0.

(2.11.23)

In reference [6], the bound (2.11.23) is stated with (cid:76)0 replaced by (cid:76)(E) and with s depending

on E. However, it is easy to see from the from the arguments given there that s can be

56

chosen locally uniformly in E, see [6, Equations (12.79) and (12.80)]. Moreover, by making

use uniform positivity of the Lyapunov exponent obtained in corollary 2.20 we reach the

inequality (2.11.23). The Green’s function decay follows from the bounds (2.11.22) and

(2.11.23).

2.11.3 Proof of Lemma 2.21

For simplicity we set C = 0. The general statement will follow by considering the related

sequence bn := an + C. Given integers L and (cid:96) with L >> (cid:96), our goal is to bound

above in terms of

a(cid:96)
(cid:96) . As a initial step, observe that by (2.11.17) we have

aL
L from

aL ≤ aL−δ log L−(cid:96) + a(cid:96).

Iterating the above procedure k + 1 times for

k = k(cid:96),L := (cid:98) L − 2(cid:96) − δ log L

δ log L + (cid:96)

(cid:99)

(2.11.24)

we obtain

aL ≤ (k + 2) a(cid:96)

In the above iteration we have made use of the fact that in the assumption (2.11.17) the

remainder r can be adjusted as long as it satisfies the inequality given there. Thus,

aL
L

≤ (cid:96)(k + 2)

L

a(cid:96)
(cid:96)

(2.11.25)

Before proceeding with the proof, a few remarks are in order. Firstly, nothing is achieved
by holding (cid:96) fixed and letting L → ∞ directly on equation (2.11.25) since this only yields
the upper bound of zero. A second attempt would be showing that letting (cid:96) → ∞ (hence,
L → ∞ as well) implies that the ratio k(cid:96)

L converges to one. However, as

q(cid:96),L − (cid:96)
L

≤ k(cid:96)
L

≤ q(cid:96),L

for the choice

q(cid:96),L =

1 − 2 (cid:96)

L − δ log L

L

1 + δ log L

(cid:96)

57

(2.11.26)

(2.11.27)

we see that k(cid:96)

L converges to one as (cid:96) → ∞ only along a subsequence where

→ 0 and

(cid:96)
L

log L

(cid:96)

→ 0.

(2.11.28)

Taking this into account, let ε > 0 be given and (cid:96)1 be an initial scale to be determined. Let

L >> (cid:96)1 be a positive integer to be interpreted as a larger scale. Iterating equation (2.11.25)

throughout a sequence of scales

(cid:96)1 < (cid:96)2 < ... < (cid:96)NL

satisfying, for some p > 0,

p log(cid:0)(cid:96)j

and

≤ L < (cid:96)NL+1 < ...
(cid:1) .
(cid:1) ≤ log (cid:96)j+1 < p2 log(cid:0)(cid:96)j
∞(cid:88)
(cid:33) NL−1(cid:89)

(cid:96)j
(cid:96)j+1

< ∞

(cid:18)

q(cid:96)j ,(cid:96)j+1

+

2(cid:96)j
(cid:96)j+1

j=1

(2.11.29)

(2.11.30)

(2.11.31)

(2.11.32)

(cid:19) a(cid:96)1

(cid:96)1

.

j=1
we reach, for q(cid:96),L defined in (2.11.27),

(cid:32)

≤

aL
L

q(cid:96)NL

,L +

2(cid:96)NL
L

Since q(cid:96)j ,(cid:96)j+1
chosen (independently of L) so that

→ 1 as j → ∞, due to (2.11.31), we conclude that the value of (cid:96)1 can be
NL−1(cid:88)

(cid:33)

(cid:32)

(cid:19)

(cid:18)

log

q(cid:96)j ,(cid:96)j+1

+

2(cid:96)j
(cid:96)j+1

+ log

q(cid:96)NL

,L +

2(cid:96)NL
L

< ε.

(2.11.33)

j=1

Thus

aL
L

≤ eε

a(cid:96)1
(cid:96)1

.

(2.11.34)

Moreover, the above conclusion holds for any integer (cid:96)1 sufficiently large, as long as L >> (cid:96)1.

In particular, we can also require that

Combining equations (2.11.34) and (2.11.35) the proof is finished letting ε → 0.

a(cid:96)1
(cid:96)1

≤ inf

n

an
n

+ ε.

(2.11.35)

58

2.11.4 Proof of lemma 2.22

E
[0,n+m+r]

We will show that the following inequality holds

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≤ CDecE
where CDec = CAPeC(η,g,(cid:107)F(cid:107)∞)e−ν(cid:48)r(m2+n2), E

[0,n]

(cid:16)| ˆG(0, n; z)|s(cid:17) E

[0,m]

(cid:16)| ˆG(0, m; z)|s(cid:17)

(2.11.36)

[0,n] denotes the expectation with respect

to the variables U (0), ..., U (n) and CAP is, up to a multiplication by a constant independent

of m, n and r, the constant obtained on the a priori from lemma 2.4. Denote

ˆω = (ω(1), ..., ω(n + 1), ω(n + r), ..., ω(n + r + m))

(2.11.37)

and let us change variables according to

ˆω (cid:55)→ ˆU .

(2.11.38)

We remark that the variables ω(n + 2), ..., ω(n + r − 1) are fixed in this process.

Note that by lemma 2.4 and a geometric resolvent expansion we have

E(cid:16) ˆG(0, n + m + r; z)|s(cid:17) ≤ CAPE(cid:54)=n+1,n+r

(cid:16)| ˆG(0, n; z)|s| ˆG(n + r + 1, n + r + m; z)|s(cid:17)

.

(2.11.39)
where E(cid:54)=n+1,n+r indicates that the variables U (n + 1) and U (n + r) were integrated out.
Observe that the corresponding Jacobian has the structure

 (cid:65)n×n

(cid:74) =

(cid:66)n×(m+r)

(cid:67)(m+r)×n (cid:68)(m+r)×(m+r)



where

Moreover,

(cid:65)jk = δjk + g

∂Veff (j)
∂U (k)

, (cid:67)jk = g

∂Veff (n + j)

∂U (k)

.

, (cid:66)jk = g

∂Veff (j)
∂U (n + k)

 (cid:73)r×r (cid:81)r×m

(cid:79)m×r (cid:80)m×m



(cid:68) =

59

where (cid:73)r×r is the identity matrix and

(cid:80)jk = δjk + g

∂Veff (n + r + j)
∂U (n + r + k)

, (cid:81)jk = g

∂Veff (n + j)
∂U (n + r + k)

, (cid:79)jk = g

∂Veff (n + r + j)

∂U (n + k)

.

By the Schur complement formula

det (cid:74) = det (cid:65) det (cid:80) det

(cid:73)m+r×m+r − (cid:68)

−1(cid:67)(cid:65)

−1(cid:66)

(cid:16)

(cid:17)

(cid:16)

det

(cid:73)m×m − (cid:80)

−1(cid:79)(cid:81)

(cid:17)

.

(2.11.40)

where, according to the estimate (2.7.1), the matrices (cid:66) and (cid:81) have entries which decay

exponentially away from their lower-left corner. Likewise, the entries of (cid:67), (cid:79) decay expo-

nentially away from their upper-right corner. It readily follows from lemma 2.15 that,

(cid:16)

(cid:73)m+r×m+r − (cid:68)

−1(cid:67)(cid:65)

−1(cid:66)

det

(cid:73)m×m − (cid:80)

−1(cid:79)(cid:81)

det

(cid:17)

(cid:16)

(cid:17) ≤ C(η, ν, g,(cid:107)F(cid:107)∞).

therefore, for C as above,

det (cid:74) ≤ C det (cid:65) det (cid:80).

Let ρ(l) = ρ (U (l) − gVeff (l)). We obtain a decoupling estimate by observing that, setting
U (l) = 0 for l ≥ n + r + 1 would only alter

Veff (j) and

∂Veff (j)
∂U (k)

by at most a factor which decays as C(η, g,(cid:107)F(cid:107)∞)e−ν(r+|j−k|) for 1 ≤ j, k ≤ n. This follows
from the exponential decay on lemmas 2.8, 2.13 and 2.14. Similarly, we can set U (l) = 0 for
l ≤ n and this only changes

by at most a factor which decays as Ce−ν(r+|j−k|) for n + r ≤ j, k ≤ m + n + r. The above

process yields two independent measures

Veff (j) and

∂Veff (j)
∂U (k)

(cid:18)

(cid:19)

(cid:89)

[0,n]

0≤l≤n

dπ0

[0,n] := det0

I + g

∂Veff (j)
∂U (k)

ρ0(l) dU (l)

(2.11.41)

and

dπ0

[n+r+1,n+m+r] := det0

(cid:18)

I + g

∂Veff (j)
∂U (k)

(cid:19)

(cid:89)

[n+r+1,n+m+r]

n+r+1≤n+r+m

60

ρ0(l) dU (l).

(2.11.42)

Using lemma 2.15 we then arrive at an inequality of the type

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≤ CAPeC(η,g,(cid:107)F(cid:107)∞)e−νr(m2+n2)

E
[0,m+n+r]

(cid:90)

| ˆG(0, n; z)|s dπ0

[0,n]

(cid:90)

×

|G+(n + r, n + r + m; z)|s dπ0

[n+r+1,n+m+r].

Rewriting the above conclusion in terms of expectations we obtain

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≤ CDecE

[0,n]

(cid:16)| ˆG(0, n; z)|s(cid:17)|

E
[0,m+n+r]

× E

[n+r+1,n+m+r] (G+(n + r, n + r + m; z)|s) .

where CDec = CAPeC(η,g,(cid:107)F(cid:107)∞)e−νr(m2+n2). At the expense of increasing CAP the above

expectations can be replaced by expectations over the full probability space, which yields

the desired conclusion by translation invariance.

2.11.5 Proof of (2.11.22)

We shall modify the proof of lemma 2.22 to obtain a “super-additive” estimate of the form

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≥ C(s, E)eϕ(s,z)re−C(η,g,(cid:107)F(cid:107)∞)e−ν(cid:48)r(m2+n2)
(cid:16)| ˆG(0, m; z)|s(cid:17)

(cid:16)| ˆG(0, n; z)|s(cid:17) E

×E

[0,n]

[0,m]

E
[0,n+m+r]

where the constant C(s, z) can be chosen locally uniform in z and s ∈ (0, 1). Since the argu-

ment is very similar to the one in the proof of (2.11.36), we only explain the key modification
which consists in obtaining a lower bound for E

as follows.

[n+1,n+r]

(cid:16)| ˆG(n + 1, n + r; z)|s(cid:17)

We start by writing

| ˆG(n + 1, n + r; z)|s = | ˆG(n + 1, n + 1; z)|s| ˆG(n + 2, n + r; z)|s.

(2.11.43)

Using Jensen’s inequality we have that, for any ε ∈ (0, s),

(cid:16)| ˆG(n + 1, n + 1; z)|s(cid:17) ≥ En+1

(cid:16)| ˆG(n + 1, n + 1; z)|−ε(cid:17)−s

ε .

En+1

(2.11.44)

61

where En+1 denotes the conditional expectation with respect to U (n + 1). Making use of
the discrete Riccati equation [6, Proposition 12.1] we obtain

Equations (2.11.43), (2.11.44) and (2.11.45) together with lemma 2.5 yield a lower bound

.

(2.11.45)

(2.11.46)

(2.11.47)

(cid:16)| ˆG(n + 1, n + 1; z)|−ε(cid:17)

En+1

= En+1

(cid:16)| ˆG(n + 1, n + r; z)|s(cid:17) ≥ C(z, s)E

E
[n+1,n+r]

(cid:16)|U (n + 1) − z − ˆG(n + 2, n + 2; z)|ε(cid:17)
(cid:16)| ˆG(n + 2, n + r; z)|s(cid:17)

[n+2,n+r]

which in combination with (2.11.21) implies that, after a suitable adjustment of the constant

C(s, z),

E

[n+1,n+r]

(cid:16)| ˆG(n + 1, n + r; z)|s(cid:17) ≥ C(z, s)eϕ(s,z)r.

Equation (2.11.43) follows from the above inequality combined with a decoupling esti-

mate analogous to the one in the proof of (2.11.36). Again, choosing r comparable to
max{log m, log n} we obtain

(cid:16)| ˆG(0, n + m + r; z)|s(cid:17) ≥ C(s, z)eϕ(s,z)r

E

[0,n+m+r]

(cid:16)| ˆG(0, n; z)|s(cid:17)
(cid:16)| ˆG(0, m; z)|s(cid:17)

E
[0,n]
×E

[0,m]

multiplying both sides of the above inequality by e−ϕ(s,z)(m+n) and taking logarithms we

conclude that the sequence bn = log e−ϕ(s,z)n(cid:16)E

(cid:16)| ˆG(0, n; z)|s(cid:17)(cid:17)

[0,n]

satisfies

bn+m+r ≥ log(C(s, z)) + bn + bm.

(2.11.48)

The bound (2.11.22) now follows from an application of the supperaditive version of lemma

2.21.

2.12 H¨older Continuity for the integrated density of states at weak

interaction

In this section we shall address the problem of H¨older continuity for the integrated density

of states for the Hubbard model with respect to energy, disorder and interaction. Our

62

results follow from modifications of the methods in [29] and references therein after we have

established the existence of a suitable conditional density as in lemma 2.5.

Let’s now prove theorem 2.3, starting from H¨older continuity with respect to energy,

equation (2.2.15). We proceed as in [29, Section 2]. For simplicity, we replace HHub by
H defined in (2.4.1). The arguments given below will be applicable to H↑ and H↓ and,

therefore, suffice to show the same result for HHub.

Fix an energy interval I of length ε > 0 centered at E ∈ R. The idea is to use the H¨older

continuity of N0 and the resolvent identity to reach the following inequality for ε << 1 and
|I| = ε, where we denote by PΛ(I) the spectral projection of HΛ on the interval I.

(1 − o(ε))E (TrPΛ(I)) ≤ C(I, ρ)εα|Λ|.

(2.12.1)

Dividing both sides of (2.12.1) by |Λ| and letting |Λ| → ∞ gives (2.2.15). To obtain (2.12.1)
(cid:16)
we fix an interval J containing I with |J| to be determined. We then write, with P0,Λ(J) =
P

(cid:17)

(J),

HΛ
0

Tr(PΛ(I)) = Tr(PΛ(I)P0,Λ(J)) + Tr(PΛ(I)P0,Λ(J c)).

(2.12.2)

Note

(2.12.3)
The above inequality combined with to the H¨older continuity of N0 with respect to E ∈ R

Tr(PΛ(I)P0,Λ(J)) ≤ Tr(P0,Λ(J)).

|N0(E) − N0(E(cid:48))| ≤ C(I, d)|E − E(cid:48)|α0.

yields, for |Λ| sufficiently large depending only on J,

Tr(PΛ(I)P0,Λ(J)) ≤ C(J, d)|J|α0|Λ|.

(2.12.4)

(2.12.5)

We now estimate the second term on the left-hand side of equation (2.12.2). By the resolvent

identity,

Tr(cid:0)PΛ(I)P0,Λ(J c)(cid:1) = Tr(cid:0)PΛ(I)(H − E)P0,Λ(J c)(H0,Λ − E)−1(cid:1)
PΛ(I)U ΛP0,Λ(J c)(H0,Λ − E)−1(cid:17)

−λTr

(cid:16)

.

(2.12.6)

63

Where we have written U = Vω + g

λ Veff . Moreover, using using functional calculus and that
E is the center of I, we estimate the first term on the left-hand side of equation (2.12.6) by

Now, the second term in equation (2.12.6) can be controlled by means of

(cid:16)

Tr

|J| − |I|Tr(PΛ(I)).

|I|

(PΛ(I))(HΛ − E)P0,Λ(J c)(H0,Λ − E)−1(cid:17) ≤
(cid:16)
PΛ(I)U ΛP0,Λ(J c)(H0,Λ − E)−1(cid:17)
(cid:16)
(HΛ − E)(PΛ(I))U ΛP0,Λ(J c)(H0,Λ − E)−2(cid:17)
U Λ(PΛ(I))U ΛP0,Λ(J c)(H0,Λ − E)−2(cid:17)
(cid:16)

A = −λTr

B = λ2Tr

−λTr

= A + B

.

for

(2.12.7)

(2.12.8)

(2.12.9)

Now, because U Λ is unbounded, we continue a slight modification of the argument in

[29]. The only difference is that we bound term (A) above (which corresponds to [29, (iii)

in equation (2.6)] as

(cid:16)

(HΛ − E)(PΛ(I))U ΛP0,Λ(J c)(H0,Λ − E)−2(cid:17)| ≤

|Tr

(cid:16)

PΛ(I)U Λ(cid:17)|.

|I|

(|J| − |I|)2|Tr

(2.12.10)

At this point, with an estimate analogous to the one in the proof of Proposition 3.2 in [14]

(cid:110)(cid:90) (m+1)ε

(cid:111)|Λ| ε = |I|.

(2.12.11)

we reach

E (|TrPΛ(I)Vω|) ≤ λ−1 sup
(cid:110)(cid:82) (m+1)ε
m∈N

mε

Thus, with M1(ε) := supm∈N

mε

ωjρ(ωj) dωj

λ|Tr(HΛ−E)PΛ(I)U ΛP0,Λ(J c)(HΛ

0 −E)−2| ≤
(cid:110)(cid:82) (m+1)ε
λ2|TrU Λ(PΛ(I))U ΛP0,Λ(J c)(H0,Λ − E)−2| ≤

Similarly, with M2(ε) := supm∈N

mε

ωjρ(ωj) dωj

(cid:111)

,
λ|I|

(|J| − |I|)2 (

(cid:111)

M1(ε)

λ

+

g(cid:107)F(cid:107)∞

λ

)|Λ|. (2.12.12)

(cid:19)

.

(cid:18) M2(ε)

λ

ω2
j ρ(ωj) dωj

, we estimate term (B) through

4λ2

(|J| − |I|)2

|Λ| +

g2
λ2 Tr(PΛ(I))

(2.12.13)

64

Due lemma 2.5 and the Wegner estimate (see [6, theorem 4.1]) we conclude that

Tr(PΛ(I)) ≤ C
λ

|I||Λ|.

(2.12.14)

Choosing the interval J such that |J| = εδ for δ < 1, keeping in mind the assumption
g2 < λ and combining the bounds (2.12.5), (2.12.7), (2.12.12), (2.12.13), (2.12.14) and

optimizing over δ gives δ = 1

2+α0

therefore we reach (2.12.1) for α ∈ [0,

α0
2+α0

] and (2.2.15)

is proven.

To show 2.3 we follow the proof of theorem 1.2 in [29]. We fix λ, λ(cid:48) ∈ J, g and g(cid:48) satisfying
the assumptions of theorem 2.3 and E ∈ I. As explained in [29], using H¨older continuity

with respect to energy given by equation (2.2.15), trace identities and ergodicity of Hλ,g and

TrP0ϕ(Hλ,g)(ϕ(Hλ,g) − ϕ(Hλ(cid:48),g(cid:48)))P0

(cid:17)

where ϕ is a smooth

(2.12.15)

function such that

Hλ(cid:48),g(cid:48), it suffices to estimate E(cid:16)


ϕ ≡ 1 on (−∞, E],
ϕ ≡ 0 on (−∞, E + |λ − λ(cid:48)|δ + |g − g(cid:48)|δ)c,
(cid:107)ϕ(j)(cid:107)∞ ≤ C

(cid:16)|λ − λ(cid:48)|δ + |g − g(cid:48)|δ(cid:17)−j

, j = 1, 2..., 3d + 4

with δ > 0 to be determined. The need for a high regularity of ϕ is due to the fact that

the random potential Vω may be unbounded. Let ˜ϕ be an almost analytic extension of ϕ of

order 3 + 3d. In particular, ˜ϕ is defined in a complex neighborhood of the support of ϕ and

if z = E + iη we have that

|∂z ˜ϕ(z)| ≤ |η|3d+3|ϕ(3d+4)(E)|.

(2.12.16)

By the Helffer-Sj¨ostrand formula, the quantity

(cid:16)

Tr

(cid:17)

P0ϕ(Hλ,g)(ϕ(Hλ,g) − ϕ(Hλ(cid:48),g(cid:48)))P0

65

equals

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)

(cid:16)

λ(cid:48)Uλ(cid:48),g(cid:48) − λUλ,g

(cid:17)

Rλ(cid:48),g(cid:48)(z)P0 d2z =

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)P0 d2z

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)Veff,λ(g)Rλ(cid:48),g(cid:48)(z)P0 d2z

(cid:16)

(cid:17)
Veff,λ(cid:48)(g(cid:48)) − Veff,λ(g)

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)

Rλ(cid:48),g(cid:48)(z)P0 d2z.

(cid:90)

1
π
(λ(cid:48) − λ)

C

(cid:90)

(cid:90)

C
π
(g(cid:48) − g)
g(cid:48)
π

C

π

+

+

(cid:90)

C

Since the last two terms enjoy a better modulus of H¨older continuity (since they do not

involve Vω) and can be treated as in [29], we shall only estimate the first of the above

integrals. By the resolvent identity,

Rλ,g(z)VωRλ(cid:48),g(cid:48)(z) =Rλ,g(z)VωRλ,g(z) + (λ − λ(cid:48))Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)VωRλ,g(z)+

(g − g(cid:48))Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)Veff,λ(cid:48)(g(cid:48))Rλ,g(z)
− gRλ,g(z)VωRλ(cid:48),g(cid:48)(z)(Veff,λ(cid:48)(g(cid:48)) − Veff,λ(g))Rλ,g(z).
(cid:90)

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)P0 d2z

(λ(cid:48) − λ)

π

C

The above considerations lead to a perturbative expansion of

into four terms. We will show below that each of them can be bounded in terms of powers
of either |λ − λ(cid:48)| or |g − g(cid:48)|.

We start by estimating

E

C

π

(cid:90)

(cid:18)(cid:12)(cid:12)(cid:12)(λ − λ(cid:48))2
(cid:18)(cid:12)(cid:12)(cid:12) (λ−λ(cid:48))2

E

π

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)VωRλ,g(z)P0 d2z

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

(2.12.17)

with a slight modification of equation (3.15) in [29] since Vω is unbounded. By the Combes-

Thomas bound, equation (2.12.16) and the choice of ϕ

(cid:82)C ∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)VωRλ,g(z)P0 d2z
≤ C(d)(cid:0)1 + E2(|Vω|)(cid:1)

|λ−λ(cid:48)|2

(|λ−λ(cid:48)|δ+|g−g(cid:48)|δ)3d+4 .

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

66

Similarly,

E

(cid:90)

(cid:18)(cid:12)(cid:12)(cid:12)(λ − λ(cid:48))(g − g(cid:48))

π

C
≤ C(d) (1 + E(|Vω|))

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)Veff,λ(cid:48)(g(cid:48))Rλ(cid:48),g(cid:48)(z)VωRλ,g(z)P0 d2z

|λ − λ(cid:48)||g − g(cid:48)|

(|λ − λ(cid:48)|δ + |g − g(cid:48)|δ)3d+4

.

Moreover, using lemma 2.11 with the the explicit dependence on ω given there, we obtain

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

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

that the expected value of

(cid:90)

(cid:12)(cid:12)(cid:12)g(λ − λ(cid:48))

π

C

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ(cid:48),g(cid:48)(z)(Veff,λ(cid:48)(g(cid:48)) − Veff,λ(g))Rλ,g(z)P0 d2z

is bounded from above by

C(d) (1 + E(|Vω|))

|g||λ − λ(cid:48)|(|g − g(cid:48)| + |λ − λ(cid:48)|)
(|λ − λ(cid:48)|δ + |g − g(cid:48)|δ)3+3d

.

(cid:90)

Using the same arguments as in [29, Equations 3.17 and 3.18] we see that

(cid:12)(cid:12)(cid:12)(λ(cid:48) − λ)

(cid:12)(cid:12)(cid:12)
|λ − λ(cid:48)||E(cid:0)Tr(P0ϕ(Hλ,g)Rλ,g(z)VωRλ,g(z)P0)(cid:1)| ≤ C|λ − λ(cid:48)|E (|Vω|)

∂z ˜ϕ TrP0ϕ(Hλ,g)Rλ,g(z)VωRλ,g(z)P0 d2z

can be bounded from above by

C

π

(|λ − λ(cid:48)|δ + |g − g(cid:48)|δ)

.

(2.12.18)

Finally, we conclude that

|Nλ,g(E) − Nλ(cid:48),g(cid:48)(E)| ≤ C(α0, d, I)m(|λ − λ(cid:48)|)m(|g − g(cid:48)|)

(2.12.19)

where

Choosing δ =

m(x) = xδα + x2−(3d+4)δ + x1−δ.

2

α+3d+4 we obtain, for any β ∈ [0,
|Nλ,g(E) − Nλ(cid:48),g(cid:48)(E)| ≤ C(α0, I)

2α

α+3d+4],

(cid:16)|λ − λ(cid:48)|β + |g − g(cid:48)|β(cid:17)

finishing the proof of theorem 2.3.

67

CHAPTER 3

SPECTRAL AND DYNAMICAL CONTRAST ON HIGHLY CORRELATED

ANDERSON-TYPE MODELS.

3.1 Introduction and Main results

This chapter aims at presenting and analysing examples of random Schr¨odinger operators

where contrasting dynamical and spectral behaviors can be observed.

In comparison to

the well established theory of Anderson localization, discussed below in detail, the systems

studied here will exhibit some form of long range correlations. Depending on their geometry,

the spectral properties of the models can change significantly. More surprisingly, we present

a model which exhibits purely absolutely continuous spectrum but where a ’phase transition’

can still be observed. Such phenomenon can be captured utilizing the notion of transient and

recurrent absolutely continuous spectrum due to Avron and Simon [9], which we shall also

review in the subsequent discussion. Before stating our models and main results precisely,

we shortly discuss the relevant background on correlated models.

3.1.1 Background on Correlated Models

The effects of strong correlations on localization properties of a given lattice were recently

studied in [50] where the authors consider a system consisting of a particle and a 1/2 spin

and the particle flips the spin only when it visits the origin.

Figure 3.1: 1/2 spin and single-particle model

There it is shown that resonant tunneling is compatible with correlated pure point spec-

trum the since model exhibits Green’s function decay on the graph metric, hence pure point

68

spectrum, but has eigenfunctions which are only localized in the particle position [50, The-

orems II.2 and II.5].

3.1.2 Main Results

Our goal in this chapter is to study infinite volume analogues of the model in [50]. These will

be obtained by connecting infinitely many copies of the Anderson model in two ways. We will
present the contrast between two random Schr¨odinger operators HSym,ω = −ASym,γ +Vω and
HDiag,ω = −ADiag,γ + Vω, where the random potential only depends on the first coordinate
and is defined through Vω(n) = ω(n1) for n = (n1, n2). By ASym,γ and ADiag,γ we denote

weighted adjacency operators of graphs which are infinite (in the horizontal and vertical
directions) subsets of Z2 obtained by connecting infinitely many copies of Z along a walk

which is either entirely vertical or alternates between vertical components of fixed length

(cid:96) > 0 and diagonal components.

Figure 3.2: Highly correlated Anderson-type models

(0,0)

(0,0)

The operators HSym,ω and HDiag,ω act in a similar fashion if one fixes n ∈ N and considers
the horizontal layer Z × {n} separately, when the models are easily seen to be distinct
samples of the Anderson model −∆ + Vω acting on (cid:96)2 (Z). However, as we shall make
precise below, by connecting these horizontal components in distinct ways a fundamental

difference is introduced. To state the quantum dynamical contrast, let q > 0 be given. The

69

averaged q-moments of a self-adjoint operator H are defined as

M q

where we recall that the position operator |X| acts as a multiplication operator on (cid:96)2(cid:0)Z2(cid:1)

T (H) =

(3.1.1)

0

−2t
T E(cid:104)0|eitHω|X|qe−itHω|0(cid:105) dt

e

(cid:90) ∞

2
T

via (|X|qφ) (n) := |n|qφ(n) and |n| = |n1| + |n2| for n = (n1, n2) ∈ Z2. Our first result
concerns HSym,ω where a combination of the symmetry and localization in the horizontal

direction induces ballistic transport in the vertical direction, up to a logarithmic correction.
For the sake of simplicity, we assume that the random variables {ω(n)}n∈Z are nonnegative
and bounded throughout this chapter so ω(n) ∈ [0, ωmax] for all n ∈ Z2.

3.1.3 Dynamical Contrast between HSym and HDiag

Theorem 3.1. There exists T0 > 0 for which the averaged moments satisfy

(cid:18) T
for all times T ≥ T0 and some positive constant Cδ0

(cid:1) ≥ Cδ0

(cid:0)HSym

M q
T

(cid:19)q

log T
which depends on (cid:107)ρ(cid:107)∞.

(3.1.2)

Our second result concerns HDiag,ω,γ = −ADiag,γ + Vω. Inspired by the analysis of [40]

(cid:1) cannot grow faster than a logarithmic power of T , which is an

(cid:0)HDiag,γ

we show that M q
T

analogue of [40, Theorem 1].

Theorem 3.2. There are γ0 > 0 and T0 > 0 such that whenever γ < γ0, the averaged
moments of the operator HDiag,ω,γ = −ASym,γ + Vω satisfy

(cid:0)HDiag,γ

(cid:1) ≤ (log T )q + C (q)

(3.1.3)

M q
T

for all times T ≥ T0.

We remark that γ0 will be explicit in the proof. It depends on (cid:96) and the decay rate of

the Green’s function of the one-dimensional Anderson model.

70

3.1.4 Spectral contrast between HSym and HDiag

The following is a simple consequence of theorem 3.2. Its proof and the relevant definitions

of packing measure and dimension will be given in section 3.6.1.

Corollary 3.3. Whenever γ < γ0, the packing dimension dim+

P (HDiag,ω) vanishes almost

surely.

The spectral contrast between the two models will be evident from the result below.

(cid:0)HSym

(cid:1) is purely absolutely con-

Theorem 3.4. For every ω ∈ Ω the spectral measure µδ0

tinuous and supported on a set of Lebesgue measure 4γ.

(cid:0)HSym

An interesting feature of theorem 3.4 is that, for small values of the parameter γ, the

support of µδ0
since the later contains [0, 2 + ωmax]. As we shall see below, this phenomenon is linked to

(cid:1) has Lebesgue measure much smaller than the spectrum of HSym,
(cid:1) inside [0, 2 + ωmax] is recurrent. To prove theorem 3.4

the fact that the portion of σ(cid:0)HSym

we have shown the following abstract result which, to the best of our knowledge, is new.

(cid:82)

Proposition 3.5. Let µ be a finite Borel measure which is purely singular and F (z) =

1

u−z dµ(u) its Borel transform, defined whenever z ∈ C+. Then, the limit F (E + i0) =
limδ→0+ F (E + iδ) exists for Lebesgue almost every E and

(cid:12)(cid:12){E ∈ R : α < E + F (E + i0) < β}(cid:12)(cid:12) = β − α.

(3.1.4)

As we could not find a reference in the literature with the exact statement needed, the

details of the argument are carried out in section 3.7. Proposition 3.5 is liked to a beautiful

equality by Boole [10], although with a slightly different statement. Boole’s equality [10]

and its extensions have been rediscovered or studied in various contexts by different authors

([45], [57] [20],[21],[36],[22],[52]). For the sake of completeness, we state it below

Proposition 3.6. Under the above assumptions

(cid:12)(cid:12){E ∈ R : F (E + i0) > t}(cid:12)(cid:12) =

C
t

.

(3.1.5)

71

Further historical notes on Boole’s equality can be found in [55, Chapter 5] and [6,

Chapter 8].

3.1.5 Phase transition within σ(cid:0)HSym

(cid:1)

Our next result sheds light on theorem 3.4 and provides further information on the dynamics

−itHSym. Before proceeding with the statement we recall that the notions of transient and

e

recurrent subspaces where given in the introduction of the thesis. The main point of theorem

3.7 below is that the transient and recurrent spectrum can naturally arise and coexist in a

situation of physical relevance.

Theorem 3.7. For all values of γ the spectrum of HSym is a random set with

[−2, 2 + ωmax] ⊂ σ(cid:0)HSym
(cid:1) \ [−2, 2 + ωmax) = σtac(cid:0)HSym

(cid:1) ⊂ [−2 − 2γ, 2 + ωmax + 2γ].
(cid:1) and σ(cid:0)HSym

(3.1.6)

(cid:1) ∩ [−2, 2 + ωmax) =

Moreover, σ(cid:0)HSym
σrac(cid:0)HSym

(cid:1).

3.1.6 Definition of the Models

We are going to define the graphs GSym and GDiag by specifying the non-zero matrix el-
ements of their adjacency operators on the basis {δn}n∈Z2.
In both cases the adjacency
operators are symmetric. Let Z+ = N ∪ {0} and IX be the indicator function of a set X.
The adjacency operator ASym,γ is defined by



ASym,γ(x, y) =

γ if x = (0, n2) and y ∈ {(0, n2 − 1)I{n2>0}, (0, n2 + 1)}

(3.1.7)

1 if x = (n1, n2) and y ∈ {(n1 − 1, n2)I{n1>0}, (n1 + 1, n2)}

where n1, n2 ∈ Z+ in the above definition.

In other words, two vertices x and y satisfying d(x, y) = 1 are connected only if their

second coordinate is the same or their first coordinate is zero. The constant γ > 0 is the

72

hopping strength on the vertical direction and it will play a key role later in this chapter.

On the other hand, the non-zero matrix elements of ADiag,γ are defined via



ADiag,γ(x, y) =

γ if x = (n1, (n1 + 1)(cid:96) − 1) and y = (n1 + 1, (n1 + 1)(cid:96)) , n1 ∈ Z+

1 if x = (n1, n2) and y ∈ {(n1 − 1, n2)I, (n1 + 1, n2), (n1, n2 + 1)I(cid:96)

(3.1.8)

where I = I{n1>0} and I(cid:96) = I{n1(cid:96)≤n2<(n1+1)(cid:96)} for n1, n2 ∈ Z+.

Note that ADiag,γ is obtained by modifying ASym,γ at points whose second coordinate is

a multiple of the fixed length L > 0.

3.2 Lower Bound on the Averaged Moments for HSym: Proof of

Theorem 3.1

The proof of Theorem 3.1 consists of rigorously implementing the following ideas. Firstly,

the fact that HSym exhibits some form of transport has its roots on the symmetry of the

model. We shall prove that the spectral measure µδ(0,0)
tinuous and this implies diffusion by the Guarnieri bound, which we shall revisit below.

(cid:0)HSym

(cid:1) is purely absolutely con-

Secondly, the disordered nature of the system and its connection to the Anderson model

allow us to show a horizontal localization result. In other words, transport can only occur

in the vertical direction. In particular, this will imply a rate of wavepacket spreading which

is faster than what one would expected generically. A careful analysis will then imply the

almost ballistic lower bound.

The Guarnieri bound states that if the spectral measure µψ (H) of a self-adjoint operator

H is α-H¨older continuous then

T (H) ≥ CψT
M q

αq
2 .

(3.2.1)
Recall that µ is said to be α-H¨older continuous if there exists a constant C < ∞ such
that for all intervals I with |I| < 1 we have µ(I) ≤ C|I|α. In particular, if µψ is purely
q
absolutely continuous we can directly reach M q
2 . However, the proof of (3.2.1)

T (H) ≥ CψT

73

can be adapted to incorporate the improvements due to the disorder which are specific to

our context, therefore we reproduce the main arguments below. The starting point is the

following estimate on the averaged quantum dynamics.

Theorem 3.8. (Strichartz-Last) Let H be a self-adjoint operator on a Hilbert space (cid:72) and

assume the spectral measure of H with respect to ψ is uniformly α-H¨older continuous for
some α ∈ (0, 1). Then, there exists a constant Cψ < ∞ such that for all φ ∈ (cid:72) and all
T > 0

(cid:90) T

0

1
T

|(cid:104)φ|eitH|ψ(cid:105)|2 ≤ Cψ(cid:107)φ(cid:107)2

T α

.

(3.2.2)

The above result, which is found in [6, Theorem 2.3], can be used to reach (3.2.1) as

follows. For simplicity, let HSym = H throughout this section. We firstly rewrite M q

T (H) as

(cid:88)

n

2
T

|n|q

(cid:90) ∞

0

T |(cid:104)n|e−itHω|0(cid:105)|2 dt ≥ 2e−2
−2t

T

e

|n|>N

(cid:90) T
N q (cid:88)
1 − (cid:88)
(cid:18)

0

|n|≤N

1
T

1 − N (N + 1)

2

0

Cδ0
T α

|(cid:104)n|e−itHω|0(cid:105)|2 dt
(cid:90) T

|(cid:104)n|e−itHω|0(cid:105)|2 dt
(cid:19)



= 2e−2N q

≥ 2e−2N q
≥ Cδ0

qα
2

T

where on the last step we choose N 2 comparable to T α and the constant Cδ0

was adjusted.

In our particular setup, one can improve upon the Guarnieri bound using the localization

(cid:90) T

1
T

(cid:88)

statement for the horizontal direction in lemma 3.11 which combined with theorem 3.8 yields

E|(cid:104)n|e−itHω|0(cid:105)|2 dt ≤ (cid:88)

2−s |n1|(cid:111)
(cid:17) 1
(cid:19)
2−s . Thus, it is
where C = C(γ, ωmax,(cid:107)ρ(cid:107)∞, s) = 2
natural to split the above sum into two contributions according to |n1| ≤ 2−s
µAnd

(2γ)s(cid:107)ρ(cid:107)∞ (4 + 4γ + ωmax) CAnd(s)

(cid:110)Cδ0

− µAnd

|n|≤N

|n|≤N

0

(3.2.3)

(cid:18)

min

, Ce

(cid:16)

T

π

.

log

CT
Cδ0

,

74

in which case min

T , Ce

(cid:110) Cδ0
2−s |n1|(cid:111)
(cid:16)

C, s, Cδ0

− µAnd

− µAnd

2−s |n1|(cid:111)
(cid:17)

− µAnd

=
2−s |n1|

.

= Ce

, µAnd

(cid:110) Cδ0

min

T , Ce
For T > T0

T , and |n1| > 2−s
Cδ0
µAnd

log

(cid:19)

(cid:18)

CT
Cδ0

which implies

the first contribution can be estimated from above by

Cδ0
T

(cid:93){n ∈ GSym : |n| < N ,|n1| < 2 log T}.

(3.2.4)

where N must be chosen bigger than 2 log T , in which case there are two possibilities for n
to be in the above set. The first one is that n2 ≤ N − 2 log T , when the condition |n| < N is
automatically satisfied whenever |n1| < 2 log T . Secondly, if (cid:98)N − 2 log T + j(cid:99) = n2 for some
j ∈ {1, ...,(cid:98)2 log T(cid:99)} then |n| < N is only satisfied when |n1| < (cid:98)2 log T(cid:99)− j. This reasoning
immediately translates into the bound

2 log T(cid:88)

(cid:93){n ∈ GSym : |n| < N ,|n1| < 2 log T} ≤ 2 log T (N − 2 log T ) +

(2 log T − j)

j=1

= 2 log T (N − 2 log T ) + log T (2 log T − 1)
≤ 2 log T (N − log T ) .

Therefore, the first contribution in the splitting of (3.2.3) is bounded from above by

log T (N − log T )

2Cδ0

T

.

The second contribution is estimated by

(cid:88)

|n|<N ,|n1|>

log T

2

2
T

N(cid:88)

− µAnd

2−s |n1| ≤ 2C

Ce

− µAnd
2−s j
e

T

j=(cid:98) log T
2 (cid:99)
2C(N − log T
2 )

=

T

1 − e

)

(N − log T
2

(cid:16)
− µAnd
2−s
1 − e−1

N−(cid:98) log T

2 (cid:99)−1

(3.2.5)

(cid:17)

.

In particular, the choice N =

(cid:16)

log T + η T
log T

(cid:17)

ensures that, by picking η sufficiently

small and eventually changing T0 (but with the same dependence on C, s, Cδ0

, µAnd ), the

75

same argument as in the proof of the Guarnieri bound yields the inequality

(cid:0)HSym

(cid:1) ≥ Cδ0

M q
T

(cid:18) T

(cid:19)q

log T

whenever T > T0. Theorem 3.1 is proven.

3.3 The absolute continuity of µδ(0,0) (HSym)

(cid:0)HSym

(cid:1) follows from recursive

We shall explain how the absolute continuity of µδ(0,0)

relations for the Green’s function which are available due to the symmetry of GSym. For
z ∈ C \ R, let ϕ := (HSym − z)−1δ(0,0). By definition of HSym we have

−ϕ(1, 0) − γϕ(0, 1) + (ω(0) − z)ϕ(0, 0) = 1.

(3.3.1)

hence

−GSym ((0, 0), (1, 0); z) − γGSym ((0, 0), (0, 1); z) + (ω(0) − z)GSym ((0, 0), (0, 0); z) − 1 = 0.
(3.3.2)
Thus, denoting by GAnd the Green’s function of the Anderson model on (cid:96)2 (Z), it follows
from Feenberg’s expansion, explained in the introduction of this thesis, that

Sym ((0, 1), (0, 1); z) + (ω(0) − z)−(cid:17) − 1 = 0.

GSym ((0, 0), (0, 0); z)

And(1, 1; z) − γ2G+

(cid:16)−G+

Where in a path expansion of the form G(u, v; z) = G(u, u; z)G+(v, v; z) the term G+(v, v; z)
denotes the Green’s function of the operator HSym restricted to the component of GSym
containing v and obtained from GSym by removing the edge which connects the sites u and
v. By the vertical symmetry of GSym we have that

(3.3.3)

G+

Sym ((0, 1), (0.1); z) = GSym ((0, 0), (0, 0); z)

(3.3.4)

thus we can rewrite equation (3.3.3) as

(cid:16)−G+

GSym ((0, 0), (0, 0); z)

And(1, 1; z) − γ2GSym ((0, 0), (0, 0); z) + (ω(0) − z)

(cid:17) − 1 = 0.

(3.3.5)

76

Therefore, letting w = −ω(0) + z + G+

And(1, 1; z) we have

2γ2GSym ((0, 0), (0, 0); z) = −w +

(3.3.6)

where we have chosen the branch of the square root onto the upper half-plane and the reason

for the positive sign on the second of the above terms is that GSym ((0, 0), (0, 0); z) necessarily

has non-negative imaginary part since HSym is self-adjoint. Equation (3.3.6) implies that

2γGSym ((0, 0), (0, 0); z) = −w + w

.

(3.3.7)

Therefore, using either equation (3.3.6) or (3.3.7) depending on whether |w| is small or
large, we conclude that GSym ((0, 0), (0, 0); z) remains bounded as Imz → 0 for any z ∈ C+
thus the spectral measure µδ(0,0)
proposition B.4 in [6, Appendix B].

(cid:1) is purely absolutely continuous, see, for instance,

(cid:0)HSym

3.4 Floquet Theory for HSym

The vertical symmetry of the graph GSym and the definition of the operator HSym
suggests that the use of a Fourier transform may be helpful when studying the dynamics

(cid:104)

w2 − 4γ2(cid:105)1/2

(cid:19)1/2

(cid:18)

1 − 4γ2
w2

−itHSym. Let us define

e

through

(cid:1) → (cid:96)2 (Z+) ⊗ L2 ([0, π])
(cid:70) : (cid:96)2(cid:0)GSym
(cid:114) 2
∞(cid:88)

π

n2=0

((cid:70)ψ) (n1, p) :=

ψ(n1, n2) sin (p(n2 + 1)) .

(3.4.1)

For simplicity of notation we let ˆψ(n1, p) = (cid:70) (ψ) (n1, p). It is immediate to check that

the following version of Plancherel’s identity is satisfied

(cid:104) ˆϕ, ˆψ(cid:105)(cid:96)2(Z+)⊗L2([0,π]) = (cid:104)ϕ, ψ(cid:105)

(cid:96)2(cid:16)G

Sym

(cid:17).

(3.4.2)

77

Indeed,

(cid:104) ˆϕ, ˆψ(cid:105)(cid:96)2(Z+)⊗L2([0,π]) =

=

×

=

ˆϕ(n1, p) ˆψ(n1, p) dp

ϕ(n1, n2)ψ(n1, n(cid:48)
2)

0

2
π

2
π

n1∈Z+

(cid:90) π
(cid:88)
(cid:88)
(cid:90) π
sin ((n2 + 1)p) sin(cid:0)(n(cid:48)
n1,n2,n(cid:48)
(cid:88)
(cid:96)2(cid:16)G

n1,n2∈Z+

2∈Z+

(cid:17).

0

Sym

ϕ(n1, n2)ψ(n1, n2)

= (cid:104)ϕ, ψ(cid:105)

2 + 1)p(cid:1) dp

A calculation which is almost identical to the one above gives that, for every ψ ∈

(cid:96)2(cid:0)GSym

(cid:1)

where

Using that(cid:8)(cid:113) 2

−1( ˆψ) = ψ.
(cid:90) π

(cid:70)

(cid:114) 2

(cid:70)

(cid:70)

= g.

(cid:16)
(cid:114) 2

π

(cid:17)
−1(g)
(cid:90) π
(cid:16)

0

−1(g)(n1, n2) =

(cid:70)

: n ∈ N(cid:9) is a complete orthonormal system in [0, π] one may also

g(n1, p) sin (p(n2 + 1)) dp.

π

0

π sin(np)

check that for g ∈ (cid:96)2 (Z+) ⊗ L2 ([0, π])

Therefore, (cid:70) is a unitary map with inverse given by

−1(g)(n1, n2) =

(cid:70)

g(n1, p) sin (p(n2 + 1)) dp.

(3.4.3)

(cid:17) ˆψ(n1, p).

From the definition of HSym one readily sees that

(cid:92)HSymψ(n1, p) = −∆ ˆψ(n1, p) +

(3.4.4)
where we have committed a slight abuse of notation by letting ∆ ˆψ(n1, p) = ˆψ(n1 − 1, p) +
ˆψ(n1 + 1, p). In particular, equation (3.4.4) allow us to conclude that HSym is unitarily

ω(n1) − 2γ cos pδn1=0

equivalent to an operator acting on

(cid:96)2 (Z+) ⊗ L2 ([0, π])

78

(3.4.5)

with action on each fiber given by (cid:104)p where (cid:104)p is a rank-one perturbation of the one di-

mensional Anderson model given by (3.4.4). A consequence of this fact is the following

result

Lemma 3.9. δ(0,0) is a cyclic vector for HSym.
Proof. Firstly, we claim that given n1 ∈ N, the vector δ(n1,0) belongs to the cyclic subspace
generated by δ(0,0), denoted henceforth by (cid:72)0. Indeed, by definition of the Fourier transform

(3.4.1),

ˆδ(n1,0) =

(cid:114) 2

π

sin p δn1.

(3.4.6)

By a result of Simon [54] the vector δ0 is a cyclic vector for (cid:104)p thus we conclude from (3.4.6)
that ˆδ(n1,0) belongs to the cyclic subspace, relative to (cid:104)p, generated by ˆδ(0,0). By taking the
inverse Fourier transform, it follows that δ(n1,0) ∈ (cid:72)0. Since

HSymδ(0,0) = −γδ(0,1) + δ(1,0) + ω(0)δ(0,0)

we easily conclude that δ(0,1) ∈ (cid:72)0 as well. Proceeding by induction one shows that
δ(n1,n2) ∈ (cid:72)0 for every pair (n1, n2) ∈ GSym, finishing the proof.
(cid:0)HSym
(cid:1) is absolutely continuous we readily obtain
(cid:1) is purely absolutely continuous.
Corollary 3.10. σ(cid:0)HSym

Since µδ(0,0)

3.4.1 Localization in the horizontal direction

The above considerations are particularly effective to translate dynamical localization for

the one dimensional Anderson model into horizontal localization for HSym. We start with

the following technical result.

Lemma 3.11. Given s ∈ (0, 1) we have, for all m, n ∈ Z+

(cid:32)

E

sup
|f|≤1

|(cid:104)m|f(cid:0)(cid:104)p

(cid:33)
(cid:1)|n(cid:105)|

| ≤ (2γ)

2

2−s ((cid:107)ρ(cid:107)∞ (4 + 4γ + ωmax) CAnd(s))

1
2−s e

− µAnd

2−s |m−n|

.

(3.4.7)

79

The above supremum is taken over all Borel measurable functions bounded by one.

Proof. It suffices to show that for every L ∈ N (3.4.7) holds with (cid:104)p replaced by its restriction
to (cid:96)2 (Z+ ∩ [0, L]), denoted henceforth by (cid:104)L
p .

Let vp = −2γ cos p. From rank-one perturbation formulas (see, for instance, [6, Theorem

5.3] and [3, Equation(A.7)] ) we find that the spectral measure of (cid:104)L

p is given by

dµp,L

m,n(E) = −vpGL

And (m, n; E) δ(Σ(E) − vp) dE

(3.4.8)

(3.4.9)

where

Σ(E) := −

1

GL

And (0, 0; E)

and GL

And(m, n; E) denotes the Green’s function for the Anderson model on (cid:96)2 (Z+ ∩ [0, L]).

Therefore, by general structure of the spectral measures,

(cid:90) (cid:12)(cid:12)(cid:12)GL

|vp|2

(cid:12)(cid:12)(cid:12)2

And (m, n; E)

δ(Σ(E) − vp) dE ≤ 1.

(3.4.10)

A second observation is that the analogue of equation (3.4.8) for m = n reads

In particular

dµp,L

m,m(E) = δ(Σ(E) − vp) dE.
(cid:90) ∞
−∞ δ(Σ(E) − vp) dE = 1.

(3.4.11)

(3.4.12)

Combining equations (3.4.8), (3.4.10) and (3.4.12) with H¨older’s inequality (applied to

the exponents (p, q) = (2 − s, 2−s
that for all intervals I ⊂ R

|vp|sE

(cid:20)

m,n

E(cid:16)(cid:12)(cid:12)(cid:12)µp,L
(cid:12)(cid:12)(cid:12) (I)
(cid:17) ≤
(cid:12)(cid:12)(cid:12) (I)
E(cid:16)(cid:12)(cid:12)(cid:12)µp,L
(cid:17) ≤ (2γ)

m,n

Thus,

1−s)) and Jensen’s inequality for expectations, we conclude

(cid:18)(cid:90)

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

I

And (m, n; E)

(cid:19)(cid:21) 1

2−s

δ(Σ(E) − vp) dE

.

(3.4.13)

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

(cid:18)(cid:90)

I

s

2−s(cid:107)ρ(cid:107) 1
2−s∞

E (|GAnd (m, n; E)|s) dE

(cid:19) 1

2−s

.

(3.4.14)

80

Since the operator (cid:104)p is bounded, with operator norm less or equal than 2 + 2γ + ωmax,

the inequality (3.4.14) suffices to conclude the proof of lemma 3.11. We mention that by

introducing an integrable weight, one could also handle the case of unbounded operators.

For further details we refer to [3, Equations (A.13)-(A.18)].

The localization statement for (cid:104)p described on Lemma 3.11 can be immediately translated

into a (horizontal) localization statement for HSym with the use of Plancherel’s identity

(3.4.2) and the fact that

ˆδ(m1,m2)(n1, p) =

The precise statement reads

(cid:114) 2

π

δm1(n1) sin (p(m2 + 1)) .

(3.4.15)

Lemma 3.12.

E

(cid:32)

(cid:12)(cid:12)(cid:104)δm1,m2, f(cid:0)HSym

(cid:1) δn1,n2(cid:105)(cid:12)(cid:12)(cid:33)
(cid:0)4γ2(cid:107)ρ(cid:107)∞ (4 + 4γ + ωmax) CAnd(s)(cid:1) 1

sup
|f|≤1

2−s .

with C = 2
π

≤ Ce

− µAnd

2−s |m1−n1|

.

(3.4.16)

3.4.2 Transient and recurrent components: proof of theorem 3.7

According to [9, Proposition 3.1] in order for an absolutely continuous measure µϕ to be
transient it is necessary that µϕ = f (E) dE where f ∈ C∞ (R). Moreover, if f is also
assumed to be compactly supported this condition also sufficient. Since the operators studied

here are bounded, this distinction will not be relevant to us. Let

We claim that S cannot be the support of a smooth function and this implies that σ(cid:0)HSym

S = supp(µδ(0,0)

) ∩ [−2, 2 + ωmax).

(cid:1)∩

[−2, 2 + ωmax) is recurrent. The claim is a consequence of the lemma below.
Lemma 3.13. Let I ⊂ [−2, 2+ωmax] be a closed interval. Then, I∩Sc has positive Lebesgue
measure.

81

Proof. Let µ+

1 be the spectral measure of the Anderson model on the half line [+1,∞) ∩ Z,
And, associated to δ1. Since, almost surely, H+
And has dense point spectrum in
1 (I) > 0. It follows from [53, Theorem 1.5] (theorem one in

denoted by H+
[−2, 2 + ωmax) we have that µ+
[52], when restated in terms of Borel measures on the real line, would also suffice) that

In particular, by choosing t sufficiently large we may conclude that

lim

And(1, 1; E)| > t}(cid:12)(cid:12) > 0.
t→∞ t(cid:12)(cid:12){E ∈ I : |G+
(cid:12)(cid:12){E ∈ I : |ω(0) − E − G+
And(1, 1; E)| > 2γ}(cid:12)(cid:12) > 0.
(cid:18)

(cid:19)

(3.4.17)

(3.4.18)

since the above set does not belong to supp

µδ(0,0)

(this follows from the formula given

at the end of section 3.3. An alternative argument is provided by equation (3.4.21) below)

the lemma is proven.

To show the remaining portion of theorem 3.7, having a formula for the spectral measure

will be useful. Let HL

µδ(0,0)
explained in the previous section, for any L > 0 we have

Sym denote the restriction of HSym to (cid:96)2 ({0, ..., L} × Z+). As
(cid:17)

(cid:17)

(cid:16)

(cid:16)

(cid:104)δ(0,0), f

HL

Sym

δ(0,0)(cid:105) =

(cid:90) π
(cid:90) π

0

0

2
π
2
π

=

(cid:104)L
p

δ0(cid:105) sin2 p dp.

(cid:90) ∞
(cid:104)δ0, f
−∞ f (E)dµp,L(E) sin2 p dp.

dµp,L(E) = δ (ΣL(E) + 2γ cos p) dE

(3.4.19)

where

and ΣL(E) = −

. In particular,

1
(0,0;E)

GL

And

(cid:16)

(cid:17)

(cid:104)δ(0,0), f

HL

where we have used the fact that, in this case,(cid:82) 1−1 δ(Σ(E)+v) dv holds for E ∈ supp(µδ(0,0)

dE.

Sym

(3.4.20)

).

δ(0,0)(cid:105) =

1
πγ

1 − ΣL(E)2
4γ2

Letting L → ∞ in the above equation we conclude that

dµδ(0,0)

(E) =

1
πγ

1 − Σ2(E)

4γ2 dE.

(3.4.21)

(cid:115)

(cid:90)
{|ΣL(E)|<2γ} f (E)
(cid:115)

1{|Σ(E)|<2γ}

82

Since Σ(E) is a smooth function on [−2, 2 + ωmax]c, it follows that σ(cid:0)HSym
such that σ(cid:0)HSym

(cid:1) ∩ [−2, 2 +
(cid:1) ∩ [−2, 2 + ωmax]c (cid:54)= ∅ whenever γ > γω. Indeed, this follows from the

The above analysis also shows that for any ω ∈ Ω there exist a hopping parameter γω

ωmax]c is transient, finishing the proof of theorem 3.7.

formula (3.4.21) since Σ(E) is increasing on [−2, 2 + ωmax]c. The critical values is then
where Ec = 2 + ωmax. For completeness we provide a proof that this number
γω =

|Σ(Ec)|

2

is non-zero almost surely.

Lemma 3.14. The Green’s function for the half-line Anderson model satisfies

−Σ(E) = ω(0) − E − G+

And(1, 1; E).

Moreover, at the edge Ec = 2 + ωmax we have

Σ(Ec) (cid:54)= 0

almost surely.

(3.4.22)

(3.4.23)

Proof. By definition of the half-line Anderson model H+

And, letting ϕ = (H+

And − z)−1δ0 we

obtain

−ϕ(1) + (ω(0) − z) ϕ(0) = 1.

Since ϕ(0) = G+

And(0, 0; z), ϕ(1) = G+

And(0, 1; z) and, by Feenberg’s expansion,

G+
And(0, 1; z) = G+

And(0, 0; z)G+

And(1, 1; z)

and equation (3.4.22) is proven.

For the sake of contradiction, assume that

Then, by (3.4.22)

P (Σ(Ec) = 0) = δ > 0.

P(cid:0)ω(0) − E − G+

And(1, 1; E) = 0(cid:1) = δ > 0

(3.4.24)

And(1, 1; Ec) is independent of ω(0) this would contradict the fact that the distribu-

Since G+
tion of ω(0) has a density. We conclude that P (Σ(Ec) = 0) = 0.

83

3.5 Absence of Diffusion for GDiag,γ: Proof of theorem 3.2

A key step to show absence of diffusion for HDiag is proving that, on expectation, the

fractional moments of the Green’s function of this operator decay exponentially. We start

with a particular case.

Lemma 3.15. Let s ∈ (0, 1) be given. For all m ∈ Z+, r ∈ {0, 1, ..., (cid:96)} and z ∈ C+ we have

(cid:0)|GDiag ((0, 0), (m, m((cid:96) + 1) + r); z)|s(cid:1) ≤ CAP (CAPγs)m .

E
ω(0),...,ω(m)

(3.5.1)

where CAP is the constant in the a priori bound (3.8.4).
Proof. Fix r ∈ {0, 1, ..., (cid:96)}, z ∈ C+ and let xm,r = (m, m((cid:96) + 1) + r). Let us proceed by
induction in m. When m = 0 the statement reduces to the a priori bound verified in the
appendix, equation (3.8.4). Suppose that the desired conclusion holds for some m ∈ Z+ and

recall the factorization

G(cid:0)0, xm+1,r; z(cid:1) = γG(cid:0)0, xm,(cid:96); z(cid:1) G+

(cid:0)xm+1,0, xm+1,r; z(cid:1) .

(3.5.2)

Taking absolute values on both sides of (3.5.2), raising to the power s and taking expec-

Integrating out the variables ω(0), ..., ω(m), using the inductive assumption and the fact that

tations we reach

E(cid:0)|G(cid:0)0, xm+1,r; z(cid:1)|s(cid:1) = γsE(cid:0)|G(cid:0)0, xm,(cid:96); z(cid:1)|s|G+
(cid:0)xm+1,0, xm+1,r; z(cid:1)|s(cid:1) .
(cid:0)xm+1,0, xm+1,r; z(cid:1) only depends on ω(j) for j > m, we obtain from (3.5.3) that
E(cid:0)|G(cid:0)0, xm+1,r; z(cid:1)|s(cid:1) ≤ (CAPγs)m+1 E(cid:0)G+

(cid:0)xm+1,0, xm+1,r; z(cid:1)|s(cid:1) .

G+

(3.5.3)

(3.5.4)

Making use of the a priori bound (3.8.4) one more time finishes the proof.

Making use of the Green’s function decay for the one-dimensional Anderson model we

readily obtain that

84

E (|G ((0, 0), (m + j, m((cid:96) + 1) + r); z)|s) ≤ CAPCAnd (CAPγs)m e−µAndj.

(3.5.5)

Pick γ small enough so that

CAPγs < e−µAnd((cid:96)+2).

(3.5.6)

Then, (3.5.5) implies
Lemma 3.16. For any s ∈ (0, 1) there exist a constant CDiag such that for all n ∈ GDiag
and z ∈ C \ R

E(cid:0)|GDiag (0, n; z)|s(cid:1) ≤ CDiage

−µDiag|n|

(3.5.7)

with CDiag = eµAnd(cid:96)CAPCAnd and µDiag = µAnd.
Proof. The above argument immediately yields the desired conclusion for z ∈ C+. By
symmetry, reasoning along the same lines we obtain (3.5.7) with |GDiag (0, n; z)| replaced by
|GDiag (n, 0; z)| and z ∈ C+. Since GDiag (n, 0; z) = GDiag (0, n; z) the proof is finished.

3.6 Upper bounds for quantum dynamics revisited

We are now ready to prove theorem 3.2. Our approach is a combination of the fractional

moment method and techniques used by Jitomirskaya and Schulz-Baldes in [40]. Let q > 0

and, to simplify the notation, let H = HDiag throughout this section. Recall that

−2t
T E(cid:104)0|eitHω|X|qe−itHω|0(cid:105) dt.

e

(3.6.1)

(cid:90) ∞

0

M q

T =

2
T

The main advantage of working with time-averaged moments is that they can be directly

related to the Green’s function via the following formula.

M q

T =

1
πT

|n|q

E|G(n, 0; E +

)|2 dE.

i
T

(3.6.2)

R

To verify that (3.6.2) holds, following the proof of lemma 3.2 in [42], one can observe that

(cid:90)

(cid:88)
(cid:90) ∞

n∈Z

−2t
T |(cid:104)0|eitHω|n(cid:105)|2 dt

e

0

85

is the L2 norm of the function

− t
T

g(t) = e

restricted to the positive real line. Let

(cid:90)

R

eiE(cid:48)t dµ0,n(E(cid:48))
(cid:90)

e−iEtg(t) dt

ˆg(E) =

1√
2π

R

(cid:90)

be the Fourier transform of g. It is easy to check that

dµ0,n(E(cid:48))
E(cid:48) − E + i
T
By definition of the spectral measure µ0,n we discover that

i√
2π

ˆg(E) =

R

(3.6.3)

.

(3.6.4)

ˆg(E) =

i√
2π

G(0, n; E − i
T

).

(3.6.5)

Equation (3.6.2) then readily follows from Plancherel’s identity.

Returning to the proof theorem 3.2, another useful observation is that, as in to [40,

Proposition 1], it suffices to show the desired upper bound for the related quantity

M q

T (E0, E1) =

whenever α > 1 and σ(cid:0)HDiag

(cid:90) E1

E0

1
πT

(cid:88)
(cid:1) ⊂ (E0, E1).

|n|<T α

|n|q

E|G(n, 0; E +

)|2 dE

i
T

To this end, we shall combine the estimate |G(n, 0; E + i

T )| ≤ T to the fractional moment

bound showed in the previous section to reach

E|G(n, 0; E +

Thus, for any N ∈ N we have that

(cid:88)

|n|>N

E|G(n, 0; E +

i
T

i
T

)|2 ≤ CDiage

−µDiag|n|

T 2−s.

)|2 ≤ CDiagT 2−s (cid:88)

|n|>N

−µDiag|n|
e

.

(3.6.6)

By following exactly the same steps in [40] we can now finish the proof. Namely, we start

with the general result

86

|n|q 1
πT

E|G(n, 0; E +

)|2 dE ≤ N q.

i
T

Lemma 3.17. ([40, Lemma 5])(cid:88)

(cid:90) E1

E0
applied with N = (cid:98)log T(cid:99) to obtain

|n|≤N

(cid:88)

|n|≤N

(cid:90) E1

E0

|n|q 1
πT

E|G(n, 0; E +

i
T

)|2 dE ≤ (log T )q.

(3.6.7)

On the other hand, using equation (3.6.6) followed by the explicit bound on the exponentially
decaying sum [40, Lemma 2] we obtain, with E+ := E1 − E0

(cid:88)

|n|>log T

(cid:90) E1

E0

1
πT

)|2 dE ≤ CDiagE+T 1−s (cid:88)

E|G(n, 0; E +

i
T

|n|qe

−µDiagN

|n|>log T
−µDiag(log T )

((log T ) + (µDiag)−1)q+1

≤ CDiagE+T 1−se
× 2(q + 1)!
µDiag
≤ C(s, q).

whenever s is chosen so that 1 − s < µDiag (we remark that this choice is possible since
µDiag = µAnd and the later can be taken proportional to s, see [6, Theorem 12.11 and

Equation (12.86)] ). Combining the above estimate with (3.6.7) we finish the proof of theorem

3.2

3.6.1 Paking Dimension: Proof of Corollary 3.3

Before proving corollary 3.3, recall that the upper packing dimension of a Borel probability

measure µ is defined as

where, for E ∈ supp µ,

dim+

P (µ) := µ − ess sup
E∈R

dµ(E)

(3.6.8)

dµ(E) := lim sup
ε→0

log (µ[E − ε, E + ε])

log(2ε)

.

(3.6.9)

87

If E /∈ supp µ we set dµ(E) = ∞. We shall denote by dim+
dim+

P (HDiag,ω) the packing dimension
is the spectral measure associated to HDiag,ω and the vector

P (µHDiag

) where µHDiag

δ(0,0).
Proof. (of corollary 3.3) From [35, Theorem 1] we have that, for each ω ∈ Ω,

dimP (HDiag,ω) ≤ lim sup
T→∞

log(cid:0)M q

T (HDiag,ω)(cid:1)

.

q log T

Taking expectations and a subsequence Tk → ∞ which realizes the lim sup we obtain, by
Fatou’s lemma and Jensen’s inequality,

E(cid:0)dimP (HDiag)(cid:1) ≤ lim sup

Tk→∞

log E(cid:16)

(cid:17)

M q
Tk

(HDiag,ω)

q log Tk

where, by theorem 3.2, the limit in the right-hand side equals zero.

3.7 A Version of Boole’s Equality for level sets of Heglotz func-

tions: proof of proposition 3.5

Consider F (E) =(cid:80)N

pn

un−E for real numbers {un}N

n=1,{pn}N

n=1. This is the specific

n=1

form taken by the diagonal elements of the Green’s function in finite volume. For a real

number α, let Qα be a polynomial of degree N + 1 given by

Qα(E) = (α − E − F (E))

(E − un).

Qα. Therefore, the coefficient of EN in the expansion Qα(E) = −(cid:81)N +1
n=1 vn. On the other hand, by definition of Qα, this coefficient must be α +(cid:80)N +1
(cid:80)N +1

Note that the solutions v1, ..., vN +1 of the equation E + F (E) = α coincide the roots of
n=1 (E − vn) equals
n=1 un,

thus

Replacing α by β we conclude the solutions w1, ..., wN +1 of E + F (E) = β satisfy

N(cid:89)

n=1

N(cid:88)

N(cid:88)

N +1(cid:88)

N +1(cid:88)

vn = α +

un.

n=1

n=1

wn = β +

un.

n=1

n=1

88

(3.7.1)

(3.7.2)

The set {E ∈ R : α ≤ E +F (E) ≤ β} is a disjoint union of intervals ∪N +1

n=1 [vn, wn] therefore,

we may conclude from equations (3.7.1) and (3.7.2) that

(cid:12)(cid:12){E ∈ R : α ≤ E + F (E) ≤ β}(cid:12)(cid:12) =

N +1(cid:88)

(wn − vn)

n=1

= β − α.

The following argument is inspired by the analysis in [6, Proposition 8.2] and provides a

proof which is also valid in the infinite-volume context.

Proof. Since µ is assumed to be purely singular, the boundary values F (E + i0) are real
numbers for almost every E ∈ R (see [6, Proposition B3]). Note that the indicator function
1{E : α < E + F (E + i0) < β} can be represented as

φα,β (E+F (E+i0))

for

π

φα,β(z) = ImLog(z − β) − ImLog(z − α)

(3.7.3)

and where Log denotes the principal branch of the logarithm. Let

ψη(E) =

η2

E2 + η2 .

We then have, by dominated convergence and definition of φα,β,

|{E ∈ R : α ≤ E + F (E + i0) < β}| = lim
η→∞

(cid:90) ∞
−∞ π−1ψη(E)φα,β(E + F (E + i0)) dE.

Using dominated convergence one more time and recovering φα,β from its boundary values,

(cid:90) ∞
−∞ π−1ψη(E)φα,β(E + F (E + i0)) dE = lim
η→∞
ε→0+

lim
η→∞

(cid:90) ∞
−∞ π−1ψη(E)
× φα,β(E + F (E + iε)) dE

η

ηφα,β(iη + F (iη + iε))

= lim
η→∞
ε→0+

= lim

η→∞ ηφα,β(iη + F (iη)).

89

On the other hand, by definition of φα,β we know

φα,β(iη + F (iη)) = lim

η→∞ Im

1

E − iη − F (iη)

dE

(3.7.4)

(cid:90) β

α

hence

lim
η→∞ ηφα,β(iη + F (iη)) = lim

η→∞ Im

η

E − iη − F (iη)

dE

η2 + ηImF (iη)

(E − ReF (iη))2 + (η + ImF (iη))2 dE

(cid:90) β
(cid:90) β

α

= lim
η→∞
= β − α

α

where we have made use of the simple fact that limη→∞ F (iη) = 0 and limη→∞ ηImF (iη) =
µ(R).

3.8 A priori bounds on the Green’s function

To obtain a-priori bounds for the Green’s function, we will make use of the weak L1

bound [2, Lemma 3.1]. This bound is valid for any maximally dissipative operator A on a
separable Hilbert space (cid:72) and Hilbert-Schmidt operators M1 : (cid:72) → (cid:72)1 and M2 : (cid:72)1 → (cid:72)
where (cid:72)1 is another separable Hilbert space. Recall that a densely defined operator A is said
to be dissipative if Im(cid:104)ϕ, Aϕ(cid:105) ≥ 0 for every ϕ ∈ D(A). A is said to be maximally dissipative
when it is dissipative and has no proper dissipative extension. Denoting by | · | Lebesgue
measure and by (cid:107) · (cid:107)HS the Hilbert-Schmidt norm, we have

Lemma 3.18.(cid:12)(cid:12)(cid:12){v : (cid:107)M1

M2(cid:107)HS > t}(cid:12)(cid:12)(cid:12) ≤ CW(cid:107)M1(cid:107)HS(cid:107)M2(cid:107)HS

1
t

1

A − v + i0

(3.8.1)

where the constant CW is independent of A,M1 and M2.

For a proof, we refer to [2, Lemma 3.1]. Another fact, which can be found in [2, Propo-

sition 3.2], is

90

Lemma 3.19. Let A,M1 and M2 be as above and let U1, U2 be nonnegative operators.

(cid:12)(cid:12)(cid:12){(v1, v2) ∈ [0, 1]2 : (cid:107)M1U

1/2
1

1

A − v + i0

U

2 M2(cid:107)HS > t}(cid:12)(cid:12)(cid:12) ≤ 2CW(cid:107)M1(cid:107)HS(cid:107)M2(cid:107)HS

1/2

We also make use of the Birman-Schwinger relation [2, Lemma B.1]

P (z − H − vP )−1 P =

P (z − H)−1P − v

(cid:16)

(cid:17)−1

for a projection P onto a subspace of (cid:72) and the fact that P (z − H)−1P is maximally
dissipative on (KerP )⊥, where the equality in question holds. It immediately follows that

(cid:12)(cid:12)(cid:12){v : (cid:107)M1P (z − H − vP )−1 P M2(cid:107)HS > t}(cid:12)(cid:12)(cid:12) ≤ CW(cid:107)M1(cid:107)HS(cid:107)M2(cid:107)HS

1
t

.

equations (3.8.1), (3.8.2) and (3.8.3) together easily imply following apriori bound, for

details we refer the reader to [51, appendix A]

1
t

(3.8.2)

(3.8.3)

Em,n

Λ

(cid:16)|GDiag
(cid:110)(2CW ωmax(cid:107)ρ(cid:107)∞)s

(m, n; z)|s(cid:17) ≤ CAP(s)
(cid:0)4CW ω2

1 − s

,

(3.8.4)

(cid:111)

.

max(cid:107)ρ(cid:107)2∞(cid:1)s

1 − s

CAP(s) = max

and we remark that the constant can be taken as

91

APPENDIX

92

APPENDIX

TECHNICAL COMMENTS

A.1 Conditions on the density ρ imposed in chapter 2

Assume that for some M > 0 and all v ∈ R

(cid:12)(cid:12) d

dv

log ρ(v)(cid:12)(cid:12) ≤ M.

This condition is satisfied, for instance, by the Cauchy and exponential distributions. Then

Letting h(v) = e−c|v| with c > 0 to be determined. The following estimate holds:

ρ(x)
ρ(v)

= elog ρ(x)−log ρ(v)
≥ e−M|x−v|.
(cid:90) ∞
−∞ e−(M +c)|x−v| dx

(cid:90) ∞

−∞

h(x − v) dx ≥

ρ(x)
ρ(v)

therefore, letting ¯ρ(v) =

(cid:82) ∞−∞ ρ(x)h(x−v) dx

ρ(v)

=

2

M + c

.

, we have shown that ¯ρ(v) ≤ M +c

2

for all v ∈ R.

Let us now verify that ¯ρ decays in two cases, starting with when ρ is an exponential. For
simplicity, let us ignore normalization factors and let ρ(v) = e−|v|. Then, by the triangle

inequality,

−∞ ρ(x)h(x − v) dx ≥ ec|v|(cid:90) ∞
(cid:90) ∞

−∞ e−(1+c)|x| dx

2ec|v|
1 + c

.

=

Thus, in this situation ¯ρ(v) ≤ 1+c
h(v) = e−cv2

with c > 0 to be determined. In this case, we have that

2 e−(1+c)|v|. Let us now assume that ρ(v) = e−v2

and

ρ(v)h(x − v) ≥ e−(1+2c)x2−2cv2

93

thus

(cid:90) ∞
−∞ ρ(v)h(x − v) dx ≥ e−2cv2(cid:90) ∞

−∞ e−(1+2c)x2

dx

√
πe−2cv2
√
1 + 2c

.

=

In particular, whenever 2c < 1 we may conclude that ¯ρ decays according to

¯ρ(v) (cid:46) e−(1−2c)v2

.

A.2 Derivation of the self-consistent equations

Let f (Γ) = Tr (Γ log Γ) where 0 < Γ < 1 is a symmetric matrix. Then, given H which is

also symmetric with 0 < H < 1 we have

(cid:16)

(cid:16)

(cid:17)

(cid:17)
+ H log Γ(I + Γ−1H)

.

I + Γ−1H

f (Γ + H) = f (Γ) + Tr

Γ log

Using the power expansion of the logarithm log(A) = A − 1

2 A2 + ... we may conclude that

f (Γ + H) = f (Γ) + Tr (H(I + log Γ) + O((cid:107)H(cid:107)2).

therefore, we have the following expression for the Frech´et derivative of f

f(cid:48)(Γ) = log Γ + I.

Similarly, the derivative of the map Γ (cid:55)→ f (1 − Γ) equals −(I − Γ) log(I − Γ) − I. We may
conclude that the entropy (cid:83)(Γ) = −Tr (Γ log Γ + (1 − Γ) log(1 − Γ)) satisfies

(cid:48)(Γ) = − (log Γ − log(I − Γ)) .

(cid:83)

Moreover, the energy functional

(cid:69)(Γ) = Tr (H0 − κ + λVω) Γ + g

(cid:88)

n

(cid:104)n|Γ↑|n(cid:105) (cid:104)n|Γ↓|n(cid:105) ,

can be more succinctly expressed as

(cid:69)(Γ) = Tr(cid:0)(H0 + λVω) Γ + gDiag(Γ↑)Diag(Γ↓)(cid:1)

94

from which we see that

(cid:48)(Γ) =

(cid:69)

 H0 + λVω + gDiag(Γ↓)

0

 .

0

H0 + λVω + gDiag(Γ↓)

In particular, we see from the above equations that whenever (cid:69)(cid:48)(Γ) − β−1(cid:83)(cid:48)(Γ) = 0 the

exp(cid:0)−β(H0 + λVω + gDiag(Γ↓)(cid:1)

0

 .

matrix Γ must be such that

Γ(I − Γ)−1 =

from which we conclude that

 exp(cid:0)−β(H0 + λVω + gDiag(Γ↓)(cid:1)
(cid:0)1 + exp(cid:0)βH↓(cid:1)(cid:1)−1

Γ =

0

0



(cid:0)1 + exp(cid:0)βH↑(cid:1)(cid:1)−1

0

where H↓ = H0 + λVω + gDiag(Γ↓) and H↑ = H0 + λVω + gDiag(Γ↑).

95

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