TH 5518 1 LIBEARY ’ imam... as; state 82.3 2.5 3:13 L M - w— This is to certify that the dissertation entitled Application of the Green's Function Monte Carlo Method to Hamiltonian Lattice Field Theories presented by David William Heys has been accepted towards fulfillment of the requirements for Ph.D. Physics degree in Major professor lxwddIL Shmp ya, a, my MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop—to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. APPLICATION OF THE GREEN'S FUNCTION MONTE CARLO METHOD TO HAMILTONIAN LATTICE FIELD THEORIES by David William Heys A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1984 Oct/(o - ”fix f, f ‘(fiqu ‘~._, ABSTRACT APPLICATION OF THE GREEN'S FUNCTION MONTE CARLO METHOD TO HAMILTONIAN LATTICE FIELD THEORIES by David William Heys The Green's function Monte Carlo (GFMC) method is adapted for application to Hamiltonian lattice gauge theories, and is applied to the SU(2) and U(l) models. The method is a Monte Carlo method for finding the ground state of a quantum mechanical system with many degrees of freedom, by iteration of an integral equation of which the ground state is an eigenstate. An interesting aspect of the method is the use of an importance sampling technique that makes use of variational wave functions to reduce fluctuations and accelerate convergence of GFMC estimates of various quantities. The calculations have been restricted, by the availability of computer time, to estimates of simple quantities, the ground state energy per plaquette and the mean plaquette field, on a small lattice (3 x 3 x 3). There is no difficulty, subject to the availability of computer time, in computing other quantities or in using larger lattices. The results are interpreted in terms of the phase structure of the two groups; the SU(2) model exists in a single quark confining phase for all values of the coupling constant whereas the U(1) model in 3+1 dimensions undergoes a phase transition from a confining phase at strong coupling (924“) to a non-confining phase at weak coupling (92*0). The method is not restricted to gauge theories and is also applied to the Hamiltonian XY model in 1+1 dimensions. The results obtained on this model are interpreted with regard to the Kosterlitz-Thouless phase transition. ACKNOWLEDGEMENTS I would like to thank Dr. D.R. Stump for introducing me to lattice gauge theories and for suggesting this particular research topic. His guidance and encouragement have certainly been invaluable in helping to bring this work to fruition. I am also indebted to Dr. W.W. Repko with whom I have had a number of interesting and enlightening conversations on various subjects. .My colleagues and friends A. Abbasabadi, E. Garboczi, H. He and M. Thomsen have been a great help in making my stay at Michigan State a pleasant one for which I will always be grateful. Conversations with them, not necessarily concerning physics, have always been stimulating. Very special thanks are due my wife, Susan, whose constant support and encouragement, though not often acknowledged, are truly appreciated. Her patience and understanding, especially during the writing of this manuscript, have been nothing less than amazing. Finally, I would like to thank Dr. B.H. Wildenthal for his assistance in bringing me to Michigan State and for his support during the first few months here. ii TABLE OF CONTENTS List Of Figures ......0....0.........O..............0.................v Chapter 1: Intrwuction .......0...................0 ..... 0.0......0... 1 Chapter 2: The Green's function Monte Carlo method ............. ...... 5 2.1 Introduction ......................... ..... ................ 5 2.2 Algorithm 1: Unweighted ensemble ..... . ..... .............. 10 2.3 Algorithm 2: Weighted ensemble ....... ..... ............... 13 2.4 Importance sapling 0..........00.....0........0.0.0...... 17 Chapter 3 The SU(2) lattice gauge theory in 3+1 dimensions ......... 26 3.1 Definition of the model .................................. 26 3.2 variational calcu1ation ..0............................... 29 3.3 . GFMC calcu1ation ......... ..... ........................... 38 Chapter 4: The U(l) lattice gauge theory in 3+1 dimensions .......... 45 4.1 Detinition Of the mel .........0.................0.....0 ‘5 4.2 Variational calculation .................................. 47 4.3 GFMC calculation 0....0..00.........0..00......O00000..... 52 iii Chapter 5: n-space formulation of the U(1) lattice gauge theory ...00.....0......0....0..00...0.0.0.0.............. 60 5.1 The n-space muations .0....................O............. 60 5.2 Variational calculations ........ ......... . .......... ..... 63 5.2.1 Disordered wave function .. ............. . ..... .. 63 5.2. Gaussian wave function ......................... 65 5.2.3 Variational results ............................ 66 5.3 GFMC calculations ......0................................. 71 Chapter 6: The Mltonianxywel ....00.00..0.....0..0...0.......0 77 Chapter 7: Summary and conclusions ...................... ... ......... 85 Appendix A: Green's function Monte Carlo calculations on the SU(2) and U(l) lattice gauge theories ............ ....... 88 Appendix B: Application of the Green's function Monte Carlo method to the compact Abelian lattice gauge theory ..... 128 List Of References .......0...................00.....0..0.......0... 140 iv Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10: LIST OF FIGURES Variational estimates of (a) the magnetic energy, and (b) the electric energy versus the variational parameter 6 for the SU(2) theory .......................... Variational estimate of the ground state energy per plaquette versus k for the SU(2) theory ................... ° Variational estimate of the mean plaquette field versus 1 for the SU(2) theory ..................... ........ GFMC estimate of the ground state energy per plaquette versus k for the SU(2) theory ........... ..... ... . GFMC estimate of the mean plaquette field versus 1 for the SU(2) theory .........0.0.......0....00........O... Variational estimates of (a) the magnetic energy, and (b) the electric energy versus the variational paramEterafor the U(l) theory ........................... . Variational estimate of the ground state energy per plaquette versus 1 for the U(l) theory .................... Variational estimate of the mean plaquette field versus k for the U(l) theory .............................. GFMC estimate of the ground state energy per plaquette versus k for the U(l) theory .................... GFMC estimate of the mean plaquette field versus 1 for the U(1) theory ................O..................... Variational estimates of the ground state energy per plaquette versus 1 for the n-space formulation Of the 11(1) theory .........O............................. : a/ah versus 1 for the variational wave function ¢2[n] ............................................... Variational estimates of the mean plaquette field versus 1 for the n-space formulation of the U(l) theory .....O.........0..........0....00.0..00............ V 33 35 37 42 43 49 50 52 55 58 67 69 70 Figure Figure Figure Figure Figure 14: 15: 32: GFMC estimates of the ground state energy per plaquette versus A for the n—space formulation of the "(1) theory 0.0.0.........OOOOOOOOOOOO.....OOOOOOOOOOO 72 GFMC estimate of the mean plaquette field versus l computed using the trial wave function ¢2[n] for importance sapling .........OOOOOOOOOOOOOOOO.......00.... 74 GFMC estimate of the mean plaquette field versus A computed using the trial wave function ¢l[n] for immrtance sapling ......OOOOOOOOOOOO......OOOOOOOOOOOOOOO 76 GFMC estimate of the expectation value of l-cosB(p) for the two dimensional 0(1) theory using the harmonic wave function ¢2 for importance sampling ....... 138 GFMC estimate of the expectation value of l-cosB(p) for the two dimensional U(l) theory using the uncorrelated wave function wl for importance sampling ...................................... ........ .. 139 vi CHAPTER 1 Introduction The candidate theory of strong interaction physics is quantum chromodynamics (QCD), a gauge theory based on the non-abelian group SU(3). Due to the remarkable property of asymptotic freedom [1] possessed by non-abelian gauge theories, short distance phenomena in QCD can be adequately understood in terms of perturbation theory. However, a number of important strong-coupling phenomena, such as the meson and baryon masses and quark confinement, can not be treated in this way since the perturbation series is not convergent for strong coupling. Lattice gauge theories were invented to study such non-perturbative aspects of gauge theories. In a lattice theory the space-time continuum is replaced by a lattice of discrete points at which the various matter fields of the theory are defined. The inverse lattice spacing provides a natural ultraviolet cut off, so that renormalization effects are finite and numerical calculations can be performed with no divergent results. 0f 1 course, in order to make contact with the real world, the cut off must eventually be removed, i.e., the lattice spacing must be taken to zero so that the continuum theory is recovered. There are two complementary formulations of lattice gauge theories. In Wilson's approach [2] the Feynman path-integral of the theory, which in the continuum is a functional integral, is replaced by a lattice approximation involving only ordinary multiple integrals. In this formulation both space and time coordinates are treated as discrete. 0n the other hand, in the Hamiltonian formulation of Kogut and Susskind [3] time remains continuous and one deals with a lattice version of the Hamiltonian of the theory. The two formulations can be shown to be equivalent by means of the transfer matrix [4]. Both versions of the theory have been the subject of intense study in recent years using a variety of techniques: perturbation expansions [5], the renormalization group [6], mean field theory [7], the variational principle [8-10], and Monte Carlo methods [11]. An excellent review of the current status of lattice gauge theories may be found in Ref.[12]. More elementary reviews covering lattice gauge theory basics are Refs.[13,l4]. The Monte Carlo calculations, which so far have only been applied to the Euclidean path integral formulation, have provided by far the most exciting results to date. Such calculations have given us evidence of confinement in SU(2) and SU(3) gauge theories [ls-17], chiral symmetry breaking in QCD [18,19], numerical evidence for quark deconfinement at finite temperature along with rough estimates of the deconfinement temperature [20], and some crude but promising estimates of glueball, meson and baryon masses [18,21,22]. It is natural, then, to try to develop a Monte Carlo method for Hamiltonian lattice gauge theories in the hope that the above calculations can be checked in a completely independent way. This work is a first step toward that goal. The particular Monte Carlo method used here is the Green's function Monte Carlo (GFMC) method. This is a numerical technique for studying properties of the ground state of quantum systems with many degrees of freedom. It was originally developed for application to quantum many-body problems [23-25]. In this work the method is adapted for application to lattice gauge theories. Perhaps the most interesting aspect of the GFMC method is an importance sampling technique. This technique makes use of an approximation of the ground state wave function, usually derived from a variational calculation, to bias the Monte Carlo procedure; this reduces the fluctuations associated with stochastic sampling and also accelerates the convergence of Monte Carlo estimates of various quantities. In principle the final results are independent of the particular importance function used, though in practice it should closely approximate the ground state. By observing how a particular variational wave function behaves as an importance function it is possible to determine how well it resembles the ground state. In this way one can learn something about the structure of the ground state wave function. In contrast the path-integral Monte Carlo method provides only numerical results, and does not easily yield any information regarding the structure of the ground state. The possibility of obtaining analytic information from the Monte Carlo calculation provides one of the strongest motivations for this work. CHAPTER 2 The Green's function Monte Carlo method 2.1 Introduction Consider the Hamiltonian H = Ho - A 31 (2.1) where Ho and M1 are positive definite operators and A is a positive coupling parameter. Furthermore assume that no has a zero eigenvalue. (This can always be arranged by simply adding a suitable constant to the Hamiltonian.) The restriction to positive definite operators is not an essential feature of the GFMC method and, in fact, some of the most fruitful applications of the method have been to systems involving non-positive definite operators [24]. In all the applications to be discussed here the operators H0 and H1 are positive definite and so we need not consider the more complicated general case. The interested reader should consult Ref.[25] for more information pertaining to the use of non-positive definite operators. The first step of the GFMC method is to write the eigenvalue equation H|w> = E|w> (2.2) as an integral equation. To this end rewrite Eq.(2.2) as (HO-E)|w> = AH1|¢> (2.3) and now consider E to be the known quantity and A to be the desired eigenvalue. This is the reverse of the usual situation in which the coupling parameter A is known and the energy E is the unknown eigenvalue. Clearly the eigensolutions are the same regardless of which variable is used as the eigenvalue, as are all observables of the system. Now introduce a set of basis states {|x>} where the label x represents all the parameters needed to uniquely specify a state. Two different basis sets immediately suggest themselves: the eigenstates of H and the eigenstates of H 0' 1' Consider the case in which |x> is an eigenstate of H , i.e., l Hllx> = Hl(x)|x> (2.4) Introducing the Green's function operator G as the inverse of (HO-E), i.e., (Ho-E)G = l , (2.5) we may rewrite the eigenvalue equation Eq.(2.3) as ¢(X) A I dX' G(x,x') Hl(X') W(X') (2.6) where w(x> and G(x,x') = . Clearly, in order for the Green's function to exist the operator (HO-E) must be non-singular, i.e. must have no zero eigenvalues. This is obviously true for E} the set of eigenstates of H0, i.e., Ho|x> = H0(x)|x> . (2.7) In this case, Eq.(2.3) may be written, with obvious notation, as [HO-E]w(x) = A I dx' Hl(x,x')w(x') . (2.8) Defining a new function x(x) as x(x) = [Ho(x)-E]w(x) . (2.9) Eq.(2.8) becomes x(x) = A I dx' Hl(x,x') l x(x') . (2.10) [Ho(x')-E] It is seen that the integral equations Eq.(2.6) and Eq.(2.10) are both of the same form: F(x) = A I dx' K(x,x') V(x') F(x') . (2.11) The form of the functions F(x) and V(x) and the kernel K(x,x') depends on whether |x> is an eigenstate of no or of H1. (r) It is easy to show that the sequence of functions {F (x)} defined iteratively by F(r+1)(x) = A(r) I dx' K(x,x') V(x') F(r)(x') , (2.12) where A(r) is an arbitrary parameter which may change from one iteration to the next, converges to the ground-state eigenfunction of Eq.(2.ll). Suppose that Fn(x) is an eigensolution of Eq.(2.ll) with corresponding eigenvalue An, (recall that A is now the eigenvalue), i.e., Fn(x) An I dx' K(x,x') V(x') Fn(x') . (2.13) Since {Fn(x)} is assumed to be a complete set, we may write (0) F (x) = E cn Fn(x) (2.14) so that (l) _ (0) F (x) - E AA cn Fn(x) (2.15) n and (r) - (i) F (x) - E "A CD Fn(x) . (2.16) A r n where the product runs over i from 0 to (r-l). If the eigenvalues are ordered such that 0< AO< A1< ... then as r*~ the sum on the right hand side of Eq.(2.16) will be dominated by the n=0 term, so that up to a normalization constant lim F(r) 1".“ (x) = Fo(x) . (2.17) where F0(x) is the ground-state eigenfunction of the Hamiltonian. In practice, the integrals involved in the iteration of Eq.(2.ll) are multidimensional and can not be performed exactly. The GFMC algorithm provides a means to carry out the iteration stochastically so that after many Monte Carlo iterations one obtains an ensemble of basis configurations sampled randomly from the probability density F0(x). (We assume that the eigenfunctions Fn(x) satisfy the normalization condition IFn(x)dx = l.) A discussion of how to calculate interesting quantities such as the eigenvalue A0 and ground-state expectation values will be deferred until the next section where the concept of importance sampling will be introduced. Here we content ourselves with a detailed description of the basic Monte Carlo procedure. Two algorithms will be described one 10 of which utilizes an ensemble of unweighted configurations and another which utilizes an ensemble of weighted configurations. Each one has its advantages over the other: the first algorithm is more straightforward to implement whereas the second algorithm, though being more complicated than the first, is for that very reason of wider applicability. In what follows it will be assumed that K(x,x‘) and Fo(x) are both positive functions. It should also be realized that the function V(x) is automatically positive because the operators H and H are positive 0 1 definite and because we restrict our attention to E<0. 2.2 Algorithm 1: Unweighted ensemble Write the kernel appearing in Eq.(2.ll) in the form K(x,x') = k(x,x') Z(x') (2.18) where the function k(x,x‘) is normalized such that I k(x,x')dx = l (2.19) for all x'. This is always possible in principle by letting 2(x') = I K(x,x‘)dx (2.20) but in practice this decomposition may not be the most convenient because the integral in Eq.(2.20) may not be tractable. 11 (r Suppose that E ) = {x0; o=1,2,...,N(r)} is an ensemble of points in parameter space, representing a set of basis states, sampled randomly (r) ) from the probability density function F (x) / IF = A(r) a 2(xo) V(xo) . (2.21) This can be done in any number of ways, the simplest of which is to set v0 = integral part [ + £ ] (2.22) where z is a uniform random deviate in the interval (0,1). If r is the largest integer for which r < then the probability that va=r is clearly given by p(va=r) = r + l - (2.23) and the probability that va=r+l is p(va=r+l) = - r . (2.24) The expected value of Va is therefore r(r+1-) + (r+1)(-r) = as required. Note that the possibility vo=0 is allowed. 12 Now, randomly select v0 new points from the conditional probability density k(x,xa). The details of the sampling procedure depend on the functional form of k(x,x') and no general method can be given. The new points chosen in this way constitute the (r+l)st ensemble E may, therefore, be considered to be randomly sampled from the probability density function F , (2.29) where the angle brackets denote the expected value. For each point x(r), a new point xér+l)i a s chosen randomly from the conditional (“Uni") O probability density function k(x ) and this new point is given weight w(r+1)= A(r) w(r) (r+1) a a ' (r) (r) 2(xo x0 ) V(xa ) . (2.30) The weighted points generated in this way constitute the (r+l)st ensemble E = < z k(x,x(r))w(r)k(r)2(x,x(r))V(x(r)) > o a a a o a a a 14 i‘r) I dx' k(x,x')2(x,x')V(x') < g 6(x‘-xér))wér) > (r) (r) k I dx' K(x,x') V(x') F (x') = r"+1) = N wa / g w , (2.33) 0 where N is the desired ensemble size. As shown in the discussion following Eq.(2.21) this may be most simply done by setting no = integral part [ + £1 , (2.34) 16 where i is a uniform random deviate in the interval (0,1). Note that, as in algorithm 1, the possibility v0 = 0 is allowed. If 2 l the point is split into v0 identical points each with weight wo/va. This has the effect of splitting points with large weight, i.e. greater than Zwo/N, into a number of points with weights less than ZZwa/N. If <1 then if va-o the point is eliminated from the ensemble, but if va=l the point survives and its weight is increased to two/N. It is easy to see that this branching and truncation procedure maintains the ensemble size at approximately N, and also forces all weights to be approximately equal. In practice the factor Z(x,x') is usually approximately equal to 1, so that in calculating wér+l) from wit) by Eq.(2.30) the factor k(r)V(x(r)) plays a dominant role. The following practical Monte Carlo O algorithm then proves useful. (r) (1) Multiply the weights by the corresponding factor k(r)V(xa ). (2) Carry out the branching and truncation procedure. gr)} using the conditional (3) Generate the points {x§r+1)} from {x (r+l)'x(r) O O probability density k(x ). (r+1)'x(r) O O (4) Multiply the weights by the corresponding factor 2(x ). The reason for this rather strange order of events is that the factor k(r)V(xér)) causes the greatest change to the weights, so by first multiplying the weights by this factor and then performing the branching and truncation of the ensemble immediately afterwards, but (r+l)} before generating the new points {x0 one ensures that the weights 17 of the points in the final ensemble will be approximately equal. This would not be the case if the steps outlined above were carried out in the perhaps more obvious order (2),(3),(l),(4), where steps (1) and (4) would be combined into a single step. 2.4 Importance sampling So far two algorithms have been described for carrying out the iteration indicated by Eq.(2.12) and it has been shown that F(r)(x) 4 F0(x), the lowest lying eigenfunction of Eq.(2.1l), as rze. To be of any practical value there must be some way to compute quantities such as the eigenvalue l0 and ground-state expectation values. The precise details of such calculations depend on which algorithm one is using to implement the GFMC method. Since the situation is somewhat more complicated for algorithm 2 than for algorithm 1, we will focus our attention on that case. Corresponding results for algorithm 1 are easily derived. The eigenvalue )0 may be estimated in a very F(r) straightforward way. For large r, (x)=aF0(x) where a is some proportionality constant, so that, using Eq.(2.13), Eq.(2.12) becomes (r+1) (r) (x) = A") F Ao F (x) . (2.35) If this equation is integrated over all parameter space then one obtains the simple result 18 (r) A =xm , (2.36) (r+l)> tot where (r) = z w(r) wtot a a r (2.37) and we have used the defining equation Eq.(2.29) for F(r) (x). Since the expected values are not known, Eq.(2.36) suggests the following Monte Carlo estimator for l : 0 ' (r) iéeSt) = )(r) Wtot (2.38) W(r+l) . tot This estimate, known as the growth estimate because it is computed from the growth of the total weight of the ensemble from one iteration to the next, suffers from three sources of error: (1) Convergence error due to the fact that Eq.(2.36) is valid only in the limit rec. (2) Random sampling error due to the stochastic nature of the GFMC method. (3) Systematic error due to replacing the ratio of expected values in Eq.(2.36) by the ratio of the values themselves in Eq.(2.38). In principle the convergence error is not a significant problem éest) settles down to fluctuate about some constant value this error can be eliminated. In since by simply waiting until the quantity l practice, however, one may have to carry out many thousands of 19 (est) iterations before he converges. Similarly, the random sampling error may, in principle, be reduced by increasing the ensemble size, but in practice it may be that in order to reduce the fluctuations to an acceptable level, an unmanageably large ensemble is required. This turns out to be the case in all calculations of practical interest. The third source of error, the systematic error, can not be estimated but can at least be bounded by making use of the theorem 2 a. min(a./b.) s ——1 s max(a./b.) . (2.39) J J z b J J - 1 Specifically min(w(’)/w(“1)) 5 fig:- < max(w(’)/w(“1)) (2 40) tot tot ' tot tot ° tot so that if the random error can be reduced then the systematic error will also be reduced. All of these sources of error may be greatly reduced by means of a technique called importance sampling. In this technique, a function, called the importance function, is used to bias the diffusion of points in parameter space in favor of regions where the importance function is largest. If the importance function is suitably chosen, this biasing causes points to cluster in regions where the eigenfunction F0(x) is large, and so the ensembles generated by the Monte Carlo procedure 20 provide much better representations of F0(x) than if no importance sampling is used. To see how to implement this technique first multiply Eq.(2.ll) by a function, the importance function, FI(x). Then the new function P(x) defined by P(x) = FI(x)F(x) (2.41) satisfies the equation F1(x) K(x,x') P(x) = k I dX' V(x') P(x') , (2.42) PI(x) which is of the same form as Eq.(2.ll) but with F(x) replaced by P(x) and K(x,x') replaced by FI(x) K(x,x') KI(x,x') = . (2.43) FI(x) This means that the two algorithms presented earlier may still be used to iterate Eq.(2.42). Furthermore, the eigenfunctions of Eq.(2.42) are clearly simply Pn(x) = FI(x)Fm(x) (2.44) with corresponding eigenvalues lo and iteration of Eq.(2.42) converges to Po(x). Suppose we let FI(x) = V(x)Fo(x) . (2.45) Of course this is not possible in practice since the eigenfunction is 21 not known. Iteration of Eq.(2.4l) generates a sequence of functions {P(r)(x)} defined by P A = i“) -—-595- (2.50) 0 tot for all r, i.e., there is no convergence error. The Monte Carlo estimator Eq.(2.38) still suffers from random sampling error and systematic error but because the distribution of points in parameter space is biased, the random error and hence, because of Eq.(2.40), the systematic error may both be expected to be greatly reduced. 22 In practice the importance function is chosen to be FI(x) = FT(x)V(x) (2.51) where FT(x) is a function optimized by the variational principle, which should resemble as closely as possible the eigenfunction F0(x). With this choice of importance function it is reasonable to expect the (est) )‘0 estimator to converge much faster than when no importance sampling is used, and also that the statistical fluctuations of ASESt) will be much smaller. This is in fact the case. A better estimator, known as the variational estimator for reasons which will soon become apparent, which does not suffer from any kind of error when the optimum choice of importance function FI(x)=V(x)F0(x) is used may be derived from the eigenvalue equation Eq.(2.3) for the ground-state |w> of the Hamiltonian, (Ho-E)|¢O> = ionllwo> . (2.52) Suppose for the moment |x> is an eigenstate of H0. With this choice of basis one has FI(x) WT(x)[HO(x)-E] (2.53) and P0(x) = WT(x)[HO(x)-E]wo(x) . (2.54) where wT(x) is a variational wave function which should approximate w0(x). From Eq.(2.52) it is clear that 23 -1 = I G“ dx' WT(x)Hl(x,x')wo(x') lo (2.55) I dx WT(X)[HO(X)'E]WO(X) which, using Eq.(2.54) leads to the Monte Carlo estimator (est) -1 - 1 I dx WT(x)Hl(x,xo) Wtot WT(XO)[HO(xo)-E] The corresponding estimator when |x> is an eigenstate of H1 is W k(ESt) = -—l- 2 ° I dx w (x)[H (x,x )-E5(x-x )1 . (2.57) 0 w(r) a H (x )W (x ) T 0 a 0 tot l o T o If ¢T(x) = w0(x), corresponding to the choice FI(x) = V(x)F0(x) discussed earlier, then (h395t>)'1 = x’1 (2.58) 0 with no error of any kind. It is found that the variational estimator has much smaller fluctuations than the growth estimator and so this is the one that will be used in the applications to be described later. The technique of importance sampling also provides a very simple means of estimating ground-state expectation values. Again for the moment suppose that |x> is an eigenstate of H and write 0 ¢T(x) = 40(3) + en(x) (2.59) where all functions are normalized and e is small if wT(x) is a good variational wave function. Then, it is easy to show that for any operator A ll'lfillllli‘x 24 <¢ lhlw > __0___£’_=2_2_0_-_2_2_ (2.60) <¢TIW0> to order 62. The quantity on the left is the desired expectation value, and the second term on the right is the expectation value of A in the variational state IwT>. The first term on the right, known as the mixed expectation value, may be estimated from a Monte Carlo calculation as <6 |A|¢ > A(x ) 1 _T___9_,_§__2_--/§-————. (2.61) [Ho‘xa)’E] [80(xa)-EJ The corresponding result when |x> is an eigenstate of H1 is A(x ) l —3——°— z r a /§ . (2.62) < wT|w0> ° 31(xo) 51(xa) The symbol a in these last two equations indicates that the estimate has a systematic error associated, as usual, with replacing the ratio of two expected values with the ratio of the valued themselves. But, again the systematic error is bounded by the statistical fluctuations of the estimate so that if the fluctuations are small so is the systematic error. In any case, in quoting results, allowance may be made of the systematic error by simply increasing the error bars of the various quantities by a factor of V2. We see, then, that the quantities appearing on the right hand side of Eq.(2.60) may be calculated and so one may in this way obtain estimates of various expectation values. Of course, such estimates are only reliable if the variational expectation value and the mixed 25 expectation value are not very different, i.e. if e in Eq.(2.59) is small, since otherwise the 0(e2) terms will make a significant (perhaps dominant) contribution. It would be much better if there were some way to compute expectation values which did not rely on the accuracy of the variational wave function wT(x). Such a method does exist [25] but is very demanding on computer resources if any precision is to be achieved and will not be considered further. CHAPTER 3 The SU(2) lattice gauge theory in 3+1 dimensions 3.1 Definition of the model The Hamiltonian of a lattice gauge theory with a unitary gauge group is - 2 2i - ”r H - 2 23(4) + E [ d grr(up +,UP) ] , (3.1) l,a d where d is the dimension of the particular group representation being used, the plaquette variables Up are defined in terms of the group elements U(l) residing on the links of the lattice as _ 'r T up - U(11)U(12)U ((3)0 (44) , (3.2) where 11,12,13, 4, are the links defining plaquette p, and the electric field operators Ea(1) are defined by the commutation relations 26 27 [Ea(l),U(l')] = - TaU(l)6(1,1') , (3.3) where Ta is a generator of the representation. In this chapter we shall concentrate on the SU(2) lattice gauge theory, in particular we shall consider the fundamental representation for which d=2 and Tr(U;) = Tr(UP). The Hamiltonian may then be written as 2 H = (ZaEa(1) + A E ¢(p) , (3.4) where the gauge invariant plaquette variable ¢(p) is defined by ¢(p) = l - gTr(Up) . (3.5) The parameter k is related to the conventional coupling constant 9 by l = 8/94 . (3.6) The group element U(1) may be parametrized in a number of ways. For example, in terms of the three component gauge field Aa(1) (a = 1,2,3), U(!) = exp[ tiaaAa(l) ] , (3.7) where 0a is a Pauli matrix. Another useful parametrization is u = a0(!) + i3o3(z) , (3.8) where 3 is a 3-vector and a + a = l . (3.9) The (real) numbers a“ (u = 0,1,2,3) all lie in the domain (-l,l) and may 28 be thought of as the components of a Euclidean 4-vector. Then Eqs.(3.8) and (3.9) indicate that there is a one-to-one correspondence between the elements of SU(2) and the points in the space 53, the three dimensional surface of a four dimensional sphere. In fact the connection lies much deeper; the geometry of the SU(2) group manifold is identical with that of S3, i.e., the two spaces are isomorphic. We will not prove this assertion here but point out an important implication: the invariant group integration measure is simply the volume element in the space 53. .This may be seen most easily by introducing a third parametrization of the group element U(1) in terms of three angular variables w(1),0(1),¢(1) with domains (0,n). (0,n), (0,2n) respectively. These variables are just the spherical coordinates in four dimensions. In terms of these variables, 4-) U(1) = cosw(1) + iaon(£)sinw(1) , (3.10) where 3 is a unit 3-vector with polar angles (9,¢), i.e., 3 = (sinecosd, sinOsind, cose) . (3.11) Using standard techniques [26] the invariant measure of the group may be found to be d0 = sinzw sine 61,) do do , (3.12) 2n2 which is the volume element in S3 with the total volume normalized to unity. This means that if f(g) is some function defined on the group and fp(w,0.¢) is the corresponding function on the parameter space 53, then 29 I d9 f(g) = I fp(w,e,¢) sinzy sine d) d6 do . (3.13) 2n2 This result is important for the numerical calculations to be described shortly where one needs to know how to carry out group integrations in the parameter space. 3.2 Variational calculation The simplest gauge invariant wave function is of the form vTIA] = g u(¢(p)) . (3.14) where u is an arbitrary function of the plaquette variable ¢(p). This wave function is disordered in the sense that there are no explicit correlations between the variables ¢(p) on different plaquettes. For small values of x the Hamiltonian is dominated by the electric energy term 2 32(1). This is a sum of single link operators and so for small A the link variables 0(1) will be completely uncorrelated. Then, the plaquette variables will also be uncorrelated and so for small A one should expect the wave function given in Eq.(3.l4) to be a good representation of the exact vacuum state. In the present calculations we do not attempt to optimize the functional form of u(¢(p)) but simply use u(¢(p)) = exp [ -2d¢(p) ] , (3.15) where B is a variational parameter chosen to minimize the energy 30 I do (Tm a me £0 = 2 . (3.16) I an thnj This is a sum of two terms: the magnetic energy given by x I an wiIAJE ¢(p) = 2 I do wT[A] AB mag (3.17) and the electric energy, which, using the hermitian character of Ea(1), may be written as I do WT2[A] z {w;1[A] 23(1) wTIA]}2 Eel "’ 1,3 0 (3e18) I do w§[A] 1 defined above could be evaluated analytically as functions of the variational parameter 5, it would be a If the quantities Emag and Be straightforward matter to minimize the energy £0 with respect to 6 for any given value of the coupling parameter I. Unfortunately, analytic expressions can only be derived in the two limits 6+0 and 6+0. For small 6 one may use the so called Euclidean strong coupling expansions [27], similar to high temperature expansions used in statistical mechanics, to evaluate the integrals. The results of such a calculation are: E /n = 1 - p + 263/3 + 0(65) . mag P (3.19) 2 4 6 Eel/Np = 36 - 26 + 0(6 ) : where up is the number of plaquettes in the lattice. For large 3 the wave function is sharply peaked in the region of configuration space 31 where all the group elements U(1) are close to the identity, i.e., Aa(l) * 0. Then, with negligible error in the limit, the range of integration of the field variables Aa(1) may be extended to (-¢,°), and the integrals in Eqs.(3.l7) and (3.18) become straightforward gaussian integrals which can be easily evaluated. Then, one finds that for 5*a, E /N = '2 magp _4_]fi-+O(B)l (3.20) _ _ -1 Bel/Np - 36 % + 0(fl ) Of course, one really needs_to compute these quantities for intermediate values of B. To do this a Monte Carlo method is used. The limiting expressions given in Eqs.(3.19) and (3.20) then provide useful checks on the accuracy of the Monte Carlo results. In this particular calculation, Creutz's heat bath algorithm, described in detail in Ref.[15], is used to generate an ensemble of field configurations {A(r); r=l,2,...,N} from the probability density w§[A]/Ido¢:[a]. Then, any expectation value of the form I an (imam <0> = 2 (3.21) I do wT[A] may be computed as the expected value of the ensemble average of 0[A], i.e., = (r) / OEA ] \ where the angle brackets on the right hand side denote the expected 32 value which may be estimated as the average over many different ensembles. Using this method, Emag and Eel are computed for many different values of 6 and the resulting data are fitted, by means of a least squares analysis, to the specific functional forms (fit) E = exp [ f (a) ] . mag m (3.23) (fit) _ where fm(3) and fe(6) are polynomials in 6 of sufficient degree to give good fits to the data. The xZ-test, with a significance level of 5%, is used to judge the goodness of fit. The particular functional forms in Eq.(3.23) are chosen, largely by trial and error, to give good fits with as few adjustable parameters as possible. For 0 S 6 S 1.2, which is the region of interest, with 65 data points, fm(a) must be of degree 5 and fe(6) must be of degree 6 in order to pass the xz-test. Figure 1 shows graphs of Emag/Np and Eel/Np as functions of B. The points are Monte Carlo results and the solid lines are the fitted functions defined in Eq.(3.23). Only a sample of the Monte Carlo results are plotted. The dashed lines are the large- and small-B limits given by Eqs.(3.19) and (3.20). The fits are clearly very good as is the agreement between the Monte Carlo results and the two limiting curves 0 33 Figure 1: Variational estimates of (a) the magnetic energy, and (b) the electric energy versus the variational parameter 6 for the SU(2) theory. 34 In Figure 2 the variational estimate of the ground-state energy per plaquette as a function of A is compared to the large-A and small-A limits computed in perturbation theory: Eo/N z x - 13 + 11 )4 + 0()6) as x»o , P 12 14976 EO/NP z c(n)(2A)l/2 + 0(1) as Ado, for an n x n x n lattice; the constant c(n) is weakly dependent on n, e.g., c(3) = 1.181 , c(w) = 1.194 . (3.25) The constant term in the large-A limit derives from the four-field coupling of the fields in the small field approximation of the theory and at the present time has not been calculated. The agreement between the variational results and the small-A limit comes as no great surprise since, as mentioned earlier, the variational wave function is expected to be a good approximation of the exact ground state wave function for small A. Also the apparent disagreement between the variational results and the large-A limit is not meaningful since a constant term is yet to be added to the perturbation theory result. The agreement may then improve or worsen and one does not know, a priori, which it will be. For this reason, perhaps a more interesting quantity to look at is Emag/Np = <¢(p)>. For the exact ground state this is related to the energy 30 by 35 5.0 1.5 Figure 2: Variational estimate of the ground state energy per plaquette versus A for the SU(2) theory. 36 R; < (p) > = 1'— -—O (3.26) N A P so that the small- and large-A limits are: 3 5 < o(p) > = l - A + 11 A + O(A ) as A+0 , 6 3744 (3.27) < ¢(p) > = c(n)(2A)-l/2 + 0(A-l) as A+e. These are plotted in Figure 3 along with the variational results. Again one sees excellent agreement between the variational results and the small-A perturbation theory result. However there is a slight discrepancy between the variational results and the large-A limit, the difference increasing somewhat as A increases. This is an indication of the inadequacy of this uncorrelated variational wave function as a model of the vacuum state of the theory. Other quantities, such as the string tension and the excitation energy of the theory, provide much more sensitive tests of the accuracy of the variational wave function than the mean plaquette field <¢(p)> calculated here. Calculations of these quantities have been carried out [9,10] and clearly show the failure of the variational wave function to model the vacuum state of the theory at large—A. Those calculations, although of some interest, do not concern us here since we are mainly interested in the variational calculation as a means of providing an importance function for use in the GFMC calculations to be described in the next section. 37 1.0 r- I I 1 j T 1 T 'I I I 1 I j I A 0.8r \ / “ A 0.6 '" K\ / q ’23. e\\\. ./ 5 * ° W ‘ e \/ e 0.4 .. 4 e Figure 3: Variational estimate of the mean plaquette field versus A for the SU(2) theory. 38 3.3 GFMC calculation The Hamiltonian defined in Eq.(3.4) can not be used as it stands in a GFMC calculation because it is not of the form H0 - AH1 with H0 positive definite operators. However, a slightly modified Hamiltonian and H1 2 HGFMC = Z Ea(1) - AM (3.28) 1,a where M = E [ l + %Tr(Up) ] (3.29) is of the required form and the GFMC method, described in detail in chapter 2, may be used to compute various ground-state properties. Notice that HGFMC differs from H only by a trivial constant term so that the eigenstates of the two Hamiltonians are identical. As shown in chapter 2, the ground-state energy of H GFMC is negative and so it is useful to write the eigenvalue equation as _ _ 2 HGFMCH’) — Q |¢> . (3.30) The connection between 02 and 80, the ground-state energy of H, is _ _ 2 EO - ZNPA Q . (3.31) The non-normalizable basis {|[A]>} is used, where a state in this basis is determined uniquely by the set of gauge fields Aa(1) (or, alternatively, the angle variables 0(1), 9(1), ¢(1)) on all the links of the lattice. In this basis, the operator M defined in Eq.(3.29) is diagonal and the equation to be iterated by the GFMC algorithm is 39 (cf. Eq.(2.6)) w[A] = A I do' G[A,A'] M[A'] w[A'] , (3.32) where . _ 2 2 -1 . G[A,A ] - <[A]| [ Z Ea(1) + Q ] |[A ]> . (3.33) 1,a The only aspect of the GFMC algorithm not covered in chapter 2 was the very crucial problem of how to sample field configurations from the (unnormalized) probability density G[A,A']. This matter is discussed at length in the paper reproduced in appendix A. Basically the idea is to write the Green's function as G[A,A'] = I; dt exp(-tQZ) <[A]| exp[—tZE:(1)] |[A']> (3.34) and to use the function exp(-th) to sample t, then, conditional on this choice, to use <[A]|exp[-tZE:(1)]I[A']> to sample field configurations [A]. The state |[A]> may be written as a direct product of single link states |A(1)> so that the matrix element appearing on the right hand side of Eq.(3.34) may be written as a product of single link matrix elements. Now since Q2 is typically large, the variable t will be small. In this limit it is possible to obtain an explicit expression for the matrix element. The result is z er[-(6s)2/t] , (3.35) (wt)3/2 where (as)2 is the metric in the parameter space 53 (as)2 = (6¢)2 + sin24(59)2 + sinzwsin29(6¢)2 (3.36) 64 = W' - w ; 60 = 6' - 0 ; 5o = o' - o . (3.37) The problem of sampling G[A,A'] then reduces to that of sampling the gaussian function exp(-652/t) for which several methods are available. Of course, there is a complication involved in the sampling of this gaussian due to the fact that the parameter space is curved. The precise details of the sampling procedure may be found in appendix A. Before going on to discuss the results of the GFMC calculation, one further point should be noted. In order to obtain statistically significant results, it is essential to use some form of importance sampling in the manner described in Sec. 2.4. The details of how to carry out the sampling procedure in this case may again be found in appendix A. All the calculations presented here were carried out on a 3 x 3 x 3 spatial lattice. An ensemble of approximately 100 configurations was used; the ensemble size changes slightly with each iteration. The results given are averages over 600 Monte Carlo iterations. The first few hundred iterations, during which convergence takes place, are discarded. To calculate the quantities presented here required approximately 3.5 hours of computation time for each value of Q2 considered (recall that 02 rather than A is the input parameter to the GFMC' algorithm) on a CDC Cyber 750 computer at Michigan State University. 41 Figure 4 shows the ground-state energy per plaquette EO/Np as a function of the coupling parameter A. The dashed curve is the variational bound obtained in the previous section and the crosses are GFMC results obtained using importance sampling based on the variational wave function wT[A]. The GFMC points agree very well with the variational bound at small A; this is to be expected since the variational wave function is an accurate representation of the exact vacuum state for small A . As A increases the GFMC points begin to lie lower than the variational bound, the difference increasing with increasing A. Again this is as expected; the variational wave function is known to become less accurate as a model of the ground state as A increases and so the exact vacuum [energy should be lower than the variational bound. As in the previous section, a more interesting quantity to look at is the mean plaquette field <¢(p)>. This is shown in Figure 5. Again the dashed curve is the variational estimate and the crosses are GFMC estimates; these GFMC results were computed from the mixed expectation value, Eq.(2.62). The solid curves are the large- and small-A limits given by Eq.(3.27). The GFMC points tend to lie below the variational curve for small-A and are inconsistent with the small-A perturbation theory curve. Ordinarily this would be taken as evidence that the variational wave function wT[A], used for importance sampling, is not a good lrepresentation of the vacuum state. However, in this case, wT[A] is believed to accurately describe the exact ground state for small A and 42 l I 1 l 1 ‘r 1 T l 1 f l T I I P" 1 6-D .- ’/‘-' ,a’ .— z/ _ I/’ I )- ’/’ + .4 ’l I” 4 5 — x _. - Ill * I o. - ,I’ .. Z ,I’ \ F’ I,’+ .. C3 ,’ L1J 3.0 r- I/ — [I L [(l . )- / - / / 1.5 c I - I _ / l I! / h- , III / J J J J J J J J J J J J J i Figure 4: GFMC estimate of the ground state energy per plaquette versus A for the SU(2) theory. 43 1.0 1 1 1 I 1 l I l I j 1 I 1 1 0.8 - " A 0.6 F" " ’3. 1 \ <§a + \\\\ 0.4 - - 0.2 .— t I J J J J I J J J 4 J J J 0 3 6 9 12 15 Figure 5: GFMC estimate of the mean plaquette field versus A for the SU(2) theory. 44 to become increasingly worse as A increases. The results on the energy per plaquette shown in Figure 4, where the GFMC points are in close agreement with the variational bound but begin to deviate as A increases, are consistent with this view. The poor accuracy of the GFMC results for small A in Figure 5 may be due to the failure of the small time step approximation used to calculate the matrix elements of exp(-t£E:(1)). The time t is sampled from the probability density exp(-th) and so is of order l/Qz, which increases as A decreases. The approximation will, therefore, be least valid for small A. This explanation could be checked by subdividing every time step into intervals smaller than some fixed at, and then observing how the results change as 6t decreases. On the other hand, since the wave function is disordered for small A, one might expect that errors in the sampling procedure would be unimportant. Another possible explanation of this- discrepancy is that the calculation may not have been carried out for enough iterations to deduce a meaningful estimate of the uncertainty indicated by the error bars. Succesive GFMC ensembles are highly correlated and fluctuations of measured quantities extend over many iterations, so it is possible that the 600 iterations used to compute the GFMC results are dominated by one very long flucuation which causes the estimates to be too small. If this is correct it is not clear why the same problem does not occur for the larger values of A. Clearly further investigation is needed to clarify the situation. CHAPTER 4 The U(l) lattice gauge theory in 3+1 dimensions 4.1 Definition of the model In this chapter calculations on the U(l) lattice gauge theory will be described which parallel those on the SU(2) model discussed in the previous chapter. By studying this somewhat simpler model it may be possible to resolve some of the questions raised by the GFMC calculations on the SU(2) model. Also, Monte Carlo calculations in the Euclidean path-integral formulation of the theory have clearly demonstrated that the vacuum state of the U(l) model undergoes a second order phase transition in four dimensions from a charge confining phase at strong coupling (92%“, A40) to a non-confining phase at weak coupling (9290, Ase) [28], and it will be interesting to see if evidence of this transition shows up in the Hamiltonian formulation of the theory. 45 46 The elements of the group U(l) may be parametrized as U = exp(iA) (4.1) where the gauge field A lies in the domain (0,2n). With this parametrization the plaquette variable Up may be written as UP = exp[iB(P)] (4.2) where B(p) is the lattice curl of A at plaquette p: B(P) = A(11) + A(12) - A(13) - A(14) . (4.3) The links 11, 12, 13, 14, define the plaquette p. An explicit expression for the electric field operator 2(1) may be obtained from the commutation relation (cf. Eq.(3.3)) [B(1).u<1'>] = - U(1) 6(1.1') . (4.4) and is found to be _ a (4.5) 3") “ i an<1> The Hamiltonian Eq.(3.l) may then be written as 32 H = - Z 2 + A E ¢(p) , (4.6) 1 BA (1) where the gauge invariant plaquette field ¢(p) is defined by ¢(p) = l - cos B(p) , , ' (4.7) 47 and the constant A is related to the conventional coupling constant g by 4 A = 2/g (4.8) The group manifold is one dimensional and the invariant measure is easily shown to be do e g% . (4.9) so that group integrations may be carried out in the parameter space by simply integrating over the gauge fields A(1). 4.2 Variational calculation As in the previous chapter, we will use the disordered wave function 4 [A] = E u(¢(p)) . (4.10) with the specific choice u(9(p)) exp[ -&B¢(p) ] . (4.11) The factor of t in the exponent is chosen purely for aesthetic reasons. The calculation proceeds in precisely the same way as the SU(2) variational calculation, the only difference being in the choice of gauge group. 48 Figure 6 shows graphs of E /N and E /N as functions of the magp elp variational parameter 6. The points are a sample of the Monte Carlo variational estimates and the solid lines are the functions Eéggt)(fi) and Eéilt)(6). The functional forms given in Eq.(3.23) again give good fits to the data with the least number of parameters: in the region 0 S 5 S 1.8 with 92 data points, fm(fi) is a polynomial of degree 7 and fe(a) is one of degree 8. More parameters are needed in this case than in the SU(2) calculation because a larger range of B is covered and more data points are used. The dashed lines in Figure 6 are small- and large-B limits derived from Euclidean strong coupling expansions and the Gaussian approximation respectively. The functions describing these curves are, for small 8, 3 5 E /N = l - 3/2 + B /16 + 0(6 ) a mag p (4.12) _ 2 _ 4 6 Eel/Np - fl /2 B /16 + 0(5 ) , and for large B, 3 -2 Emag/Np %3 + 0(6 ) r (4.13) _ _ -1 Figure 7 shows the variational estimate of the ground state energy per plaquette as a function of the coupling parameter A and compares these results to the large- and small-A limits derived from perturbation theory: 49 1.0 i “v 1 1 i j 1 j T fi 1 1 W W 1 I j 1 r 1 0.8 - W t 1 a \ z 0.6 [- \\\\ q \E )- W “J 0.4 - a \\\ I \\‘ 1 ‘\ ‘~~~~ 0.2 _ ~“§~. - 1 J J l L J I J J J J J l J J 1 JJ 1 Figure 6: Variational estimates of (a) the magnetic energy, and (b) the electric energy versus the variational parameter a for the U(l) theory. 50 2.0 1.5 /’N LLJ 1.0 0.5 J J I J J J 1 l J Jl l I l l 0.00 0.75 1.50 2.25 3 00 3.75 A Figure 7: Variational estimate of the ground state energy per plaquette versus A for the U(l) theory. 51 II E /NP A - A2/8 + 3A‘/10240 + O(A6) as A40 , (4.14) /2 d(n)(2A)l - d2(n)/4 + O(A—1/2) as Aaw, M \ z I! for an n x n x n lattice. As in the SU(2) large-A limit, the constant d(n) depends weakly on the lattice size, e.g., d(3) = 0.787 , d(“) = 0.796 . At small A the variational estimates are in excellent agreement with perturbation theory as expected, but at large A the Variational estimates lie significantly higher than the perturbation theory result. This clearly indicates the inadequacy of the simple uncorrelated wave function Eq.(4.10) as a model of the vacuum state of the theory at large A. The same conclusion also follows from a consideration of the variational estimate of the mean plaquette field <¢(p)> as a function of A. This is shown in Figure 8 along with the large- and small-A limits determined using Eq.(3.26): <4(p)> a 1 - A/4 + 3A3/2560 + 0(A5) as A90 , (4.15) <4(p)> z d(n)/(2A)l/2 + 0(A’3/2) as lee. 4.3 GFMC calculation The discussion of section 3.3 leading to Eq.(3.32) is applicable almost without change to the U(1) model, so that the equation to be 52 Figure 8: Variational estimate of the mean plaquette field versus A for the U(1) theory. 53 iterated by the GFMC method is MA] = I do G[A,A'] MA‘] 4411‘] (4.16) where M[A] = E [ l + cos B(p) ] (4.17) and ' 32 2 -1 G[A,A'] = <14]! {-2 2 + 0 1 IIA']> . (4.18) 3A (1) 2 32 = I dt exp(-tQ ) <[A]| exp[t Z 2 ] |[A']> . (4.19) an (1) The state |[A]> may be written as a direct product of single link states |A(1)> so that the matrix element appearing in the integrand of Eq.(4.19) may be written as 82 n l exp[t 2 1 8A (1) ] |A'(1)> . (4.20) In appendix A it is shown that in the small-t limit this single link matrix element may be written as exp[-(44)2/4t1 . (4401/2 32 3A2(1) z (4.21) The problem of sampling field configurations from the Green's function G[A,A'] then reduces, as in the SU(2) case, to sampling a gaussian distribution. The precise details of the calculation, 54 including the use of importance sampling based on the variational wave function Eq.(4.11), may be found in appendix A. As in the SU(2) case these calculations have been carried out on a 3 x 3 x 3 spatial lattice, with an ensemble size of approximately 100 configurations which changes slightly with each iteration. The results given are averages over 600 Monte Carlo iterations and required approximately 200 seconds of computation time for each value of Q2 considered. Figure 9 shows the ground-state energy per plaquette Eo/NP as a function of A. The crosses are GFMC results and the dashed curve is the variational bound obtained in the previous section. The solid curve is the large-A perturbation theory result Eq.(4.14). At small A the GFMC results agree with the variational bound but for larger values of A the GFMC points lie significantly lower, clearly indicating that the uncorrelated wave function Eq.(4.11) is no longer a good model of the ground state at large A. In fact for large A the exact ground-state wave function can be derived. In that limit the energy is dominated by the magnetic energy and so the terms 1 - cos B(p) will be small, i.e., B(p) 4 0 as A* a. With this approximation, the Hamiltonian Eq.(4.6) is quadratic so that the ground-state wave function is a gaussian in the gauge fields A(1): w[A] = eXP {-94 Z Ak(§) Mkk.(§.§') Ak.(§') 1 (4.22) where 55 I 1 I j 1 1 ‘r ‘1 I 1 V I 1 j 2.0 - + P * q / . 1.5 — ,/ .. a. z’ I O z 4 [,1 _ \ + D / LIJ 1.0 _ Ix _ 7/ . ,I .. l / II 0.5 — f .- l / h I, 1 / l J l l 1 JL Jl J l L J J _I I 0.00 0.75 1.50 2.25 3.00 3.75 A Figure 9: GFMC estimate of the ground state energy per plaquette versus A for the U(1) theory. 56 a g (“(2) . (4.23a) 4-9 .5 4+4 ”kk'(x'x') = l. 5 mkk.(q) 9391 211 9°(x-x') J . (4.23b) 3 q n n 2» 4*» f (q)5kk. - fk(q)fk,(q) 4 .(q) = , (4.23c) mkk f(3) fk(§) = 1 - exp( 21; qk ) , (4.23d) n e 4 2 f(q) = z |fk(q)l . (4.23e) k for an n x n x n lattice. The sum in Eq.(4.22) is over all I, 1', k, k'. In these equations a link 1 is defined by two indices § and k; the link lies between the lattice sites at g and g + 3k where 3k is a unit vector. At large A then, the ground-state wave function explicitly couples links which are widely spaced; this type of coupling is absent in the simple disordered wave function Eq.(4.11). There appears to be an abrupt crossover point at A z 1.2 in Figure 9 where the GFMC results begin to deviate markedly from the variational bound. This may be taken as evidence, albeit inconclusive, of a phase transition at that point. For A < 1.2 the ground state of the theory resembles the disordered variational wave function as indicated by the close agreement between the variational results and the (in principle) exact GFMC results, but for A > 1.2 the disordered wave function is no longer accurate and the ground state is more closely represented by the gaussian wave function Eq.(4.22) with its explicit couplings between widely separated links. 57 It is interesting to compare this result to the corresponding result for the SU(2) model shown in Figure 4. In that case the deviation of the exact GFMC results from the variational curve simply increased very gradually as A increased. This is consistent with the fact that there is no phase transition in the ground state of the SU(2) theory. The GFMC results at large A in Figure 9 appear to lie significantly lower than the large-A perturbation theory curve. It may be that by using a poor importance function one obtains inaccurate estimates of the energy with an ensemble as small as 100 configurations; the GFMC method relies heavily on the law of large numbers of probability theory and so it is conceivable that small ensembles result in inaccurate GFMC results. To study this systematically would require repeating the calculations for different ensemble sizes and observing how the results vary as the ensemble size increases. Such an undertaking would clearly be very demanding on computer time. Furthermore, it may be that if the importance function is too inaccurate the ensemble size necessary to obtain good results is unmanageably large. With these points in mind, the problem of ensemble size dependence of the GFMC results has been left for future investigation. Figure 10 shows the mean plaquette field <¢(p)> as a function of A. Again the crosses are GFMC results based on the mixed expectation value Eq.(2.62), the dashed curve is the variational estimate, and the solid curve is the large-A perturbation theory result Eq.(4.15). Notice again the abrupt deviation of the GFMC points from the variational curve at 58 1.0 0.4 Figure 10: GFMC estimate of the mean plaquette field versus A for the U(1) theory. 59 A z 1.2 indicative of the phase transition in this model at that point. Recall that the GFMC results for this quantity are not exact and can only be trusted if they do not differ very much from the variational estimates. Thus, the lack of good agreement between] the GFMC results and the large-A perturbation theory curve is not surprising. As in the corresponding SU(2) results shown in Figure 5, the GFMC points at small A are noticeably low; they are inconsistent with the known small A behaviour given by Eq.(4.15). The comments made at the end of chapter 3 concerning this discrepancy are also valid here. However, for the point at A a 0.75 the calculation was repeated breaking each time step t into ten smaller substeps with no noticeable change in the result. Of course, this is by no means intended to be a complete study of the problem, but it does tend to cast some doubt on the explanation of the discrepancy as a failure of the small-t approximation. As stated at the end of chapter 3, further investigation is clearly necessary to resolve the problem. CHAPTER 5 n-space formulation of the U(1) lattice gauge theory 5.1 The n-space equations The calculations on the U(1) lattice gauge theory described in the previous chapter used a basis set in which the plaquette fields ¢(p) are diagonal. Using such a basis the problem of how to use the gaussian wave function given in Eq.(4.22) as an importance function presents so far insurmountable difficulties and one is restricted to using the simple disordered wave function Eq.(4.11) which gives poor results at large A. A different formulation of the problem is possible which allows one to use a basis in which the electric field energy is diagonal and also to use as importance functions both disordered and gaussian wave functions. 60 61 Write the wave function w[A] in the manifestly gauge invariant form 41A] = : exp[ i g n(p)B(p) 1 ¢[n] . (5.1) where n(p) are integer-valued plaquette variables. It is necessary to restrict n(p) in this way to ensure that the wave function w[A] is periodic in the gauge fields A(1). Equation (5.1) resembles a Fourier series expansion in the magnetic field variables B(p). This is not quite the case, though, because not all of the fields B(p) are independent. In fact the sum of the B(p)'s over any closed surface in the lattice must vanish in accordance with Gauss' Law. Inserting Eq.(5.l) into the eigenvalue equation, the corresponding eigenvalue equation for the function ¢[n] may be shown to be S[n]¢[n] + A{Z K[n.n'] ¢[n'] = E¢[n] (5.2) n! where the operator S[n], which comes from the electric field energy, is S[n] = Z n(p)n(p')A(p,p') . (5.3) PP' A(PIP') = Z 33(P) 33(P') . (5.4) 1 BA(1) 8A(1) The operator K[n,n'], which comes from the magnetic field energy, is K[n.n'] = E. {6[n.n'] - %6[n.n'+5pp.] - 15[n.n'-5Pp.]} . (5.5) The function 6[n,n'] = 1 if n(p) = n'(p) for all p and is zero otherwise, 6[n,n'+5pp,] = 1 if n(p) = n'(p) for all p # p' and ‘62 n(p') = n'(p')+l, and similarly for 8[n,n'-6pp.]. If instead of the Hamiltonian defined in Eq.(4.6) one uses HGFMC defined by HGFMC = H - ZNPA (5.6) then a slightly different equation for ¢[n] than Eq.(5.2) is obtained: S[n]¢[n] + x z G[n,n'] ¢[n'] = -024In1 (5.)) {n'} where G[n,n'] = K[n,n'] - 2Np6[n,n'] (5.8) and Q2 is related to E0 the ground-state energy of H by Q2 = 2N A - E (5 9) P 0 . . Equation (5.7) is of precisely the same form as Eq.(2.8) so the GFMC method can be used as described in chapter 2 to compute various quantities. As in the previous two chapters we shall restrict our attention to the ground-state energy per plaquette Eo/NP and the mean plaquette field <¢(p)>. 63 5.2 Variational Calculations 5.2.1 Disordered wave function The simplest variational wave function to try is ¢[n] = g u(n(p)) . (5.10) The energy E0 = Eel + A Emag , (5.11) where 5.1 = z ¢'In1 S[n] ¢[nJ / z |¢In1|2 . (5.12) {n} {n} r = 2 (I’m K[n.n'] an) / z |¢[n]l2 . (5.13) mag {nn'} {n} must be minimized with respect to the choice of the single plaquette function u(n(p)). Using Eq.(5.10), the energy E is found to be 0 4 E n2IU(n)|2 + A g n'(n)[u(n)-tu(n+l)-sU(n-1)] g Iul2 EO/Np = . (5.14) The correct functional form of u(n) may be determined by comparing this expression for the energy to the corresponding result for the quantum pendulum. The quantum pendulum is defined by the Hamiltonian qu = - 82/302 + qu(l - c056) (5.15) 64 where 9 is an angular variable which lies in the domain (0,2n). The wave function wqp(9) may be written as a Fourier series wqp(9) = E v(n) exp(inO) , (5.16) and then the energy is easily shown to be E n2|v(n)|2 + A9P E v*(n)[v(n)-tv(n+l)-tv(n-l)] - . (5.17) qp E |v(n)|2 Comparing this expression to Eq.(5.l4) it is clear that u(n) = v(n) (5.18) and E A N = 4 E A 4 .1 0( )/ p qp( / ) (5 9) It is easy to show that the variational estimates of the energy per plaquette for small- and large-A are EO/N = A - A2/8 + 7A‘/204a + O(A6) as A+0 , p (5.20) 2 E /Np = (2201/2 - 1/4 + 0(A-l/ ) as A9“. 0 For comparison, the corresponding limits obtained from perturbation theory are (see Eq.(4.14)) Eo/N = A - A2/8 + 3A4/10240 + 0(A6) as A+o , p (5.21) EO/Np = d(n)(2A)l/2 - d2(n)/4 +0(A'1/2) as A+o. Notice that the expressions in Eqs.(5.20) and (5.21) have the same small-A limit but that the perturbation theory result for large A is 65 considerably lower than the variational result. This is as expected since the magnetic field variables B(p) are disordered in the wave function in Eq.(5.l) with ¢[n] = ¢l[n], so that this wave function should be a good approximation of the exact ground state for small A. The optimized function u(n)=v(n) is rather too cumbersome to use for importance sampling in a GFMC calculation. Instead we shall use a simpler choice 2 u(n) = exp( -an ) . (5.22) It is a straightforward matter to minimize the energy Eo/NP given by Eq.(5.14) with respect to a for any given A. The results are almost indistinguishable from those obtained using the optimal choice over the range of A considered. 5.2.2 Gaussian wave function In the gaussian approximation, valid at large A, the ground state wave function may be shown to be ¢2[n] = eXPI-tapg.n(p) M(p,p') n(p')] (5.23) where a = (2/A)l/2 . (5.24) . 4 4 4 4 4 4 4 4 If the plaquette p, hav1ng corners at the sites x, x+ei, x+ei+ej, x+ej, is denoted by the two indices § and k where 31, 33' 3k constitute a 66 right handed set of unit vectors, then the matrix M(p,p') in Eq.(5.23) is 4 4 4 , 4 4 4 ”n'(3'x') = .1— 7- mkk-“D epr a q-(x-x') ] . (5.25) n3 q n where 24 *4 4 f (q)5kk. - fk(q>fk.(q) .9 f(q) mud-q.) = (5.26) and fk(q), f(q) are given in Eqs.(4.23d) and (4.23e). Now, if instead of having a fixed by Eq.(5.24), we allow it to be a free parameter then the wave function Eq.(5.23) may be used in a variational calculation, the parameter a being chosen to minimize the energy E0. As in the earlier chapters it is necessary to use a Monte Carlo method to compute the quantities Ee and Ema . In this case the 1 9 Metropolis Monte Carlo algorithm [29] is used to generate configurations from which Ee and Ema are computed. 1 9 5.2.3 Variational results Figure 11 shows the ground-state energy per plaquette as a function of the coupling parameter A. The crosses (+) are the results of the variational calculation using the wave function ¢1[n] given in Eqs.(5.10) and (5.22). The circles (°) are computed using the wave function ¢2[n], Eq.(5.23). The solid and dashed lines are small- and large-A perturbation theory results given by Eq.(5.21). At large A the 67 1.5 1.0 EU/Np 0.5 Figure 11: Variational estimates of the ground state energy per plaquette versus 7\ for the n-space formulation of the U(l) theory. 68 wave function ¢2[n] is clearly the better one; this is to be expected since ¢2[n] has built into it the explicit couplings between different plaquettes appropriate to the large A limit, which are absent in ¢l[n]. Notice, too, that for small A the two variational estimates are almost identical. It is at first sight surprising that the wave function ¢2[n], which is constructed to be a good approximation of the ground state for large A, should also be quite accurate at small A. Upon further consideration, however, it is seen that if the variational parameter a is chosen to be very large then the function ¢2[n] is sharply peaked in the region n(p)=0 for all p. In terms of the magnetic field variabled B(P), the state is completely disordered, and so we see that ¢2[n] should also be accurate at small A. It is interesting to see how the variational parameter a in the wave function ¢2[n] depends on A. In Figure 12 the quantity a/ah, where ch = (2/A)l/2, is plotted as a function of A. The variational parameter exhibits very striking behaviour at A = 1.1, indicating the presence of a phase transition. Because ¢2[n] is a good representation of the ground state at both large and small A, then the fact that a phase transition is present strongly suggests that the exact ground state of the theory also must exhibit a phase transition. The presence of a phase transition in the state described by ¢2[n] is also evident from the variational estimate of the mean plaquette field <¢(p)> = Emag/Np' This is shown in Figure 13. Again the crosses (+) are variational estimates using ¢1[n], the circles (0) are obtained using ¢2[n], and the solid and dashed lines are small— and large-A perturbation expansions. 69 2 6 1 I I 1 1 I T I T r 1 ...... 2.2 - _ o o I: P O 1 ts .' \ L _ as 1.8 .0 + «t in 1.1. b i _ .0. 00...... l- + .....Ooj .4 .1 I I J. cl 1 #1 I 7 Figure 12: a/ah versus A for the variational wave function ezln]. 7O 1.0 Figure 13: Variational estimates of the mean plaquette field versus A for the n-space formulation of the U(1) theory. 71 5.3 GFMC calculations The GFMC method with importance sampling may be applied precisely as described in chapter 2 to the present example. It proves most convenient to use algorithm 1 presented in Section 2.2. Due to the particular form of the function G[n,n'], which has only a small number of non-zero terms, it is possible to compute directly the normalization integral denoted by Z(x) in chapter 2 and defined in Eq.(2.20). Furthermore, again because of the special form of G[n,n'], the configurations n sampled from the kernel conditional on n' will differ from n' by at most one unit at a single plaquette, i.e., n(p)=n'(p) for all p, or n(p)=n'(p) for all p#p0 and n(p0)=n'(po)1l. Since all the matrix elements are known, it is a simple matter to sample the kernel as a discrete probability distribution. All the results described below were obtained on a 3 x 3 x 3 lattice and used an ensemble of approximately 100 configurations. The results are averages over 1000 GFMC iterations and required approximately 100 seconds of computation time for each value of 02 used. Figure 14 shows the ground state energy per plaquette Eo/NP as a function of A. The crosses (+) are GFMC estimates using the disordered wave function ¢1 for importance sampling, and the circles (0) are the results obtained using ¢2. The solid and dashed curves are the variational bounds, obtained in the previous section, using ¢1 and 02 respectively. The GFMC results obtained using ¢2 for importance ' sampling interpolate smoothly between the known small- and large-A 72 0.5 c 0.0 0.5 1.0 1.5 2.0 2.5 Figure 14: GFMC estimates of the ground state energy per plaquette versus A for the n-space formulation of the U(1) theory. 73 limits. The results obtained using ¢l, however, fail to be accurate for A > 1.3 and continue to lie close to the corresponding variational estimate; these GFMC estimates are not consistent with the variational bound obtained from the trial function ¢2. It appears that the disordered state is metastable with respect to the GFMC iteration, at least for the ensemble size used here, and cannot converge to the actual ground state. This may be interpreted as evidence for a phase transition in the ground state of the theory. The trial wave function o1 is qualitatively different from the true ground state wave function for A > 1.3 where the ground state is described well by the function ¢2 with its explicit long range couplings between different plaquettes. So when this function is used for importance sampling it fails to direct the diffusion into regions of configuration space where the exact ground state wave function is greatest. Apparently, though, there is still a low energy state resembling the disordered phase which is metastable with respect to the GFMC iteration. This metastability is due to the fact that ¢1 biases the ensemble of configurations in favor of those lying in the region of configuration space dominated by this low energy state. The cross over from the disordered phase described by $1 to the harmonic phase described by ¢2. the two phases being qualitatively different, is a signal for the phase transition. This conclusion is further supported by the calculation of the mean plaquette field <¢(p)>. Figure 15 shows this quantity as a function of A for the GFMC calculation using ¢2 for importance sampling. The crosses are GFMC estimates based on the mixed expectation value Eq.(2.62) and the circles are variational estimates based on the trial 74 1.0 0.4 Figure 15: GFMC estimate of the mean plaquette field versus A computed using the trial wave function ¢2[n] for importance sampling. 75 function oz. The solid curves are small- and large-A perturbation expansions. The variational estimate at small A differs slightly from the correct small-A limit. The GFMC method provides a correction to the variational results which is consistent with the perturbation theory result. Notice that in the region of the phase transition, A z 1.2, the GFMC estimates differ considerably from the variational estimates indicating that in this narrow region the function ¢2 is not a very accurate representation of the exact ground state wave function; presumably the nature of the phase transition is different in the exact ground state and the harmonic state ¢2. Figure 16 shows the mean plaquette field computed using ¢l for importance sampling. The failure of this disordered wave function to accurately model the ground state of the theory and also the metastability discussed earlier are evident at large A. It is interesting to compare the results of the present chapter to those of the previous one where the calculations were performed in the space of states in which the magnetic energy is diagonal. The agreement between the two sets of results is striking. The fact that these two very different formulations of the same problem give very similar results gives us considerable confidence in the GFMC method. Further discussion of the results of this chapter may be found in appendix B where the calculations described here are compared to similar calculations on the U(1) model in 2+1 dimensions. 76 1.0 0.4 Figure 16: GFMC estimate of the mean plaquette field versus A computed using the trial wave function (:1an for importance sampling. CHAPTER 6 The Hamiltonian XX model Because of the unconventional nature of the n-space formulation used in the previous chapter it would be useful to apply the same technique to study a different model. The Hamiltonian XY model admits such a treatment. The reprinted paper in this chapter describes calculations on the XY model which parallel those of the previous chapter. Again it will be found that the n-space formulation of the problem leads to a very simple implementation of the GFMC algorithm. 77 PHYSICALIEVIEWD W21NUM8 78 151mm. l9“ Application of the Grem’s-fimction Monte Carlo method to the Hamiltonian 1? model David W. HeysandDanielR. Stump WdflpicsaadAMy, lithium University, ”MIMIC!” (Received lZSqtuahulm) AnapplicationoftheGreen's-ftmction MonteChrlomethodtotheHsmiltai'nnXYmodelis dmaihed. hnpanncesamplingisimplanmtedwhhtwouislwsvefmcdau—aiecurupcndiaa msdisudaedmusndonewhkhmcupaatathecmrdadcmdaivedfrcmthespbnnap- proaimsticnofthemodel. Optimaltrialfunctionsarewtsinedfrmthevsristicnslprindple. The MuteCarbrmulusnhtapretedwithnpldwtheMu-Mphncuwfion. I. INTRODUCTION The (item’s-function Monte Carlo (GFMC) method is a numerical technique for studying properties (i the grotmdstateofsquantumsystanwith manydegreesof freedom. quantum many-body pnibltxns.“2 We dmcribcd an appli- ation of this method to the chut-Susskind Hamiltonian formulation of the compact U(1) lattice gauge theory tn 2 and3spetisldimensionsinspreviouspaper.’ lnthispa- peweshalldescrihesimilsrcslculstionsfortheHsmil tonian formulation of the XY model. TheXYmodel,alsocalledtheclsasicslplsnsrspin model, dacrihes classical twodimensional spins located cm a two-dimensional cubic lattice with s nesrest~neighbor interaction mergy proportional tog-.5". The aim ofclss- sicsl statistical mechanics is to compute the prutim function Z- 2 exp —B£§('i)'§(‘s’+£) "Li (1.!) Animportsntfmturedthismodelisthexcsterlitz- Thculessphssetransition,‘ whichscparatcsaphasein whichthesumoverststesisdominatedbyspin-wsvefluc- tusticmcfsnorderedststesothatthespindirectionssre highlyccnelated,andsdisorderedphsseinwhichthe correlationhetwemspindirectionsissmall. Thisphsse transitionisdrivmhysninterestingmechanism: mrticer hthespinfiddwhichsncoupledinpsinatlcwtan- mummhindtoprnducesdisadaedstateatsclitio calvalueoffl. Topological configurations thstproduce ha-mgedisorderofthefieldsmsyalsoherdevsntto thetransitionfromsnorderedtosdisordaedvaeuum stateinlatticepuge theories.’ TheXYmodelisimpor- tsnttothelstticepugetheotistssthesimplcstessmple ofthismechsnism. lnthisworkwesreinterestedinthis modelssatatinggrotmdfortheGFMCmethod. TheMetmpolisMontem'loslgct-ithmhsshemap- pliedtothe computationofthepartitim function (I. ll.‘ TheHsmiltonianformulationd‘theXYmodelctmsists dammmflsmiltonianthatdescribessuie- dimmsionslchsind’intaactingspim.’Theseccnddi- mmsionistime. Thectmnectionhetwemthisformulation ndthatd'Equdl'nthatthepsnitionfunctionisslat- It was originally developed for application to - ticespproaimstiond’theFeynmsnpsthinteualofthe quantumsystem. Fortheaskeofcompletmmswederive thiscmnectionintheAppmdixofthispaper. ltisthequsntumHamiltonisntowhichweapplythe GFMCmahod. Animpatantandevmamtislmspectd’theGF-‘MC methodistheuseofimpcrtsncesampling. Animpor- tance function, which should resanble the ground-state dgmfmcumismedtohisstheMonteCarlossmplingin favor of regions of configuration space where the wave function is greatest. The variational principle provides a wsytoconstructusd’ulimportsnceftmctions. lntheXY- model cslculaticns,ssintheU(l)-puge~theory calcula- tionspresmtedinourpreviwspsper,weusetwoimpcr- nucefunctions. Theftrst describessdism'deredstste; im- portsncessmplingwiththisftmctionisgoodatweskcou- plinghutheccmesincressinglywoneasthecouplingin- erases. The second is derived from the spin-wave ap- puimationofthemmdstatesndyieldsgoodimpor- tsncessmplingathcthstrongandwmkcouplings. The variationalcalculstionthstoptimizesthetrislftmcticnis done analytically for the disordered state, but numerically ft! the spin-wave state, by the Metropolis Matte Carlo method. Thevariationslrmultssreinterestingintheir ownrightsstheygivesomeindicaticnofthenatureofthe mmdstatesssfunctiond'theconplingcmstsntThm theGFMCcslculsticnseatmdtheacauacyofthevaris- tionalmlculations. Theoutlineofthispapaisasfollows. Wedefinethe HamiltonianXYmcdelandeaplsinourapplimtiond'the GFMCmethcdinSecJ]. We dacrihethevsriational calculation that yidd trial functions for the GFMC im- portsncesamplinginSean. WediscnsstheGFMCre- cultsinSecJandmskesanemmarizingtansrhin Sec.V. II. DEFINITIONOPTHBUODEL mmdmnw is’ H--‘ 2 ——12[l+ul(0,—0,+.)], (2.!) [Cl ac" l-l withthepaiodichcundarycmidition Gnu-0,. HereO, banan‘levarishlethatdeftnesthedirectionoftheith I704 22 APPUCAHONOFflIEGREEN'S-PUNCTIONHONTECARID... spin; thus its range is (-w,rr), and wave functions are paiodic in 0, with period 21:. H is defined such that the ground-state clergy is negative; we let -Q2 dmote this energy. In the calculations to be discussed, we formulate the eigenvalue problem in the space of variables cmjugate to 9,; specifically, we write the ground-state a'genfunction as N fielszdfiilexp :2 more...) . (22) 1’ III wherepeiiodicityina, requiresthatthevsriablen, bean integer. Then the ii-space a'genfunction d(ii) obeys the equation -Q’¢(m=sm¢(a)—izx(a.a'ma'), (2.3) it. where N S(3)= 2 (Hi—"(+4? (2.4) [all and N l K(ii,ii')= 2 warn-,- ii,ii'+e,) l-l +-§-5('fi,it'-é})] , (2.5) whereé‘, istheN-ccmponentvectorwithjthcompcncnt 5”. To put the a'genvalue equation into a useful form, we define umslausmmm ; (2.6) this function satisfies the equation 1(a)=a.2x(a.a')[c’+5(a')1-‘X(a'). (2.7) to TheGFMCmethodappliestosnequationofthisform. The method consists of simulation of a diffusion process with branching. The branching probability is proportional to [Q’+S(ii')]" and the diffusion is govaned by K(ii,ii'). We refer to Klfiji') as the (item’s function, slthcughinthisproblanit'uaorinu'oducedastheinvaae dan operator. TheGFMCmethcdismostpowerfulwhencombined n'th an importance-sampling tmhnique.’ In very large system this technique is necessary for obtaining accurate rmults. We implement importance sampling by introduc- ing a trial wave function dfl'n’), which should be an ap- proximation ofthe actual eigenfunction. That we define thcfunction Hilby Nil-Orfilflh’) . (2.8) This obeys the equation flilskz flx(3,i')[Q’+S(fi')]"F(h") . 1" ‘14?) (2.9) 79 1785 whichistheequationtowhichweapplytheGFMCdif- fusion process. Now the diffusion is govancd by the listed Grem's function dfiiillflifri ')Nrfii '). The GFMC method is based on iteration of Eq. (2.9). To iterate the equation we must take Q2 to be the given quantity, and regard A as the eigenvalue to be determined. Then iteration yields a sequence of functions Wis), Fmta), . . . ,F"(ii) dd'rned by 47(3) ——x(a,a')[Q’+S(a')]-' ”(3') Ftr+ll(a)=A(rlz is xl’Wii') . (2.10) where the constant A‘" may vary from one iteration to the next. It can be shown that F‘"('r'i) approaches the ground-state eigenfunction with enemy -Q as r—no, in- dependent ofthe initial function F°’(ii); and that the nor- malizationcbcystherelaticn _ Ftr+ll(-fi) Am hm ———-— , .... rmta) 1 whereAistheccuplingccnstantforwhichtheground- stateenergyis -Q . Constant normalization ofthefunc- 6?? 1""(3) (after convergence to the limit) requires 1' IA. The GFMC algorithm for solving Eq. (2.9) is a simula- tionofadiffusicnprocesswithbrsnching. Attherthstep oftheprocesswehaveanensemble floffteldccnfigurao ticns 7,-[3'5 a-l,2,3, . . . ,N,]; let 13(3) denote the probability distribution of 7,. The nest msemble I’Ngisobtained from 0’,intwosteps: (i)Esch i;branchsintok,newpoints,wherek,is snintegerpickedbyarandomprocesssuchthattheex- pectedvslueofkfis (2.11) -, mamas» 13'19’+5(n.)l" —' . § ‘143'.’ Thepcssibility k,-0isallcwed. Hmii",which maybe thoughtofssaguessofthevalueofkcsnvsryfromone ita'ationtothenext. (iilThenmchofthek, pcintsismovedfrom ii',toa newconfigurationh’ehosenfromtheprobability distribu- tion (2.12) mammal/man . (2.13) gunman/man I‘ Notethat theform d'K(3.i') bplia that 'n’ differs fromh”,byatmostcneunit. 1'hemsanble5’,+.istheresnltd’pmccssingsllofthe guns of f, in this way. The probability distribution 7-H” I786 , N. . r) r...(ii)=A.‘,’—z ‘7 ” K(h’.h"l[Q’+S(ii’)]" ”NH 3' dflii') XP,(ii') . (2.14) That is, the evolution of P,(ii) is the same as Eq. (2.10) with air—1'— =1"). (2.15) H») Thedore. P,(ii) approaches the eigenfunction 17(3) as r—scc. Also.sinceP,(ii)andP,+.(ii)havethessmenor- malization, specifically 23?,(i)=l for all r, aftea suf- ficient number of stepsin the diffusionweshall have tr) N’ M _N In}. . '+l (2.16) This provide an etimate of the eigenvalue A after each ite'ation. Note that A3" controls the sin of the ensenble; in practice we readjust the value of A1," every few itea- tions so as to keep the ensemble size approximately con- stant. Thus the simulation yields an etimate of A and a sequence of ensenble of ii-space configurations with probability distribution F (ii ). Use of the trial function ‘7 is called importance sam- pling. 'I‘hediffusioninthespaceofiiconfigurations is controlled by the biased Green‘s function 47(3)K('ri,ii ')ldrlii '). The factor ”(3)/ma ') biase the diffusion in favor of move ii '-oii in directions that increase drfii). Ifdr is an approximation ofthe ground— stste eigenfunction, then this bias acceleate the conver- gencetcthe ground statcsndreducefluctuationsofthe etimate of the eigenvalue A. The importance-sampling technique also provide a way toetimateexpectation valueofopeatorsinthegrormd state, provided a; is a good approximation of the e'ger- functiond. Ifd; diffesfromdbyansmmmtofcrdee, thentoordee’wehave (11.4] )g (‘l4|5r)_“r|4 l‘r) (217) (H‘) (N‘r) (¢r|¢r) ' ' Theleft-handsideisthedesiredeapecmionvalueofan opeatorA. Theseccndtermontheright-handsideis simply the expectation value in the trial state. The first term on the right-hand side, which iscslled themixed ex- pectationvsluecanbeetimatedss (“A m) ‘ watchman-‘).... (uh) acumen-').... whee ( ). denotes the average of the erclcsed quantity ove the ensenble geneated by the GFMC diffusion. Since Eq. (2.17) is only valid to order 8, this estimate is not trustworthy if (A )1 and (A ) are very diffeent. The trial function e; is ordinarily obtained from a vari- ational calculation. Thus the GFMC method can be , (2.18) 80 navmwnzvsawnnmnswm a thoughtd'eanextetsiond‘thevariationslprinciple, thatimprovestheaccuraeyofnumeicsletimate. The GFMC detemination of the e'genvalue A is in principle exact,evetifdrisnotagocdapproximationofd;but thatisonlyforslargeecughenenbleandinpractice thecalculstionsarenotfesibleifdrdiffesfromdtco much. Expectation value computed from the mixed ea- pccmicc valuearevalidtoordeMr-dlznoiq. (2.17) give the orde-(dr-dlcorrectiontotheordinaryvariso tionaletimate. Inaddition,tthFMCapproachcsnins dicatewhetheavariationalwavefunctionisanaccurate rqn'eertationdthegroundststebytetingwhetheit wa'kswellasanimportance-samplingftmction. ltcsnbe prove, for example, that fluctuations in the measurement ofAbyEq.(2.16)approachzeoasthetr-ialfunctionap- proachetheexacteigenfunction. Inthenextsectionwedescribethetwotrialfunctionsto beusedforimportanccsamplingintheGFMCcalcula- tions, and variational calculations which (ptimiae the choiccoftheefunctions. III. VARIATIONAI. CALCULATIONS Weshallconsidetwotrislwsveftmcticnstoapproxi- matethegrmmdstateoftheXYmodel. Theftrstisde- finedasafunctioninthespaceofeconfigurationsas _, )v '|(0)-nfl(9;-0‘+|); l-l themegy(¢)|H|¢i)istobeminimizedwithrespectto thechoiceofthefunctionuhu). Itcanbeshownthatthe minimumeiegyisobtsinedifutvlisthegmtmd-state eigenfunction of the Hamiltonian of a quantum pendu- lum, (3.1) a! h I—z—a‘”2 +Ml—cosm) s whee -aga)gw. Theruultingvsristionslboundonthe eiegyperspinis 2 —% 5 “21+“! 9 (3.2) (3.3) wheeeoisthesmsllesteigenvalueofh. Weshallpreent ourresultsintemsofanotheenegyEmrathethan -—Q’,definedby Eo'm-Qz: aotethatEoistheground-stateeiegyof N a: N -‘2'§‘7+1‘2‘[l—cos(9,-0H.)]. 'I'hcvartahmal' . m. oonhselmfi. IS (3.4) (3.5) 50 — 390 . N (3.6) Thesmall-andlarge-Alimitsofeosre l8 2: 256 eozAm—%+O(A'm) as A—m . FcrccmpariscntheelimitsfcrEo/Nareessilyshcwntc be 2 coal-17+ +O(A‘) as A-oO , (3.7) A2 SA‘ . N~A-— 0(A ) A-OO , 50/ +— 768 + II (3.8) so/Nzimdm—-,-d1(N)+0(A-"’) as A»... . whee V: w - . 11’ d(Nl- N [l—ccsN] sin” ’(3.9) forachainostpinswithpeicdicbctmdsryccnditicns; thevalue cfd(N) is approximately 0.90 fcrNgreatethsn 10. Thus so and 50/1)! have the same small-A limit, but so is greste than Eo/N for large A. The trial function (I, describe a discrdeed state of the spins. Specifically, the correlation betweei spins separat «I by a distance I: is, for this wave function, e . k (¢)l°°‘(9i+r-91)l¢i)-=[I_'dwu2(m)ccaw (3.10) which decrease exponentially with k. We expect in to be a good approximation of the eigenfunction for small A, where the ground state is disordered in this way. But it can already be seen by comparing the limiting forms (3.7) and (3.8) that (1, become less accurate as A increase. Thesecondtrisl wave function isdeignedtobesccu- rateinthelsrgeA limit;itturmcuttobesccurateat smallAaswell. Itisdeftnedintheccnjugatespacecfii configurations as ‘2fillsflp (3.") - i0 2 "1AM? u' wheea is the variational paramete,snd m 2 A (3.12) Themativaticnfcrthisfcnnisthatwitha-litdupli- catethegrcundstated'thesfin—wsvespprcaimaticncf themodel, whichiskncwntobetbee’geistateinthe hrge-A limit. The spin-wave szpproximsticn consists of replacing l—ccs(A9)by -(A9) tn the Hamiltonian, and extendingtherangeofo, frcm(—tr,rr)to(—co, an). The reulting model is solvable since its Hamiltonian '3 qua- dratic; itsgrcund stateisd, with as], but wheethe variable a, take a continuum of value. We emphasize thatthetrialfuncticndzisnctanaiveharmcnicspproxio maticn,becausethen,arereuictedtointegevalue;this isnecessyytoprcsevethepeicdicitydthewavefmc- tiarinOspace. Weevaluatetheexpectaticnvalue(h|1!|h) numei- cally.usingtheMetrcpclisMcnteCsrioslgorithmtcgei~ eate O ”I d MW li,,il;,ii,...,l.] 'lth Aw 3- in? —w—r) w’,-f N P, AWCAflONOPITIEGREEN'S-WNCTIONWCARID... 81 rm My th'stnhuticn 4,1. and etimating the expats- ticnvaluebytheaveagecfthecpeatcrovertheecon- figurations. Thisisdcnefcr many value of the variation- alparametea. 'I'heresultingdatacntheenegyasa functicncfoistheifittoapclyncmisldsufl'tcieitly hrgedegreetogiveagccdftt. Andftnallyweminimize thepolynomislwithrepecttca. Figurelissgraphofthevalueofathatminimizethe elegy,assftmcticncftheccuplingccnstant A. Thee- rmbarsarecslculatedinastnigbd’crwardwayfrcmthe standardercrsinthepclyncmislccefftcimtsfcundby thelest-square fitmenticnedintheprevicuspsrsgrsph. 1hecalculaticnisfcrachsind50spins,withpeiodic bctmdsryconditicn. Asantidpated.aspproachel.thespin~wavevalue,at largeA. AsAdecreseJrincresseandscdzbeccme moreshsrply peakedatii=0, which implieamcredisor dedstateinaspace. Theeisafairlydramaticvana ticnofafcrAnearl. Asimilsrvariaticnalcalculatico fortheU(l)lattice gauge theoryinthreedimensicns. dis- cusaedinkef.3,hasadisccntinuityinthevaluecfaasa function of A.indicatingsphase transitioninthat model. FigureZshcwsthevsriaticnalbctmdsonEo/Nasa functioncfA.fcrbcthtrislfuncticns¢.andd;,alcng with the large» and small-A limits give: in Eq. (3.8). Clearly the trial function ‘2 deived from the spin-wave approximation is more accurate than the disordeed func- ticndr,fcrA>l; itseiegyspprcachethecorrectlsrge-A limit, as it must by construction. The spin-wave function isalsoagccdapproximaticnatsmallA,wheeitsenegy is tally slightly large than that of the discrdeed state. Bothfuncticns approach theccrrectsmsll-A limit. Thetwottialfunctionshanddzareanslogsofthetri- al functions that we used in U(l)-lattice~gauge-thecry cal- culaticns.3 The analog of 1’) is a product of single» plaquettefunctions,sndthesnslcgcfd;deivefromthe free~fteld lurmcnic approximation of the U(1) gauge theory. InthenextsecticnwedecribethereultschFMC elculaticnsthatttaetheetwotrialfuncticns for impor- tancesampling. 0"! 1.10 l a 0" l ‘. i "o ....... 0.1. e a v 1 fl PIG. l. VariatienlparameteavscouplingemetantA. I788 I.” ‘( mfiufi 7‘17’1 In ' also FlG.2.Variationaletimatedthegronnd-ststeenegype spinvscouplingccnstantA.'I'hesolidanddssbedcurveare peturbation expansions forsmallandlarge A. respectively. The a'uaes(+)andcircles(0)arevariationalestimateswithtrisl wave functions in and”. repectively. Errorban aremuch amallethanthesizecfthepcints. IV. MONTE CARLO RESULTS Figure 3 is a graph oon/N. the grctmd-state elegy pe spin of the Hamiltonian (3.5), as a function of the cou- pling parameter A, from Greei‘s-function Monte Carlo calculations with importance functions in and ‘2. The curvesarethe variationalboundscbtsinedinSec.III,snd the points are the GFMC results. The GFMC calculs~ tions used an etsenble of approximately lm configura- tions; this essenble size change with each iteration. The results in Fig. 3 are aveage over 8m iterations. Each GFMC point required approximately 90 sec of computa- ticntimeonaCDCQbeflOccmputeatMichigsn State Univesity. m V v V i ‘7 v V a v A FIG. 3. Mmteercetimated’thegrcund-stateeiegype qiavsccuplingccnstsntA. Thesolidandthshedcurvesre variationsletimatewithuialwavefunctionsd,andd;.repec- lively. Thecrcsset+lsnddrcletolarethteeroresults with importsncefnnctionsd.andd;.repectively. 82 nAvmerzvsAwnnAmenswm n ThereultsshownarefmachainofSOspinswith .peicdicbcundaryccnditicn AsAvariefrchtcucthe elegy intepolate between the small-A asymptotic behavior described well by the discrdeed wave function w. and thehsrmcnic spin-wave behavior described by”. Thecr'cssovefrcmoneformtothectheoccursforA~l. Thetwo MonteCarloetimateareslmcst equal,and are ccnsistert with the variational bounds. However, theeisatendercyfcrtheGFMCetimateobtsinedwith thediscrdeedfuncticndfioliehigheinenegythan thstobtainedwith dzinthe region Azl. Furthemore, thefcrme etimate have greate uncetainty,asindicated bytheercrbars,thsnthelatte,forwhichtheerorbsrs aremuchsmallethanthesizeofthepcintplcttcd. Thee ercr bars come only from the fluctuation associated with stochastic sampling. Thee two terdetcie are not tmex- pected;theyreflectthefactthat¢.isnctagcodapproxi- mationcfthegroundstateforAzl,wheethespinssre mcreccrrelatedthanintl. Itisinteetingtoccmparetheereultstotheanalo- gous calculations for the UH) lattice gaugetheory in 3 and2spatisl dimensions. Inthethree-dimensicnsl model, the Monte Carlo reults obtained using the discrdeed wave function for importance sampling are definitely dif- feent than thosecbtsined with the harmonic wave func- tion.intheregionoflargeA;infactthefcrmereultssre inconsisteit with the variational bound provided by the harmonic wave function. We inte'pretthisssevidenceof thephssetrsnsitioncfthethree-dimeisicnsl U(l)gauge theory: thediscrdeed stateismetsstablewith repectto ther-‘MC diffusionproces. Inccntrast,theMonte Car- loreultsarethesamefcrthetwoimpcrtsnceftmcticnsin thetwo-dimeisicnslmcdel;thisisccnsistettwiththefact thsttheeisaophasetransiticninthetwo-dimmsicnsl model. OurXYmodelresultsshowevidenceoftheKcste-litz- Thcules phasetransition,inthatthediscrdeedfuncticn dcenotprovide effective importance sampling for Azl. Thediscrdeedstateisnctmetsstableasitisinthe three~dimetsional U(1) gauge theory,buttheeiegyeti- mateobtainedwiththediscrdeedimportancefuncticnis slightly large, and has large fluctuations, than that ob- tained with the spin-wave function in this region. The diffeence between the 11' model and the U(1) gauge model is explained by the fact that the Koatelitz- ‘l'houlesphsaetransiticnissninftnitecrdetransiticn, whilethegaugesmcdeltransitionissaeccnd-ordertransi- ‘I‘heKcstelitz-‘I'houlwrenormslizationogrcupcslcula- ticnpredictsthstthephssetransiticndtheXYmodel cccursatA-l.02;thispcintisdiscussedbtieflyinthe Appeidix. That valueis pefectly consistent with thein- tepretsticn ofour results given above. ForAsl.02the groundsflteisdiscrdeedaofiactsasaneffectiveimpcr- tancefuncticn;butfcrA>l.02thespindirectionsare moreocn'elstedthanindlsothisfuncticngiveweske hnpcrtanceasmpling. FigureflanndflblshcchnteCsrloetimateofthe cmrdaticnfuncticncfneighbclingspins r-(I-”(aj-a‘+|)) . (‘on Q APPLICATION OF THE GREENS-FUNCTION MONTE CARID . . . F164. The expectation value of l-ccs(0,-0,—+,) vs cou- pling constant A. The curve are peturbsticn expansions. The triangle (A) are simple expectation value in the variational wavefunctions,andtheeosse(+)areMcnteCarloetimate dthe mixed expectation value, F4. (2.17). The trial functions ared;fcr(a)and¢.for(b). Notethat”isrelatedtotheeiegy£oby _l_dE__9_ N TA ' 'I'heMcnteCarIOpointsinF'tgs.4(a)and4(b)arecbtained from the mixed expectation value, i.e., liq. (2.17), for the importance functions d, and in. repectively. The curves m the graphs are from small- and large-A perturbation theory. Here thee are marked diffeence betwee) the Monte Carlo reults. In particular, the GFMC etimate cfrobtsinedwiththedisordeedimportancefuncticn have large uncetainty and differ significantly from the mdinary expectation value in m, in the region Ag]. Again, this is precisely what we expect from calculations 7': (4.2) with an importance function that does not approximate . the ground-state eigett'unction. It is interesting to note that the GFMC and variational etimate of 7 obtained with the spin-wave function 4); are almost equal for all A, suggetingthatdzisquiteagocdrepreentsticnofthe dysfunction 83 I789 V. SUMMARY Inthispapewedecribereultsofanapplicaticncfthe Green's-function Monte Cario method to the Hamiltonian XY model. Thee calculations are parallel to calculations decribedinanerliepapefortheccmpactUllllattice gauge theory in 2 and 3 spatial dimensions. Intheemcdelsanimportsntissueistheexisteiceand natureofaphssetransitionseparatingadisordeedphase and a phase in which the model is accurately described by its harmonic approximation. We find that the GFMC re sults give a good indication ofsuch a phee transition. In particular, we can judge whethe a wave function reen- ble the ground-state eigenfunction by its pe'formance in rducing fluctuations when used in the importance- sampling procedure. In our calculations the disordeed trial function peforms poorly for value of the coupling cmstsnt for which the harmonic wave function approxi- mate the ground state For the threedimetsicnal com- pact U(1) gauge theory the inadequacy of the disordeed trial function is obvious: it yields elegy estimate that are greater than the variational bound provided by the harmonic wave function, at least for the ensenble size that weusein the GFMC diffusion. FortheXI’mcdel thisinadequacyismoresubtlebutcanbeseeninthelarge fluctuations of elegy etimate. The GFMC method cffes a second way to jrdge whethe a trial function represents a good approximation of the ground state, based on the mixed expectation value, i.e., Eq. (2.17). If of approximate d then the mixed ex- pectation valueofan cpeatcrA isnearly equal totheex- pectation value of A in ‘7; if thee two quantitie are quite diffeert, the) or cannot be a good approximation ofd. Thus, for example, the increasing diffeeice between thetwoestimateofi’eAincressebe-ycnd l inFig. 40>), is soothe indication that the disordeed wave funCo tion doe not reemble the eigenfunction for Ag 1. The Monte Carlo results imply by thee conside'aticns thatthegroundstateoftheXYmcdel changefroma discrdeedstatetoastatebettedecribed byshsrmonic wave function for Aal. This value ts in agremient with the Kcste'litz-Thoules renormalization-group mislysis, which predicts a phase transition at A- 1.02. APPENDIX TheocnnectionbetwemtheHsmiltcnisntZJHndthe partition function (1.1) of the classical XY model deive from the Feynman path integral of the quantum problen. The path integral for the Hamiltonian H is, with basin-cum. z: IdOfltk“, (Al) whee 40m) denotes integration over paths' in the space of 0cmfiguraticns,andA istheimaginary-timeacticn A= fair: +A[l-al(9,+.-0;)]. (A2) l7!) Wencwconsideadiscreteapproximaticncfthetime coordinateletrtakethevalue t,=aj, j=0,l,2,3. . .. (A3) withintevalctobespeeifiedlate. Ifoissrnallccm- paredtothetime ove which 9,(t)varietheiwe msyre- placetheintegraloverbyasumovej,andthetime deivativebyadiffeence;i.e., Idt-vaz , 1 (A4) d0, I . . dt -0 a[9(1,j + I)-9(t,j)] , whee 0(i.j)=0,(t,). Again for small a, we may assume that 0(i,j + l )—0(i,j) is small and approximate [0(7.j+1)-01i.j))’=211-cct191i.j+1)—ou,j)n . (A5) With thee substitutions the action become A - E 3'?) 1-ccs[9(i,j+1)—ou,j)n +cA)l-ccs[9(i+l,j)—9(i,j)]l . (A6) Atthispointweletheintevalcbe(l/2A)m;thet DAVIDW.I-IBYSANDDANIELR.S’IUMP 84 IS II! A = 2 {2-m[0(i.j+ l)-0(r‘.j)] U A 2 —ce[0(t'+l,j)-0(i,j)]] . (A7) Thelattice“pathintegral”ove9(iJ)ispreciselythepar~ tition function (1.1) for classical statistical mechanics of theXYmcdel,wheethedir-ectionofthespinat(i,j)is defined by the angle 0(i.j), and the invese tenpeature is in (A8) 2 This deivaticn of the connection betweei the one- dimensicnsl quantum problem and the two-dimensional classical statistical mechanics problen is the invese of the usual deivaticn,’ which starts frun the partition function anddeivetheHamiltcnianHasthetransfematrixin the limit that one ofthe dimensions become continuous. The Kcstelitz~Thculess phase transition occurs at in- vese tenpeature £22.24”, according to a renormalization-group calculation.‘ Theefore, by Eq. (A8) the critical vslueofA is 1.02. This value is pe'fectly ccnsistert with the results (1' the GFMC calculations decribed in Sec. IV. 3.. ACKNOWLEDGMENTS ‘I‘hisworkwassuppcrtedbytheNaticnalScieicchtm- datitm under Grant No. PHYBI-OISZO, and by a grant fromControlDataCcrpcraticn. 'M. H. Kale, Phys. Rev. m ”9] (I962); Phys. Rev. A 2., 250 (I910); M. H. Kale. D. Leveque. and L. Velet. it'd. 2, 2178 (I974). 3D.M.CepeleyandM.H.Rsle,ialcntchrlclletlsodsia mam) Physics, Topics in Oman Physie. Vol. 7, dited by thindespringefliewYork. I979). Thisrefeesccisa emnpreheiaive review (I the Green's-function Monte Carlo method. 3D. W. Heys and D. R. Stump, Phys. Rev. D21, 2067 (I983). ‘1. M. Rmtelitz and D. I. Thoula, I. Phye C 5 I181 (I913); I. N. Ketelitz. M. 1, I046 (I974). ’A. N. Palm, Phys. Lett. 22B, 79 (I977); Nucl. Phys. 3129, 42911977”. Banks, It. Mym. and I. chut, W. m 493 (I977); D. R. Stump, Phys. Rev. D 21, 972 (I981). ‘1. Tdochnik and G. V. Outer, Phys. Rev. B n, 3761 (I979). 7E Fndkin and L. Sneakind. Phys. Rev. D 11, 2637 (I978); I. “In, Rev. Mod. Phys. 11, 659 (I979). 'Theimpcrtancesamplingprocelmethtwemeism'milsrto thatdecribedinthepapebyRalos,Leveque,sndVe-Ie (Rd. I). Scab Ref. 2. CHAPTER 7 Summary and conclusions The Green's function Monte Carlo (GFMC) method has been adapted for application to Hamiltonian lattice gauge theories, and has been applied to the SU(2) and 0(1) theories. The results obtained so far are restricted, by the availability of computer time, to estimates of simple quantities, specifically the ground state energy per plaquette Bo/Np and the mean plaquette field <¢(p)>. on a 3 x 3 x 3 lattice. This lattice is small compared to those used in path-integral Monte Carlo calculations, but the average quantities calculated here are rather insensitive to lattice size. This is indicated by perturbation theory calculations: for small A the results are independent of lattice size, and for large A the results are only weakly size dependent. The GFMC calculations use a variational wave function as an importance function to bias the Monte Carlo sampling procedure in favor of regions of configuration space in which the wave function is large. Thus, if the variational wave function is a good approximation of the 85 86 exact ground state wave function, the fluctuations of GFMC estimates are greatly reduced and the rate of convergence of the estimates to their asymptotic values is increased. By comparing the GFMC results to the variational results one can obtain some indication as to how accurately the variational wave function models the exact vacuum state. Some care is necessary, however, when interpreting the results in this way. If there is considerable disagreement between the variational and GFMC results then it is clear that the variational wave function is not a good representation of the ground state.. The converse is not true. If the GFMC results lie close to the variational results one cannot conclude that the variational wave function is a good representation of the ground state. Calculation of other quantities might reveal a considerable disagreement. A good example of this kind of behaviour is provided by the SU(2) results obtained using a disordered variational wave function discussed in chapter 3. There it was found that the GFMC estimates were close to the variational results even at large A where the variational wave function is known to~ be inaccurate from variational estimates of the string tension [9] and mass gap [10] of the theory. To conclude from the quite close agreement between the GFMC and variational results on the energy per plaquette and the mean plaquette field that the variational wave function is a good approximation of the exact ground state wave function would clearly be quite wrong. 87 The results on the U(l) model in the n-space formulation using a disordered trial wave function showed similar behaviour. In that case, though, the approximate agreement between the variational and GFMC results was due to metastability of the disordered state with respect to the GFMC iteration. Presumably, because of this metastability, any quantity computed by the GFMC method using the disordered importance function would give results close to the variational results. If the large A limit were not known, it would be very difficult to discover such metastability. Perhaps by increasing the ensemble size to a sufficient level the metastability could be removed, but in view of the computational effort required this is probably not a good way to proceed. A better approach would be to use a different importance function to check the results, but this, of course, is not possible when one only has a single variational wave function available as is the case for the SU(2) theory. In conclusion, the GFMC method is a potentially powerful tool for use in lattice gauge theories but it appears to be necessary tohave available at least two variational wave functions, or at least to know the limiting behaviour of the theory for large and small A, in order to interpret the results correctly. Future work should therefore be devoted to the development of more accurate variational wave functions for non-abelian theories, by incorporating into the wave function explicit couplings between different plaquettes. The resulting variational wave functions, although interesting in their own right, would be very useful as importance functions in the GFMC method. APPENDICES APPENDIX A Green's function Monte Carlo calculations on the SU(2) and U(1) lattice gauge theories Green’s function Monte Carlo calculations on the SU(2) and U(1) lattice gauge theories David W. Heys* and Daniel R. Stump Department of Physics and Astronomy Michigan State University East Lansing, Michigan 48824 Abstract An application of the Green's function Monte Carlo method to the Hamiltonian formulation of the SU(2) and U(1) lattice gauge theories is described. The Green's function is that of a diffusion process in the gauge group space. A small-step approximation of the diffusion distribution is used in actual calculations. Also, a variance reduction technique is implemented, importance sampling with a disordered trial wave function optimized by the variational principle. The results of computations are reported for a 3)(3)<3 spatial lattice. The quantities computed are the ground-state energy and the expectation value of the magnetic energy, as a function of the gauge coupling constant. The results are compared to variational estimates and to weak-coupling perturbation theory. * Address beginning October, 1984: Department of Applied Mathematics and Theoretical Physics, University of Liverpool. 88 89 I. Introduction The Green’s function Monte Carlo (GFMC) method is a numerical method for computation of properties of the ground state of a quantum system with many degrees of freedom. The method was originally developed for application to many-body problems in nonrelativistic quantum mechanicsl‘z. It is also applicable to the Hamiltonian formulation of lattice gauge theories defined by Kogut and Susskind 3. The Hamiltonian formulation is an approach to lattice gauge theories that is complementary to the Wilson path-integral formulation 4. The properties of the two models are expected to be qualitatively similar. Each approach has advantages. The Hamiltonian approach is a more conventional quantum mechanics construction, in which the theory is defined in terms of field operators and a Hamiltonian operator; the basic problem is to obtain the energy eigenstates. The usual approximation methods of quantum mechanics, such as perturbation theory"5 and the variational principle 6’7, can be used to study the eigenstates. This operator formulation provides a different kind of insight into the nature of the gauge theory than the path-integral, because it deals directly with the quantum states of the fields. Monte Carlo methods are suited to numerical studies of systems with many degrees of freedom. Some very important results on lattice gauge theories have been obtained from Monte Carlo calculations on the path-integral formulation of the theories 8’9. Therefore it is natural also to develop Monte Carlo methods for application to the Hamiltonian formulation of the theory. The GFMC method, which has already been applied successfully to quantum many-body problems, is an obvious method to try. The first problem to solve regarding a quantum system with many degrees of freedom is to compute properties of the ground state. That is the subject of this paper, for the SU(2) and U(1) lattice gauge theories in three spatial dimensions. Specifically, ‘we show results of GFMC computations of the ground-state energy, as 90 a function of the gauge coupling constant, and of the expectation value of a plaquette variable related to the magnetic field. These quantities are analogous to the mean plaquette action computed in the earliest Monte Carlo studies of path-integral lattice gauge theories 9; they are interesting in that they provide an indication of the transition between the strong and weak coupling limits of the theory. ‘ Our numerical results are limited to a small lattice, a 3 x3 x3 spatial lattice. This is small by the standards set by Monte Carlo calculations on the path-integral, but not small compared to other GFMC applications. The SU(2) gauge theory has 243 independent quantum variables for a 3::3:<3 lattice. There is no fundamental problem in using a larger lattice; the only limitation is the availability of computer time. The quantities described in this paper are not very sensitive to lattice size, because they are averages over the entire lattice. Thus the results are already interesting for a small lattice. We hope to use the GFMC method to study other properties of lattice gauge theories, such as the string tension or the energies of elementary excitations. We have carried out some numerical calculations of these quantities by the variational principle17’lo, but it remains for the future to extend the Monte Carlo method to those calculations. The problem presented by the Hamiltonian formulation of a lattice gauge theory is quite different than that of the path integral formulation. In the path integral, the probability distribution of the fields is given; it is emBS where S is the lattice action and B is related to the coupling constant. Then the aim of the Monte Carlo calculation is to generate a set of field configurations with this known distribution, e. g. by the Metropolis method or Creutz's heat-bath algorithm 9. In the quantum problem, in contrast, the ground-state distribution of the fields is not known. What is known is only that the wave function is the lowest eigenfunction of the Hamiltonian. The aim of the GFMC method is to generate a set of field configurations with a 91 probability distribution related to the ground-state eigenfunction. But the GFMC algorithm does not derive from an g_priori distribution; rather, it derives from the eigenvalue equation, written as an integral equation. The integral form of the eigenvalue equation resembles a steady-state diffusion problem. The origins of the GFMC method are found in techniques of Monte Carlo solution of such diffusion problems. The idea is to simulate diffusion of an ensemble of points in the configuration space. The diffusion process is defined such that the evolution of the probability distribution of the points is identical to iteration of the eigenvalue equation. Since iteration of the equation converges to the lowest eigensolution, the GFMC ensemble of points converges to a set with probability distribution equal to the ground-state eigenfunction. Perhaps the most interesting aspect of the GFMC method is the use of an importance sampling technique, in which a trial wave function is used to guide the diffusion to the significant region of configuration space. Importance sampling reduces the variance in the Monte Carlo estimates. But the technique is potentially more valuable than a mere computational trick. The trial function must approximate the ground-state eigenfunction to provide strong importance sampling. One may gain some insight into the structure of the eigenfunction by studying importance sampling with trial functions of different forms. In the calculations described in this paper the wave function is a function of the gauge field, and the GFMC ensemble is an ensemble of gauge-field configurations. In the language of quantum mechanics, we are using a basis for the Hilbert space in which the gauge-field operators are diagonal. It is possible to use instead a basis in which the electric-field operators are diagonal. In fact we did use such a basis in an earlier application of the GFMC method 11 to the compact U(1) gauge theory 12 and to the XY model 13. For that basis we constructed trial functions for importance sampling that approximate the eigenfunction in both the strong and weak coupling limits. For the gauge-field basis, however, we have 92 not succeeded in constructing a useful weak-coupling trial function. All the results reported here use a disordered trial wave function for importance sampling. The disordered wave function is an accurate representation of the ground state in the strong-coupling limit. It is a product of independent functions of the plaquette variables; thus it is gauge invariant, and has minimal correlation between the gauge fields. Comparison of the variational estimates 7 based on this trial function and the GFMC results should show how well this simple wave function represents the vacuum state. The remainder of the paper consists of Section II, on the details of our application of the GFMC method to the 80(2) and U(1) lattice gauge theories, including the implementation of importance sampling with the disordered trial wave function; Section III, on the results of computations for a 3:(3:t3 spatial lattice; and Section IV, a brief summary. We have also included an appendix on a technical point: the ”growth estimate" fails to give an accurate measurement of the eigenvalue in our calculations. 93 II. The Green's function Monte Carlo method A. Application to lattice gauge theories In this section we describe an application of the Green's function Monte Carlo (GFMC) method to lattice gauge theories. The details are described for the SU(2) gauge theory; the analogous application to the U(1) gauge theory is an obvious modification. The field variables of the SU(2) lattice gauge theory are elements of the group SU(2); an element U(i) is associated with each link 2 of the lattice. The group element U(E) may be specified in terms of a 3-component gauge field Aa(£) (where a=1,2,3) or in terms of three angular variables (w(£),8(£),¢(£)); these are defined by ' U(1) 8 exp (é-aaAaUU = cos M2.) + 1 aa na(£) sin M2) , (2.1) where Ga denotes the Pauli matrix and na(£) is the 3-dimensional unit vector with polar angles (6(2),¢(£)). The relation between the two representations is A80.) - 2 M2.) na(2,) . (2.2) Also, there is a 3-component electric field Operator 88(1) associated with each link, defined by the commutation relation 1 ”am . 0(2)] - "fan um . (2.3) The operator 38(2) is a differential operator acting on functions of the gauge fields Aa(£), or equivalently on functions of the angles (wcz).e(x>,¢<2>); in terms of A.(‘)' a 1 a Ea - ”(Ma-a — K7 (f(A)-1) AaAbTAb i 3 - TeabcAb—a-A ’ (2.48) c 94 where A =- (AaAa)1/2 , f(A) =-§‘— cot-g— . (2.413) The Hamiltonian of the SU(2) gauge theory is:3 1 2 2 4 . HKS'Tg 2E8 +-§ZZ¢(P), (205) 1 p the gauge-invariant plaquette variable ¢(p) is 1 + + ¢(P) ‘ 1--§-Tr U(11)U(£2)U (£3)U (in) (2-6) where (21,22,23,£H) are the links that define the plaquette p. We use periodic boundary conditions in the definition of the plaquette field 0(p), to minimize finite size effects in the numerical results. In our GFMC calculations we use a Hamiltonian H that differs from that in Eq. (2.5) by an overall scale factor, and an additive constant; H is H = K - Ari, (2.7a) where K - is: (2) , (2.7b) 9. 1 ' + + M - Z ( 1 +7 TrU(£1)U(;2)U (13W (2.») . (2.7c) p The relation between the coupling constants A and g is A - 8/g“ . (2.7d) The Hamiltonians are related by ..l. 2 .. HKS 2 g (H Zle) (2.8) where Np is the number of plaquettes; obviously they have the same eigenstates. We write the Hamiltonian in this form because our application of the GFMC method requires that the magnetic energy be 95 negative. The starting point of the GFMC method is an integral equation for the ground-state eigenfunction of H. Let YlAa] denote the eigenfunction, a function of all the link variables; it obeys the eigenvalue equation . - 2 f where the ground-state energy is denoted by -Q2 14. Or, Eq. (2.9) is equivalent to the integral equation HAa]-Ajdn G[A8,Aa]M[Aa]‘P[Aa]; (2.10) the functions that appear in the integral, which are functions of the full field configuration, are defined by '1 , I ([A8] | (x+Q2) |[Aa]> -c[Aa,Aa] , (2.11s) < [A8] | M | [Aa’] > = MIAa] :1 5( 118(2), Aa’(2)) . (2.11b) The integration measure for SU(2), which is expressed most simply in terms of the angular variables, is d9 - Hdw(£), E d000,)" 7—1;,— sinzupu) sin am am) dam aw); (2.12) the domain of W and 6 is (0,1) and that of o is (0,2n). The normalization of the delta function in Eq. (2.11b) is 111.1(2) a (Aa(9.),Aa'(£)) - 1 . (2.13) The function G[Aa,A;1 is the Green's function of the operator I<+Q2 , defined by (K + Q2) 0 [A8, Aa'] = n G[AaUL), 11812)). (2.14) z 96 Equation (2.10) is an eigenvalue problem, in which Q2 is the given quantity, with A and VlAa] the eigenvalue and eigenfunction to be found. The GFMC approach to the solution of Eq. (2.10) is based on iteration of the equation by simulation of diffusion. It can be shown that iteration of Eq._(2.10) converges to the ground-state eigensolution. However, it is not possible to deal directly with YlAa] because its domain is multidimensional; for the smallest lattice gauge theory, a 3x3x3 spatial lattice, there are 243 link variables. Instead, the aim of the GFMC method is to obtain a probabilistic representation of the wave function; specifically, to generate an ensemble of field configurations ENS= { A(°)(2)- -1 2 3 N} (215) a ’0 999°“09 9 0 such that the probability distribution of the configurations in ENS is proportional to YIAa] 15. The GFMC algorithm generates ENS by a process based on iteration of Eq. (2.10). The process is a simulation of diffusion with branching, in which: (i) the branching fraction f of the configuration A§°)(2) is proportional to M[A;°)], and (ii) the diffusion creates f new configurations from A§°>(£), with probability distribution G[Aa,A§°)]. Each step in the evolution of the probability distribution of the ensemble is identical to one iteration of Eq. (2.10). The probability distribution converges to the ground-state eigenfunction. The GFMC process described so far is incomplete, because applications of the GFMC method to systems with many degrees of freedom always require the use of an importance sampling trick, a technique also called directed diffusion. One implementation of importance sampling for lattice gauge theories is described in the next section. But before proceeding to that subject, it is useful to discuss the nature of the Green's function G[Aa,A;]. 97 The crucial problem that must be solved in order to apply the GFMC method to a quantum system is to find a way to sample the Green's function as a probability distribution. The first step in this lattice gauge theory application is to separate the Green's function G[Aa’Aaf] into a product of factors, each of which acts on the fields of a single link. This is accomplished by the formula -1 ' ([Aa]|(K + Q2) |[Aa]> .. - 2 .. - [0 dt e N <[Aa] le tKl [Aa'l> - <2-16> The left-hand side is the energy-dependent Green’s function G[Aa,A;1; the integrand on the right-hand side is the related time-dependent Green's function. Since K is a sum of single-link operators, the time-dependent Green’s function factorizes, as -tK ’ -tk I <[A ]| e ”A ]> - 11 (A (2)! e 9. IA (1» (2.17) a a z a a where . 2 . k2 Ea(£J , (2.18) each factor depends only on the field variables of a single link. This representation leads to a method of sampling the distribution G[Aa,A;1: first select a diffusion time interval t by a random process with probability’distribution - 2 Q2e ‘Q dt ; (2.19) then for each link select Aa(2) with probability distribution . I - 'tk , g( t, 118(2), A8 (2)) (118(2) | e z | A8 (9.) > . (2.20) Thus the problem reduces to sampling g(t;Aa,Ahf), the diffusion Green's function for the fields of a single link. Furthermore, an important simplifying approximation can be used. In a large system, the ground-state energy Q2 is large, proportional to the number of plaquettes. Then the diffusion time interval t chosen in accord with the distribution (2.19) must be small. Thus it is only 98 necessary to sample the diffusion Green's function (2.20) for a small time interval. In the small-t limit, this distribution describes ordinary free diffusion. As a first step toward understanding the Green's function g(t;Aa,Aaf) it is useful to study the analogous function for a U(1) gauge theory. The group element of the U(1) gauge theory can be expressed in terms of an angle 9, which lies in the domain (0,2n), U a e16 ; (2.21) the corresponding electric-field energy is just k - -32/an. (2.22) The single-link diffusion function for the U(1) gauge theory is < e | J” | e'> a 2 (4110-1/2 exp(-(6-6'+2nv)2/4t); (2.23) vs—m this is the Green's function of free diffusion on a circle. In the limit of a small diffusion time interval t, the Green’s function is approximately < e | e-tk | e'> s (4flt)-1/2exp(-(6-6')2/4t) , (2.24) with the understanding that when 8 diffuses outside its domain (0,2u), it is moved back inside by a shift of iZn. That is, diffusion on a circle may be approximated by free linear diffusion made periodic. To sample the distribution in Eq. (2.23) for a small time interval, let 6 8 8' + a (mod 2n) , (2.253) where s is a random variable with probability distribution (4st)-1/2 exp(-£2/4t) . (2.25b) 99 The simplification in Eq. (2.25), based on the small-step approximation, extends to the SU(2) gauge theory. However, the analysis is complicated by the nontrivial geometry of the group SU(2). To understand the nature of the Green's function g(t;Aa,Aa’) requires an insight into the geometric structure of the group SU(2), as defined by Eqs. (2.1), (2.3), and (2.12). First, an arbitrary group element U can be expressed as 0.x +1342, (2.26s) 1. where x2 + £2 . 1. (2.26b) “ . Thus there is a one-to-one correspondance between SU(2) group elements and points of the 3-dimensional surface of a sphere in four dimensions; we refer to this space as 83. The angular variables (w,6,¢) in Eq. (2.1) are simply 4-dimensional polar coordinates of a point of S3. Second, the SU(2) integration measure dm is the volume element of S3. Third, the operator k is proportional to the angular part of the d’Alembertian in four dimensions, - — ... 2 ... 4k maw(81n Waw) (2027) 3111124: (silne 336(51ne 5%)4Hs—irszH-g-E‘Z )° Therefore the Green's function g(t;Aa,A;) is the distribution for free diffusion in S3. In the limit of a small time interval t the diffusion distance must be small, and then the curvature of the space has a negligible effect. That is, the small-t limit of the diffusion Green's function g(t;Aa,Aar) is equal to that of free diffusion in the tangent space at A43 100 The small-t limit of the Green's function g(t;Aa,Ahr) is most easily written for Aa and A8' near zero, 1. e. for the corresponding group elements U and 0' near the unit element. Then the Green's function is approximately equal to that of a U(l)zens , (2.38b) where < >ens denotes the ensemble average. In principle this estimate does not depend on whether the importance function uIAa] is a good approximation of the eigenfunction, since Eq. (2.38a) is valid for any “[AaJ . The only error is statistical. However, the variance of the Monte Carlo estimate depends on the choice of ulAa]. The variance is small if ulAa] approximates the eigenfunction; in fact, if uIAa] is equal to the eigenfunction then the right-hand side of Eq. (2.38b) is equal to A for any ensemble of configurations, and so there is no variance. In practice the trial function must approximate the ground-state eigenfunction to obtain an accurate value of A. If the eigenvalue A could be computed with sufficient accuracy as a function of the ground-state energy -Q2, then certain expectation values could be deduced. For example, the ground-state expectation value of the plaquette field 0(p) defined in Eq. (2.6) 104 is related to the derivative of 02 with respect to A. The form of the Hamiltonian H implies B uni-2.93- ‘ (‘l’l¢(p)l‘¥> 2 N dx , (2.39) P where ND is the number of plaquettes. The GFMC method with importance sampling also yields a simple approximate estimate of the expectation value of an operator, based on the assumption that the trial function is an approximation of the ground state eigenfunction. Suppose the eigenfunction is VIAa] = uIAa] + slAa] (2.40) where e is small; then to order 62 the ground-state expectation value of a function ClAa] of the field variables is approximated by <91CI?) g 2 _ ° (2'418) The first term on the right-hand side, called the mixed expectation value, is computed from ensemble averages, by (uLCI‘P) : (CIAaI/M1Aa])ens “J” < 1 /M[Aa] >ens (2.4lb) This Monte Carlo estimate can have some systematic error for a finite ensemble, because it involves the ratio of two ensemble averages 18. The other term is the expectation value in the trial state. Since Eq. (2.41s) is only valid to order £2, this estimate of the expectation value of CIAa] is not trustworthy if it differs significantly’ from the expectation value in the trial state. The importance function ulAa] is normally defined to have a simple form, and optimized by the variational principle. Therefore a conservative interpretation of GFMC results is to regard them as corrections to the variational estimates of the quantities of interest. The estimate of an expectation value based on the mixed expectation value is by definition only a computation of the lowest 105 order correction to the variational estimate. The computation of the eigenvalue based on Eq.(2.38) is in principle exact; but since the statistical significance of the computed value is limited unless the trial function is an approximation of the eigenfunction, as a practical matter the computation of A also gives the correction to the variational estimate. In Section III we describe the results of GFMC calculations on the SU(2) and 0(1) lattice gauge theories. The importance function used in those calculations is a disordered trial wave function that we described in a previous paper7. For the SU(2) gauge theory it is ulAa] = exp ( 2 aMIAa] ) (2.42) where MlAa] is the magnetic energy defined in Eq. (2.7c), and a is an adjustable parameter. In Ref. 7 we described a variational estimate of the ground-state of the SU(2) gauge theory based on this trial wave function. These variational calculations are numerical; Creutz's heat-bath Monte Carlo method 9 is used to compute the expectation value of the energy in the state ulAa]. In the GFMC results described in Sec. III, the value of the parameter a is that determined by the variational principle. The remainder of this section is a discussion of details of the application of the GFMC algorithm to the integral equation (2.36). It is necessary to define a diffusion process with probability distribution GDIAa,Aa']. By Eq. (2.16), GDlAa’Aa'] can be sampled by first picking a time-interval t with distribution - 2 Qze tQ dt, and then moving AJKR) to Aa(£) according to the distribution MlAa] ulAa] ’ W m £g[t;Aa(£),Aa(2)) . (2.43) But there is a complication associated with the distribution (2.43): unlike the free time-dependent Green’s function, this 106 distribution is not normalized to unity because of the biasing factor. To take into account the normalization, it is necessary to assign a weight to each configuration in the ensemble. The weighting can be done in various ways. The most ObVlOUS way would be to use the free Green's function for diffusion, and to reweight the new configuration by the biasing factor; then the importance sampling would derive from the increase of the weight of a point that moves toward larger MIAa] ulAa]. However, we use a different weighting method that includes importance sampling as a part of the configuration move itself. Our approach relies on the fact that the time interval of the diffusive step is small, of order l/Qz, :i.e. of order l/Np where Np is the number of plaquettes. Since the diffusion time interval t is small, the configuration move Agii) + Aa(£) is small. Let (¢'(2),6'(£),¢'(2)) and (w(z),e(2),¢(2)) be the angular variables corresponding to the gauge field A;(£) and Aa(2); and let 6w(£), 68(1), 5¢(2) denote the small changes of these fields, as in Eq. (2.29). Then the ratio of the trial function (2.42) before and after the move is approximately ulAa] BMIAa'] ETA—871 E :1 exp ( 2 a( 5W(E)W (2.44) 3111A '] aMlAa'] +ee(2)—m +5¢(2)W )). Or, in terms of the 3-vector 63(2) that represents the diffusion step in the tangent space at Aa’(£), ulAa] + ’ m s :Iexp(2a58(£)'f (1) ) , (2.45) where f . BMIAa' (2) ' w '557f33' (2.46) . BMIAa’] 3).“) aMlAa’] 1 . + m) ( 9 WW“ sine’m 32"“ J' This approximation of the ratio ulAallulAgl is a product of 107 factors, each of which acts on the fields of a single link. Thus it can be combined with the Green's function to define a distribution for the change of the fields on each link. For a small diffusive move the Green’s function is approximated by Eq. (2.31). When this is combined with the factor uIAal/uIAa'], the complete distribution (2.43) can be written without approximation as MIAa] u[Aa] Mug] “[Aa’] £g(t;Aa(£)’Aa’(£)) (2.47) .. R[Aa,Aa'] 11 (nt)'3/2exp( -( 623(2) - at f’(2))2/t), 9. where MIAa] uIAa] MIAa] “[Aa] R[Aa, Aa'] - (2.48) x nexp( -2a 53(2)-¥'(1)+a2 t 134(2) ) . l . Each single-link factor on the right-hand side of Eq. (2.47) is the distribution function for a process in which the link variables first make a deterministic forced move atf'u) , and then a diffusive move 6E<2> for which the probability distribution is the Green's function of free diffusion in the tangent space. The other factor R[A8,Aa'] reweights the new configuration. Since RlAa, Aa'] is approximately equal to 1, the importance sampling in this approach is mainly due to the deterministic move, which forces the point in the direction of increased ulAa] MlAa]. The results described in Sec. III are for calculations which use the simple small-step approximation, Eq. (2.31). It is possible to improve the approximation by subdividing the diffusion time into 108 smaller intervals and letting the diffusion proceed separately for each of these intervals. However, we believe that the naive small-step approximation is sufficiently accurate. In detail, the GFMC algorithm for iteration of Eq. (2.36) is as follows. The aim is to obtain a weighted ensemble of field configurations ENS = { A§°)(£),w(°); o - 1, 2, 3, . . . , N}. (2.49) The iteration of the ensemble consists of three steps: (i) Branching Each configuration A;°)(£) in the current ensemble (0) (0) branches into f new configurations, where f is an integer chosen by a random process with expected value =_c_ll_M[A(o)] w(o) a where N (2.503) q =§L z M1A§°’1w‘°); e: 081 here N is the number of configurations in the current ensemble, and Ne is a fixed number equal to the desired mean ensemble size. Each of these new configurations is assigned weight w' where w(0)/f(0) if >1’ w, 3 ' (2050b) q if :>l , and prevents the ensemble from becoming too large by eliminating points if 1(1. (ii) Biased diffusion Then each field Aa(°)(£) moves to a new field Aa(£), by the combined deterministic forced move plus diffusive move discussed in the previous paragraph. 109 (iii) Reweighting, The weight assigned to this configuration in the new ensemble is A 0 . 75'2- RIAa,A;o)] W . . . (2.50c) This process ultimately converges to a weighted ensemble with probability distribution FlAé]. The value of the parameter A0 in Eq. (2.50c) controls the size of the ensemble. For a sufficiently large ensemble, the total weight grows during this iteration if A0 is larger than the eigenvalue, and decays if A0 is less than the eigenvalue. In practice A0 is maintained at a value such that the total weight, and therefore also the ensemble size, remains approximately constant. This property provides an estimate of the eigenvalue, which we refer to as the growth estimate. However, for a finite ensemble there is some systematic error in the growth estimate. In our lattice gauge calculations we find that the growth estimate does not yield an accurate measurement of A. This point is discussed further in the Appendix. 110 III. Numerical results In this section we describe Green's function Monte Carlo (GFMC) computations of the ground-state energy and mean magnetic energy per plaquette of Hamiltonian gauge theories for a 3:t3:<3 lattice. The SU(2) gauge theory is defined in Sec. II. For comparison we consider also a U(1) xU(1) xU(l) gauge theory. The U(1)3 gauge fields are angle variables BaU.) (with ad, 2, 3); the associated group element is 3 11(2) = n exp(iea(2)) . (3.1) a=l The U(1)3 Hamiltonian is HAb= -§ aZ/aeazu) + 7’}; (1-cos Ba(p)) (3.2) a,2 - a,p where Ba(p) is the lattice curl at plaquette p of the gauge field 6 (l). The Hamiltonian H a Ab 1 2 free-field limit is the same as that of “KS/2g where H'KS is the SU(2) Kogut-Susskind Hamiltonian, Eq. (2.5). is defined such that its harmonic, i.e. Figure 1 shows the ground-state energy per plaquette as a function of the coupling constant for (a) the SU(2) gauge theory and (b) the U(1)3 gauge theory. The quantities plotted are E/Np vs. A, where E is the eigenvalue of H for the U(1) gauge theory, Ab and E is the eigenvalue of “KS/2.82 for the SU(2) gauge theory. In terms of -Qz, the eigenvalue of H introduced in Eq. (2.9), E is E=2AN-2. p Q Actually Q2 is the input, and the corresponding A is the computed eigenvalue. The dashed curves in Fig. l are variational bounds obtained using the disordered trial function uIAa]. The 7variational calculations were described in a previous paper' . We refer to the *wave function uIAa] as disordered because the expectation value of the Wilson-loop operator obeys an area law in this state; we have calculated the corresponding string tension7. Also, this function (does not explicitly couple the magnetic fields on different 111 plaquettes, although there is some implicit coupling because neighboring plaquettes share a common link. This disordered wave function should be a good approximation of the ground-state eigenfunction for small A, so the variational bounds should be accurate estimates for small A. The solid curve is the large-A limit of the energy, 1. e. the weak-coupling limit in terms of the original gauge coupling constant g, for the U(1)3 gauge theory. This limit is derived from the harmonic approximation of the theory. Asymptotically as )-.¢., 3,5 é c(n) J21" - -;— c2(n) + 0(1/JT) (3.3) P for an n25, indicating that the variational wave function is not an accurate representation of the ground state for A>'5. In the large-A range the values of E are consistent with the free-field limit. The numerical values are consistent with the results of a previous calculationlz. The energy of the U(1) gauge theory changes abruptly from that of the disordered state to the free-field value. On the other hand, the SU(2) GFMC points do not show any abrupt deviation from the variational bound. This difference is easily explained; the U(1) model undergoes a phase transition from a charge confining disordered phase to a non-confining free-field phase AE4.5, whereas the SU(2) model does not. Figure 2 shows the expectation value of the plaquette field 0(p), defined for the two models by 1 -é-Truul)U(22)u+(23)u+(1,) for SU(2), v(p) = (3.5) 1 - cos(_6(21)4-6(22)-9(23)-9(2“)) for U(1). Again the dashed curves are the variational estimates, and the solid curves are the large-A limits given by <0(p)> - f c(n) H 2 A (3.6) where f'l for the SU(2) model and f-4/3 for the U(1) model. The crosses are GFMC estimates computed from the mixed expectation value, Eq. (2.41). Again we see very different behaviour for the two models. In the U(1) model, the mean plaquette field d(p) decreases abruptly in the region of the transition at As4.5, from values near the variational estimate down to values near the weak-coupling limit. 113 In the SU(2) model, the field ¢(p) changes gradually over the range of A considered, and does not differ very much from the variational estimate. The GFMC points in Fig. 2 tend to lie below the variational curve for small A. This tendency is more pronounced in the SU(2) model than in the U(1) model. Ordinarily this would be taken as evidence that the trial function ulAa] does not adequately describe the vacuum state. In this case, however, we expect that ulAa] does accurately describe the ground-state for small A and becomes increasingly worse as A increases. This is born out by the results on the energy shown in Fig. 1, where the GFMC points lie very close to the variational curve for small A and begin to deviate as A increases. The discrepancy may be a result of the failure of the small-time-step approximation used in calculating the matrix elements of e-tki. The time step t is of order l/Qz, which increases as A decreases; thus the approximation is expected to be least valid for small A. This explanation of the discrepancy could be checked by subdividing every time step into intervals smaller than 6t, and then observing how the results change as 6t decreases. On the other hand since the wave function is disordered for small A we might expect that errors in the sampling procedure would be unimportant. The error bars in the graph are the ordinary standard deviation for 600 iterations, but we are not certain that enough iterations have been done to deduce a meaningful estimate of the uncertainty. Since the GFMC algorithm is iterative, ensembles in the sequence are not independent unless separated by a sufficient number of iterations. This convergence problem is more serious for the SU(2) model because there the diffusion takes place in a larger space, and so requires more iterations for convergence. Further investigation is clearly necessary to clarify the situation. 114 IV. Summary and conclusions In this paper we describe an application of the Green's function Monte Carlo (GFMC) method to the SU(2) and U(1) lattice gauge theories. The numerical results obtained so far are limited, by the availability of computer time, to estimates of simple quantites, specifically the ground-state energy per plaquette E/Np and the mean plaquette field v(p), for a 3><31<3 spatial lattice. These GFMC calculations use a disordered trial function ulAa] as an importance function to bias the Monte Carlo sampling procedure in favour of regions of configuration space in which uIAa] is large. By comparing the GFMC results to the variational results based on the trial function ulAa], we can obtain some indication as to how well ulAa] describes the vacuum state. For the U(1) model our results show a clear indication of the phase transition at A554.5 separating the charge confining phase, described well by the disordered trial function, and the non-confining free-field phase. For A)>5 there is a definite difference between the variational estimates and the GFMC results. The present results are in good agreement with the results of our previous Monte Carlo study of the U(1) lattice gauge theory 12. There we formulated the problem in a completely different way. We wrote the wave function in the form VIA] ' 2 exp (1 {n(p)B(p) ) x[n(P)] . (4.1) {n(p)} P where the variables n(p) take only integer values, and applied the GFMC method to an eigenvalue equation for x[n(p)]; also we implemented importance sampling for two kinds of trial functions - a disordered wave function which accurately describes the ground state for small A, and a correlated wave function derived from the harmonic limit which is accurate for large A. That these two different studies of the U(1) gauge theory lead to similar results gives us considerable confidence in the GFMC method. 115 In one regard our earlier results on the U(1) model differ from those obtained here. In the calculations applied to the n(p)-space wave function x[n(p)] we found metastability behavior in the GFMC iteration for the U(1) gauge theory in three spatial dimensions. When a disordered n(p)-space trial function is used for importance sampling in the large-A region, the computed eigenvalue does not converge to the free-field value, but remains near the variational value. In contrast, no metastability of the GFMC iteration is seen in the Aa-space calculations. We attribute the difference to the fact that n(p) is a discrete variable, whereas Aa(2) is continuous valued. For the SU(2) gauge theory our results are consistent with the nonexistence of a phase transition in that model. The trial wave function ulAa] accurately describes the vacuum state for small A. And even for large values of A the variational estimates are approximately equal to the GFMC results, for the energy and mean plaquette field. The implication is that the ground state does not suddenly change as it does in the U(1) gauge theory. When the variational and GFMC results show considerable disagreement, as in the U(1) results, then it is clear that the variational wave function is not a good representation of the vacuum state. The converse is not true. The fact that these SU(2) GFMC results are close to the variational estimates does not imply that the variational wave function is a good representation of the SU(2) vacuum state, for if we compare the results for a different quantity, e.g. the string tension, we may find considerable disagreement. In fact we know from our earlier variational calculation of the string tension7'that uIAa] does not describe the vacuum state for large A with sufficient accuracy to reproduce the known asymptotic behaviour of the string tension derived from asymptotic freedom. In view of the comments of the preceding paragraph, it would be very interesting to calculate Monte Carlo estimates of the expectation values of other quantities. Such calculations present no particular difficulty if one is willing to use estimates based 116 on the mixed expectation value, Eq. (2.41). But these estimates are mainly useful for revealing inadequacies in the trial function, and are not accurate when such inadequacies exist. It would be much more satisfying to compute expectation values exactly, 1. e. subject only to statistical errors, rather than from the mixed expectation value, which introduces an unknown systematic error. Such a procedure does existl?2, though it is expected to be very demanding on computer resources if any great precision is to be achieved. It is interesting to compare the Green’s-function Monte Carlo method to the projector Monte Carlo method introduced by Blankenbecler and Sugar19 and recently applied to the compact U(1) lattice gauge theory in three spatial dimensions by Chin, Koonin, and Negele 20. In that method ehTH is used as a projection operator onto the lowest energy state of the system, where H-K-I-V is the Hamiltonian and T is large. The object of the projector Monte Carlo method is to obtain an ensemble of configurations with distribution WIAa]. T is divided into a large number N of small time intervals t 8 T/N, and the ensemble is generated by repeated action of the operator e-tH. Since t is small we can write "tH _, -tK "tV e e e = , (4.2) correct to order t; in the basis in which Aa is diagonal, the distribution is -tV [Aa'] . <[Aa] Ie-tHI [Aa’]> ([A8] le"K| [Aa’]> e (4.3) The technical details of a calculation with this projector method are essentially the same as those of the GFMC method. In particular, the function that governs diffusion of the configurations is < [A8] I e-tKI [Aa’]> for both methods. Therefore, it is completely straightforward to modify our GFMC program to carry out the projector Monte Carlo calculation. This would be a useful exercise as a check on the present results. 117 Appendix Iteration of Eq. (2.36) yields a sequence of functions {F(r)[Aa] } defined by F(r+1)[Aa] - A(Or) IdQ’GDIAa,Aa’]M[Aa’] F(r>[Aa’] (11.1) where Agf)is an arbitrary parameter which may change from one iteration to the next. In the limit r + 9, F‘r)[Aa] becomes proportional to FIAa], the eigensolution of Eq. (2.36); thus Eq. (A.l) can be rewritten as ( 1) A(or) ( ) r+ r Integration of this equation implies that A(r) «(”15 - 79- mm) (11.3) where W(r) is the total weight of the ensemble at step r of the iteration, and < > denotes the expected value. Equation (A.3) provides a simple way to estimate the eigenvalue from the growth or decay of the total weight during the iteration. We refer to this as the growth estimate. In practice we adjust Agf)to»maintain a constant total weight, i.e. W w(r) w(r+1) Ill (A.5) It can be shown that if this were the only source of error then A would be bounded by the inequalities min( A(or)) < A < max( A(Or) ) . (A.6) 118 Figure 3 shows the ground state energy per plaquette as a function of A for the SU(2) gauge theory. Results for the U(1) gauge theory are similar. The crosses are GFMC results obtained from the growth estimate Eq.(Am4). The curves have the same meaning as in Fig. l. The Monte Carlo results are clearly in error: they are systematically too high. This cannot be attributed to the systematic error in Eq.(AWS) since the inequalities (As6) do not hold. Rather we believe that the discrepancy is due to the failure of the trial function u[Aa] to describe the eigenfunction. This is suggested by the fact that the discrepancy increases as A increases, 1. e. as the disordered trial state becomes a less valid approximation. 119 Acknowledgements We are pleased to thank M. Creutz, J. Hetherington, S. Koonin, and J. Negele for conversations and advice regarding this work. This research has been supported by the National Science Foundation under contract PHY-83-05722, and by a grant from Control Data Corporation. 120 Footnotes 1. 8. 9. 10. 11. M. H. Kalos, Phys. Rev. 128, 1791 (1962); Phys. Rev. A _2_, 250 (1970). M. H. Kalos, D. Levesque, and L. Verlet, Phys. Rev. A _9_, 2178 (1974). D. M. Ceperley and M. H. Kalos in Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, ed. K. Binder (Springer-Verlag, 1979). J. Kogut and L. Susskind, Phys. Rev. D g, 395 (1975). K. Wilson, Phys. Rev. D _1.(_)_, 2445 (1974). J. Kogut, D. K. Sinclair, and L. Susskind, Nucl. Phys. _B_1_l_4, 199 (1976); T. Banks, S. Raby, L. Susskind, J. Kogut, D.R.T. Jones, P. N. Scharbach, and D. K. Sinclair, Phys. Rev. D _1_5_, 1111 (1977); C. J. Hamer, Nucl. Phys. M, 492 (1978). References on the use of the variational principle to study the U(1) gauge theory: S. D. Drell, H. R. Quinn, B. Svetitsky, and M.Weinstein, Phys. Rev. D_1_9_, 619 (1979); D. Horn and M.Weinstein, ibid. 25, 3331 (1982); U. Heller, ibid _2_3_, 2357 (1981). Numerical variational calculations on the U(1) gauge theory are described in our papers Refs. 7 and 12. References on the use of the variational principle to study the SU(2) gauge theory: D. Horn and M. Karliner, Nucl. Phys. B (to be published); D.W. Heys and D.R. Stump, Phys. Rev. D (to be published). For a recent review article see M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rep. _9_5_, 203 (1983). M. Creutz, Phys. Rev. Lett. _4_3_, 553 (1979); Phys. Rev. D 21, 2308 (1980). D.W. Heys and D.R. Stump, "A variational estimate of the energy of an elementary excitation in the SU(2) lattice gauge theory," in preparation. We refer to the calculations in Refs. 12 and 13 as applications 12. 13. 14. 15. 16. 17. 18. 19. 20. ,121 of the Green’s function, Monte Carlo method, although the diffusion operator used there is not a Green's function. D.W. Heys and D.R. Stump, Phys. Rev.D _2_8_, 2067 (1983). D.W. Heys and D.R. Stump, Phys. Rev. D (to be published). We have defined the Hamiltonian H in such a way that the ground-state energy is negative. In quantum mechanics one ordinarily refers to 22 as the probability distribution. In the GFMC method, however, Y itself is used as a probability distribution; this is possible because the ground-state eigenfunction is positive definite. Note that the variables Aa and w differ by a factor of 2. It can be shown that the optimal choice of FlAa], i.e. that minimizes the variance in the computed value of A, is MIAa] uIAa] HA8] where ulAa] is the best available approximation of the ground-state eigenfunction. The GFMC iteration yields a sequence of ensembles El’ 132, ...,EL. We estimate the right-hand side of Eq. (2.41b) by the average 1 L ._ 1 .....E., L 181 <1/M>1 this might make a systematic error, if the fluctuations are large, because of correlations in the values of 1 and i. R. Blankenbecler and R. L. Sugar, Phys. Rev. D _2_7_, 1304 (1983). S.A. Chin, J.W. Negele, and 8.152. Koonin, Santa Barbara preprint. 122 Figure captions Figure 1. Ground-state energy per plaquette E/Np vs. coupling constant A for (a) the SU(2) gauge theory, and (b) the U(1)3 gauge theory. The solid curves are the large-A perturbation expansion for the U(1)3 model, the dashed curves are the variational bounds, and the crosses are the Monte Carlo estimates. Figure 2. Mean plaquette field ¢(p) vs. coupling constant A for (a) the SU(2) gauge theory, and (b) the U(1)3 gauge theory. The solid curves are the large-A perturbation expansions, the dashed curves are the variational estimates, and the crosses are the Monte Carlo estimates based on the mixed expectation value. Figure 3. Ground-state energy per plaquette E/Np for the SU(2) gauge theory, computed from the growth estimate. The curves have the same meaning as in Fig. 1a. 123 FIGURE la o.Nm o.oo m.nm M.Mm ..ma o.wm 1 I h b _ _o.m b b . .~.o P D . .m.m 124 FIGURE lb P E/N m.Vm u.oo m.Mm r.mc w.Vm v.00 M.Mm .08 o.Nm 125 FIGURE 23 CDIp) 70 oh 1. Dom .l o.» T ab 1. Pm 1 o.» 1 Pm l o.» 1 n... L 126 FIGURE 21: (Dip) P Pb 127 FIGURE 3 E/NP m.o So 4. m.o 1. «.o 1 u.o 1 ”.0 I - J I 1. # h'b 1.! Br _w.m d b J r, _m.o APPENDIX B Application of the Green's function Monte Carlo method to the compact Abelian lattice gauge theory PLEASE NOTE: Copyrighted materials in this document have not been filmed at the request of the author. They are available for consultation, however, in the author's university library. These consist of pages: Appendix B, pages 128—136 Universg' Micr Ilms lntemational 300 N. ZEEB 90.. ANN ARBOR, MI 481061313) 761-4700 PHYSICAL REVIEW D VOLUME 28. NUMBER 8 128 13 camera 1983 Application of the Green’s-function Monte Carlo method to the compact Abelian lattice gauge theory DavidW. HeysandDanielR. Stump .Deparnnenr ofPhysics and Astronomy. Michigan State Uniueeiov. East lmng, Michigan “£24 (Received 15 Jane 1983) We have applied the Green's-ftmction Monte Carlo (GFMC) method to the Hamiltonian femuo lationofthecompact U(lllatncegaagetheoryinthreeandtwoispaceldimensionsmsmafllattice. 3x3x3and 5x5. TheGFMCmethod isaMonteCarlomethodoffindingthegroundstateda quantum-mechanical system with many degrees of freedom. by iteration of an integral operator of which the ground state is an eigenstate. An intereting aspect of this method is an impotence- sampling technique that make use of a trial wave function to accelerate convergence of the Monte Carlo estimates. We used two importance functions in thee calculations. which were deigned to be accurate in the small- and large-coupling limits. Thee importance functions were optimized by the variational principle; the results of the variational calcalatiom are interesting in their own right. Our Monte Carlo results exhibit evidence of the phase transition of the three-dimensional compact U(l)lattieegsugetheory. andindicatethenonexistenceofsphasetransiuoninthetwo-dimetsional theory. I. INTRODUCTION Lattice gauge theories are used to study quark confine- ment and other nonperturbative aspects of gauge theories, especially those relevant to quantum chromodynamics. There are two formulations of lattice gauge theories—the path-integral formulation' in which all four dimensions are discrete and the Hamiltonian formulation in which time terrains a continuum. These theories have been in- vestigated by a number of technique, e..g. perturbation expansions,3 menofield theory, the variational principle,‘ and Monte Carlo methods. The purpose of this paper is to describe an application of the Green’ s-function Monte Carlo (GFMC) method to the Hamiltonian formulation of the simplet lattice gauge theory, the compact U(1) gauge The U(1) lattice gauge theory is primarily interesting as a contrast to non-Abdian gauge theories. All lattice gauge theories exhibit the phenomenon of charge confine meat in the strong-coupling limit. In non-Abelian gauge theories this phenomenon persists to weak coupling, but in the (3 +1)-dimensional U(1) gauge theory there is a phase transition to a nonconfining state at a finite coupling. Thee statenents have been amply demonstrated in inve- tigations of the path-integral formulation of these lattice theories, that use the Metropolis Monte Carlo algorithm to compute the path integral.“ One goal of our GFMC calculations is to try to verify thee statements in the Hamiltonian formulation; the U(1) lattice-gauge-theory calculationstobedecribedareafirststepinthisdiree- tion. In U(1) lattice gauge theorie the transition to a noncon- fininggrumdstateoccursinthreespacedimeuiormbut not in two dimensions. This difference can be understood in terms of the behavior of long-range topological config- urations in thee models. In two space dimensions there exist vortice that maintain confinment at arbitrarily weak coupling; this was first decribed by Polyaltov m an early instanton calculation. 7 Other authors argued that 1n three space dimensions monopole undergo an ionization transition at a nonzero value of the coupling constant, below which the ground state is nonconfining.‘ Thee spatial configurations can be decribed also as time slices of spacetime configurations in the path integral of the theory.’ The reults of path-integral Monte Carlo calcula- tions have shown that the U(1) gauge theory does have a phasetransitioninthreespacedimerrsimbutnotintwo dimensions. '0 Oar Green ‘s-function Monte Carlo calcula- tions also demonstrate this fact. The GFMC method is a Monte Carlo method that re velspmpertieoftheground stateofasystem with many degree of freedom. 11 was developed to solve quantum many-body problems. and has been applied to a number of example of thee. " '2 We have applied this method to several lattice field theorie, including the U(1) lattice gauge theory. and the XY- and ngauge models. In our experience it is not difficult to put a lattice field theory into a form to which the GFMC method can be applied; infact thiscan 11suallybedoneinmorethanonewsy,and atehastheproblmofdecidingwhichonetotry. The simplest quantity to calculate in the GFMC method is the ground-state energy as a function of con. pling constant. This quantity is analogous to the average action per plaquette calculated in Monte Carlo studies of the path-integral lattice field theories. In principle the GFMC method can be extended to elcalation of other quantities, e.g.. the expectation value of a Wilson loop operator; but in practice we have not yet carried out any such calculations on the U(1) gauge theory. Perhaps the most interesting aspect of 2the GFMC method 1s an importance-sampling technique. This tech- nique, which isanesential partofthemethod, makeuse 3067 ©1983 The Ameian finial Sadety m of an approximation of the ground-state wave function. called the importance function. to bias the Monte Carlo diffusion process; this reduce the scale of fluctuations as- aociated with stochastic sampling. and so accelerate the convergence of Monte Carlo etimate to an accurate value of the computed quantity. In principle the results do not depend on the importance function but in practice it should be similar to the ground-state eigaifunction. Normally the importance function is obtained from a variational calculation. One can judge whether a wave function doe reanble the eigatfunction by determining whether it performs adequately as an importance function in reducing statistical fluctuations. Thus this approach can be combined with the variational principle in a poten- tially powerful way: two variational wave functionswith about thesameatergyertpectationvalaecanbedis- tinguislied on the basis of their performance as impor- tance functions. The realts of our calculations on the UH) lattice gauge theoryshowarathaclearsignalofthephasetransitionof this theory in three (space) dimensions. The same signal is not seen in calculations on the two- (space) dimaisional theory, as expected since this theory is not supposed to have a phase transition separating confining and noncon- fining ground state. Also. the transition in the three. dimatsional theory is not a first-order transition. Our cal- culations have been restricted to small lattice; the realts to be described are for 3x3x3 and 5x5 lattices (ranernber that the fourth dimension is a continuum). We believe that the quantitie that we have calculated are meningfal on such small lattice, and that the only affect ofalargerlatticewouldbetomalteasharpertramition between strong- and welt-coupling behaviors. Of course this would not be true of all quantitie. The paper isorgsnizedasfollows. SectionIIisssketch of the GFMC method. with importance sampling. in gar. as] tams. Section 111A define the U(1) lattice gauge theory and our approach to the application of the GFMC method to this model; Sec. [113 describes the variatitmal wave functions that we use for importance sampling. Thee variational calculations are interesting in their own right. Section IV discusses the Monte ero results. Sec- tion V lists some ctmclusicns. II. THE NUMERICAL METHOD A. TheOraar'sofaactiaa McateCarlomahad The Green’s-function Monte Carlo (GFMC) method wasdevelopedasanumaicalmethodforfindingthe groundstateofaHamiltcnianwithmanydegreesoffree- dam." LetHbeoftheform H-Hg-mz . (2.1) whaeH, andearepcsitiveopa'atorsandAisacou- plingparameter. Let —K’denotetheloweta’genvalue of H. assumed tobenegative. Theeigenvalue equation ”flu—Kw (2.2) canbewrittenasanintegralequation.» 129 DAVIDW.HEYSANDDANIELR.STUMP 2_8 calm. +x’r'u,¢ . (2.3) ThestartingpointfortheGFMCmethodistoregarqu. (2.3) as an equation for d) and A with K2 given. Next. in. trnduceacompleteset ofbasis state I17) forwhich H, is diagonal.i.e.. (rim. madman?» .. The multidimatsional quantity 1? that labels one of thee state is a configuration of the quantum variable of the model, e.g., a field configuration in a lattice field theory. In what follows we use a notation appropriate for a prob- lan in which if has discrete value. but the method ap- plie equally to continuous-valued variable. If «,7 ) -(fl' I 1’) that (2.4) #17)3126(K2;IT.E')V(17')¢(17'), (2.5) in whee GlK’;F.fi')=(fi'|(H.+K’)" In") . (2.6) The GFMC method applie to integral equations of the form of Eq. (2.5)." In applications to quantum many- hody problems. V is a potential and G a Green's function. In our applications of the GFMC method to lattice field theories we begin also with an equation of the form of Eq. (2.5). but not always one in which G is introduced as an inverse operator. In particular. for the U(l) lattice-gauge- theory calculations to be decribed. G is simply one of the operators in the Hamiltonian. The GFMC method is a Monte Carlo algorithm for solving Eq. (2.5) by iteration. Let I”:U1",;o nl,2.3.....N'l be an ensanble of configurations with probability distribution $.01"). One iteration of the equa- tion yields a new atsanble f=[fi,;a=l.2.3. ....Nl wha'e the configurations ii, are obtained from thefi',byaprocesthatccnsistsoftwosteps,branching and diffusion: (i) Each fi',branche into n, newpoints. where n, is an integer picked by a random process such that the ex- pected value of n, is town). The possibility 11,-0 is allowed. Haeristhoaghtofasanapproximationofthe e'garvalue A. _ (ii)Tharechofthen, pointsismovedfrom f1",toa new configuration [I chosen from the probability distribu- liar P(ir‘.ii",) defined by P(17.17'.)-G(17.fi“.)/2 6117.17» . at In the lattice field theorie to which we have applied this method the denominator of Eq. (2.7) is a constant. in- dqiaidart of it", This will be assumed below. Note that the processes (i) and (ii) require that V(ir‘) and 607.11") bepcsitive;itmaybenecesarytoaddaconstanttothe Hamiltonian to meet this requiranent. The probability distributionpflr'r'mfpcintsinthenewatsanblefl‘is (2.7) ”(ms-'1; 2P(fi°.i1",)lol’(fi“.) ; (2.1) ' l! APPLICATION OF THE GREENS-FUNCTION MONTE CARLO . . . m.inta'msda'r¢,(fi‘), \ ¢/(1T)=% Z'PUT. E'MOVUT' Hum" ). (2.9) Itcanbeshown that theseqaatceofaisanblegarerat- ed by iterating the proces just decribed converge to an atsanble for which the probability distribution is #17), the solution of Eq. (2.5). The parameter A0 determine how the atsanble size change on further ita'stion: after the proces has converged. so that (1, xcf=¢ in Eq. (2.9). we shall have at sva'age -1 _=_. Ear-")1 , (2.10) where it should be rananbered that the factor 2-G(".i1")isindependattofi1’ '.Thisprovideaway to determine the eigaivalae A. which in the problan dc» finedbyEq. (21) 1s thevslueofcouplingccnstant for which the ground-state aiagy is -K’. In practice. one rcadjasts the value of 10 way It itera- tions so as to keep the arsemble size approximately con- stant. The adjusted value of lo ccnvage to A time 2 17 G (17.1? '1. The method decribed here yields a numerical deter- mination of the eigenvalue 1, and a sequatcc of ensanble of configurations with probability distribution WT) (after cmvergaree). It is also possible to invert ways to extend the method to calculations of other quantitie. e.g.. expec- tation valuecfoperators.but inprscticetheerequirea hrge increase in computation time. I. Importance sampling Forproblemswith manydegreescffreedom itissdvsn- tageotn, and asaprsctical mattereven necesary.tomodi- fytheGFMCmethodbyusecfanimportancesampling technique."” A wave function dig-(17). which should reanble the ground-state eigenfunction #1? ) as closely as passibleisintrodacedbyrewritingtheintegralequation, Eq. (2.5). in tams (is new function: nmqrtpwm 12.11) as c. '1" u) in. are. no. cop run-A}; out. )V(u mp 1.12.12) ‘7“? ) Nowcnerega:dsF(i1')andAastheunknowneiga1func- tion and eigenvalue. The iterative diffusion process decribed in Sec. 11A is used again to gataate atsanble with probability distribu- tion PUT). The diffusion process is changed in two ways by the preence of the importance ftmction ¢r(17).Ftrst. the probability distribution that gova'ns diffusion d’ the particle 1s now fir”? 1607. "WWTUTWI'J 2¢r(17)0(17.17')lvr(17')l" i‘ (2.13) 130 w ‘I'heeffectofthischsngeisthatthediffasionprocessis hissedin favordmovefi'at'r'forwhichii‘isinthere- gion of configuration space where firm.) is large. If firm.) is at lest qualitatively similar to the eigatfunction d(ir‘). this biasing reduce the fluctuations due to stochas- tic sampling. and so speeds up the ccnvergaice to accurate numaical etimate. The second change in the diffusion proces is a normali- nation dfectintheaelectionofn,.thenumberofnew points gata'ated from thepoint 17;. Theexpected value d'n, should now be (ad-111417112 vrifilG(rT.17'.)lvr(17L)l"- p (2.14) Forfirlii’hal. i.e..withoutimportancessmpling. thenor malizingsumisindependentof" .inthelatticefield theoriewehaveccnsida'ed. sothisfactorwouldnotbe needed. But for a nontrivial $7117). the ccrnpatation of thenqrsmalizingsummaybethetrickiestpartofthepro- gram. Again it can be shown that the iteration conva'ge to atsanble with probability distribution Hfi). the solution 11' Eq. (2.12). Also. after ccnvagatce the expected change in atsanble size for tire iteration is givat by size N' A ° Importancessmplingrducethefiuctaationdensanble size. so that Eq. (2.15) convage more rapidly toan accu- rateestimateoflt. Theoptimal choiceofh-(ir’)canhe daived.sndturnsouttobesimflyrelatedtotheexact a'gaifunction d(fr’); this choice would actually reduce fluctuationstozero. Theimpcrtance-sampling techniqueprovideawsyto etimate ground-state expectation value that require lit- tle additional amputation. Suppose the a'gaifunction fir?) difi’a's from the trial function pm?) by a small amountcfa'derenhartoordere’wehave (w I!) “(out It.) _ (ma I») 1111) mm (mm ' (2.15) (2.16) Haetheleft-hsndsideisthedeiredepectationvalueof anopaatorA;thefirsttamontherightistwicetheaver- age ofA in the aisanble gatasted by the GFMC itera- tion of Eq. (2.12). and the other tam is simply evaluated for $7. The estimate (2.16). called the mixed expectation value. Ins statistical error from the fact that it involve stochastic sampling. plus systanatic error from the fact thstitisvslidtocrdae’only. Itistrustwcrthyailyif (fiIA Ii) and “VIA Hr) slenot toodifferart. In summary. the importancessmpling method outlined above is obviously most useful when an accurate approxi- mation of #17) is known. e.g.. from a variational calcula- tion. Then the GFMC approach calculate the corrections tothatspproximation exactly(uptostatisticalarorsdue to sampling fluctuations). But evai if iflfl') is not partic- N70 ularly accurate it may still perform the function of impor- tance sampling in reduction of fluctuations. provided the diffusion by Eq. (2.13) is biased in a qualitatively correct way. III. THE LATTICE GAUGE THEORY A. Compact U(1) lattice page theory The Hamiltonian of the U(1) lattice page theory is" H=§g22£1(11—%2[1+eeempn. (3.1) 1 3’ r where EU) is the electric field on lattice link land B(p) is the magnetic field on lattice plaquette p. If p is the pla- quette associated with site 'x' and directions ij then B(p)=.4(r+fij1-Atr,j1—A1£+f.n+a(in, (3.2) where A(i’.i) is the gauge field on the link lassociated with site it and direction 1. The fandamartal commuta- tion relation is [£111.11 (1')]: -15(I.I'). (3.3) The variable A (l) or B(p) are retricted to lie in the range (-rr,rr). In our application of the GFMC method to the U(1) lattice gauge theory we arrived at the basic integral Eq. (2.5) by a somewhat different path than that decribed in Sec. 11A. By a special choice of basis state we avoided the use of an inverse operator. i.e.. a Green’s function. in constructing our form of Eq. (2.5). Still. since the ulti- mate equation doe have that form the GFMC method sp- plie. We shall consider a basis in which the electric field en. ergv is diagonal. Specifically. let the ground-state eigeno function be written in the gauge-invariant form 1,: 2 exp [i 2 n (pimp) lélfl (1’)] . ' (w'l (3.4) where n(p) are integer-mined plaquette variable. Let —ngz/2 deote the vacuum eergy. The the eigevalue equation Hilv= -%g’deI become. in terms of the n (p)-spsce wave function. (3.5) - is’Qiéln (p)l= it’slnwléln (11)] -—I3 2 G[n(p).n'(p)]é[n'(p)]. ' la‘tpll (3.6) Here the diagonal operator S[n(p)]. which come from the electric field energy. is in» (p)]= 2 n (p)n(p')A(p.p') . (3.7a) '0'. 131 DAVIDWJIEYSANDDANIELLSTUMP 23; (3.7b) , = 38(2) any) M”) $11.11)) an!) ‘ The nondiagonal operator G[n.n']. which come from the magnetic field eagy. is G[n(p).n'(p)]= 2 [5[n (p).n'(p)] '0 + %5{n (p).n'(p)+5,o] +%6[n(p),n'(p)—5"O]I . (3.8) G[n,n'] will play the role of the Grear's function in the GFMC itaation. i.e.. of the function that controls dif. fusion of the points in the space of n(p)-configurations; but note that G[n,n'] is not the inverse ofeither operator in theoriginal Hamiltonian. To put the equation in the form of Eq. (2.5). define a new wave function X[n(p)]=-}32[Q:+S[n1p)](¢[n(p)]. The the equation obeyed by X[n(p)] is X[n(p)]=)( 2 G[n(p).n'(p)]V[n'(p)]r[n'(p)], (3.9) mm (3.10) where 122/3‘ (3.111 and V[n(p)]=(Q:+S[n(p1]l" . (3.121 We have applied the GFMC method to Eq. (3.10). The diffusion step in the GFMC iteration involve moving a point in n(p)-space from n'(p) to n(p) by the function G[n,n']; the definition of G[n,n'], Eq. (3.8). implie that n (p) and n'(p) differ at most by one tutit on one plaquette. We have obtained Monte Carlo realts only for the smallet lattice in three dimesions. of size 3X3x3. (It should be ranernhered that “time” is a fourth continuous dimesion in the Hamiltonian formulation.) Although this is a small lattice size compared to those used in stad- ie of the path-integral formulation of lattice gauge theories by the Metropolis Monte Carlo algorithm. it is not small compared to other applications of the GFMC method; it has 81 independent quantum variable in the original Hamiltonian. With the many variable it is essetial to use importance sampling in the GFMC pro- gram. The importance-sampling functions that we used we: obtained from variational calculations. decribed next. I. Variatieal calculations The first variational wave function is th- [111181111]: (3.13» I _25 APPLICA‘ITON OF DIE GREENS-FUNCTION MONTE CARLO . . . theeergy(tilchylistobeminimizedwithre to the choice of the single-plaquette function u(B).‘ The minimum occurs if 11 (B) is the ground-state eigefunction of the operator 82 ll = —4— +Ml—cosB) . 882 (3.14) where -rr 5 B 5 17. This is the Hamiltonian of a quantum pendulum. The reultlng variational etimate of the vacu- um energy per lattice plaquette is -ngQI/Nfie - 412121.40). (3.15) where e0 is the smallet eigenvalue of h. The eagy -91 is not really the natural eergy to use in decrihing our re- salts; instead we shall use £0. defined by the relation —%ng:=-%8212M'p‘50) , (3.16) Note that £0 is the smallet eigenvalue of 2£2111+12Il —cosB(p)] . (3.17) l r where i=2/g‘. The first variational etimate of £0 is 50/11" ==eo . (3.18) 11 can easily be shown that the small- and large-A limits of co are e ~A- E + 11:- +O(l.°) as A—oO °" 8 2048 ’ 13.19) eozm-%+O(A‘m)e A—.ee . For comparison thee limits of Eo/N, are. for an ananatticc. so/N 41-311 3” +O(A°)asA—o0 — 8 10240 ’ (330) Eo/N sz-Actni—%e2(n1+Oti-"11ar A-ee . whaec(n) isa dimensionles number. e.g.. c(3)=0.787. c(5)=0.795. c(eo)=0.796. (3.21) Note that co and £0 /N, have the same small-A limit. but that £0 /N, is smaller than co in the largeA limit. In the trial wave function it, the magnetic fields on dif- ferent plaquette are uncorrelated; d1. decribe a disor- dered state in B(pkspsee. In particular, the expectation value of a Wilson loop operator in the state it. decrease exponetially with the 100p area. (in: 1] cm" u,)-exp [— 2 y] . (3.22) l'EL I '51. where y-—lnf_"ds 311(B)|2e". (3.231 Since the vacuum state of the three-dimensional U(1) lat- tice gauge theory is. for welt coupling. an ordered state in which the expectation value of the Wilson loop operator 132 am decrease exponetially with the loop paimaer. the state it, should not he a good approximation of the eigestate i for weak coupling. i.e.. for large i=2/g‘. This may a1- resdybeseeinthedifferecebetweal50/N,andeoin the largel limit. The second variational wave function is writte as a probability amplitude in n (p)-space. e dz=exp —-,'-a2 n(p)M(p.p')n(p') . (3.24) N. Here a is the variational parameter and M(p.p') is the matrix that reproducethegroundstateoinnanon- compact harmonic approximation of the U(1) lattice page theory. This approximation consists of two parts: replacement of l-cosB by 112/2, and extension of the range of B from (77.17) to (-—ee.ee). The reulting model is solvable since its Hamiltonian is quadratic; the ground-state wave flmction is Eq. (3.24) with a-=1. bat wha'e the plaquette variable n(p) take a continuum of value. It should be emphasized that d, is not the wave function of a noncompact harmonic approximation in our elculations. because the variable n(p) are retricted to intqer value; the function in A (1)-space. defined by Eq. (3.4). is paiodic in A”). The matrix M(p.p') is. for an l X 11 XI! hill“. (fabu-fifU/f . Hie-13-?) fl 2 H(p.p')= :7 gexp [gal-exp(brinln). 1.0:}; )1/2; (3.25) herep.p' are the plaquette (it) and ('x",lt') with normal directions k,k'.andthesurnsovav, rtmfrom 1 ton. The harmonic wave function decribed in the previous paragraph is equivalet to the variational wave function considered by Horn and wamteih,‘ although it is writte in a somewhat diffaent form. Our calculations use the reciprocal space of field configurations n(p) conjugate to the plaquette variable B(p). Horn and Weinstein work in the space of configurations of the lattice variable A”). and maintain gauge invariance by a projection technique involving functional integration. In spite of the formal differeces,webelievethatthetwoapproschesre equivalet. We wae unable to calculate analytically the expectation value of H in the state ‘1 (Ref. 18); instead we evaluated (d; IH 1‘2) using the Metropolis Monte Carlo algorithm to geerate a set of configurations with probability distri- bution ‘2’. The result is shown in Fig. l. graphs of the value of the variational parameter a that minimize (d; :H 562) vs coupling parameter 122/g‘. Figure 11a) is for a three-dimesional lattice of size 3x 3 x 3; value of a for lattice site SXSX 5 differ very little from those for 3X3x3. Figure 11b) is for a two-dimesional lattice of size 5x5. At weak coupling. i.e.. large A. a approaches 1. the value that correponds to the harmonic approxima. tion. At strong coupling the wave function is sharply peaked at small n (p). correponding to a disordered state in the space of magnetic field configurations. The rapid variation ofa for A near 1 in the three-dimensional case is preumablyareflectionofthephasetransitimoftheUU) 133 son navm w. HEYS AND DANIEL a mm E w (o) u (b) zer art will , , P (s) . . l’ ’I . ”t. 01.0- i l , 1 LS" . .I',a 1.4:. llli . M i“ i h” > ($4,, H ”mm“ ‘ ”WM 5 .r/ me A ofs 1A0 A 154 {ofi‘shoo FITOA :04 sic A as A so i” 1.0)- 77 FIG. 1. Variational parameter :1 vs coupling constant A for [,1/ (a) three dimensions and (b) two dimesions. Error bars include °'5 ’ ’1’ / systematic error. F ,'/' o 4 J L L J J J 4 lattice page theory. The characta of the transition seen 0 °5 '° 1 ’5 2 C 2 5 in the variational calculation is consistent with a second- eo- order transition": within the accuracy of our Monte Car- lo determination of (‘2 I” M2) there is only one minimum for any A. the position of which varie continu- ously as shown in Fig. 1. For a first-order transition, in contrast, one might expect to have two local minima at different value of a such that the position of the absolute minimum change discontinuously from one to the other at some transition point A... Figure 2 shows variational etimate of £0 /N, for dif- faet value of A for both of the trial wave functions in and ‘2. along with the large- and small-A limits of E0 IN, give for three-dimensions in Eq. (3.20). Again Fig. 2(a) is for a 3X3>< 3 lattice and Fig. 2(b) is for a 5 x5 lattice. Note that in the weak-coupling region. i.e.. A) l. the second variational etimate is the better one; this is as an- ticipated since the construction of d, incorporate the correlations between magnetic fields on different pla- quette appropriate to the wek-coupling limit (with 021). Note too that the two variational etimate are al- most the same in the strong-coupling region. The variational calculations show an intereting differ- ece betwcal the two- and three-dimesional theorie: the transition between small~ and large-A behavior is much sharper in the threedimalsional theory. This agrees with the conjecture that there is no phase transition to a non- confining phase in the two-dimesional theory. On the otha hand, the variational etimate based on the harmon- ic wave function e; is better at large A. eve in the two- dimesional theory; evidetly the vacuum state is not as simple as one with no correlations between the B fields on diffaalt plaquette. This is ccnsistet with the specula- tion that it is the infiuece of long-range topological de- fects, two-dimesional vortice. that maintains disorder in the two-dimensional theory at large A (Ref. 7); the vortice live on top of an esetially harmonic wave function. IV. MONTE CARID RESULTS InthissecticnweshalldecribethereultsofGFMC elcalations for the compact U(1) lattice page theory. The basic equation that defines our GFMC algorithm is Eq. (3.10). A brief recapitulation of the method is as fol- lows: Each complete Monte Carlo itaation of Eq. (3.10) replaces an esanble s'cin;tp1;a-1.2.....N'l ch’ .ccnfigarationscftheinteger-valaedplaquatevariahle SD ENERGY N o o A l l A A I J A 0 L0 20 30 40 50 FIG. 2. (a) Variational etimate of the vacuum eergy per plaquette vs coupling constant A for the three-dimensional theory. Thesolidanddashedcarvesarefrom perturbationex- pansions at small and large A. repectively. The crease 1+ ) and circle (0) are variational etimate with trial wave func- tions a, and ‘1. respectively. (b) Variational etimate cf the vacaumea'gypaplsquetteforthetwo-dimesionaltheory. Thectu'veandpcintshavethesamemeaningasinh). n(p) by a new emanble f-1n,(p);a=1.2.....Nl. This replacanet is a two-step proces involving branch- ing. which is governed by V[n',(p)]. and diffusion. which is govaned by G[n(p).n',(p)]. The change in atsanble size N'-oN provide a measuranalt of the eigenvalue A. Importance sampling is provided by the trial wave func- tions in and ‘2 defined in Sec.lll B. Figure 3 shows Monte Carlo etimate of the eagy per plaquette Eo/N,. i.e.. the quantity defined in Eqs. (3.16) and (3.17). for three and two (space) dimesions; the lat- ticesizeare3x3x3 and 5x5. The twosctsofpointson this graph are the reults obtained with the two impor- tance functions. The curve are the ordinary variational etimate. of which individual points wae shown in Fig. 2. Thee curve are not perturbation theory curve; how- ever. the trial wave function a. is known to he an accurate approximation of the eigefanction at small A. and d; is accurate at large A. By the variational principle thee curve are rigorously upper bounds on the may £0. We shall describe the realts as estimate of Eo/It’, vs A.hutitshoa1dherccallcdthat£oistheinputquantity E 2.0- (a) r- l.5- y. y 8 w 1.0 »- 2 U 0.5 - F' O a 1 1 J a a A A _1 _1 O 0.5 1.0 1.5 2 O 2.5 A 4.0- (b) 3.0)- ’. ’ I >- ’v’ E o"... w 2 o e 0 2 Ian r- 10 - o g 4 J J. J J A .1 a Q 0 1.0 2.0 3.0 4.0 6.0 A FIG. 3. (a) Monte Carlo etimate of the vacuum eagy per plaquette vs coupling constant A for the three-dimesional theory. The solid and dashed curve are variations) etimate with trial function it and 1,. respectively. The crease ( + )and circle (Glare Monte Carlo realts with importance functions d. and ‘2. respectively. (b) Monte Carlo estimate of the vacuum eergyperplaqaetteforthetwo-dimensionsltheory. Thecarve sndpcintshavethessmemeningssinla). sndAtheunknowneigevsluethatisoutputbythe GFMC calculation. Thee calculations used “hie of approximately 1m ,ctmfigurations; this size fluctuate with ech itaation. We also checked some realts with larger esanble. Typ- ielly 1W iterations were used to obtain the etimate of Eo/N, shown. Each computation took roughly 1.5 min at s CDC Cyber 750 computer. To be sure of conva- gece we checked that the final estimate is indepaidalt of the starting alsanble; e.g.. that an initial atsanble with all value cfn(p) equal to see give thessme final vslueas one with randomly geaated value of n (p). The Monte Carlo results obtained with importance ftmctionezliecnacontinuouscurvethatinterpolatebe- twee the known small-A and large-A depedence. Thee reultscsnhethoaght ofssacalculationofthecorrection to the variational eagy. The correction is very small ex- cept whe A~l because the variational etimate with trial ftmctiondzaecuratelydecribethegrotmdststeinboth thesmall-sndlargeAlimits. TheMonteCsrlocorrecticn APPLICATION OF THE GREENS-FUNCTION MONTE CARLO . . . 134 N73 forA~1isjusteoughtopushtheeagyhelowthevsli- ational bound provided by the other trial function in. In contrast. the Monte Carlo realts with importance function in show an intereting failure: for A 2 1.3 the estimate are not consistet with the variational bound due to ‘2. We interpret this as.the stronget evidece in our calculations of the existece of a phase transition in the three-dimesionsl U(1) page theory. This conclusion is based on the following argumet: The trial wave func- tion (1.. which has no correlation between the magnetic field value on different plaquette. is qualitatively dif- ferent than the ground-state tigefunetion if A 2. 1.3. which insted has the long-range correlations associated with the matrix M(p,p') that define the harmonic wave function a, in Eq. (3.24). Therefore the importance func- tion (1. fails to direct the diffusion proces to the region of the space of configurations where the most significant ground-state configurations are located. Furthermore. the disordaedphsseofthesystan should still existasalow- eergy state concentrated in the same region of configura- tion space as the uncorrelated importance function 11.. Apparently this uncorrelated state is metastable with repect to the GFMC iteration; for a finite-esanble size and ita'ation time it cannot converge to the actual ground state a. Similar metastable state are exhibited as hys. ta'esis 100ps in Monte Carlo stadie of the path-integral formulation of lattice gauge theorie. So. for A 2. 1.3 there are two qualitatively differet low-alergy state: the actu- al ground state decribed well by the harmonic wave func- tion ‘2. and the uncorrelated state which is metastable with repcct to GFMC ita'stion with the uncorrelated im- portance function sh. The reults of the GFMC elculations on the two- dirnalsional theory are quite differ-alt. Thee are shown in Fig. 3(b). The Monte Carlo realts obtained with the two different importance functions agree with one anotha' over the attire range of A. and are consistent with both variational bounds. There is no sign of a metastable state. We take this as the bet evidence of the nonexistece of a phase transition in the two-dimesional U(1) lattice page theory. In particular. it seems that either of the trial wave functions 1:, and ‘2 resemble the a'galfunction closely eough to he used successfully as an importance function for any A. The crossover from small-A to lsrge~A behavior in the three~dimesional theory occurs continuously as a func- tionofA. Thusthephasetransitionappersnottohea first-order transition. We have also invetigated a model with a fast-order phase transition. the Zz-gauge theory in three (space) dimesions. by the GFMC method. Thae. in contrast to the U(1) theory. the slope ofthe curve ofe~ agy vs A change discontinuously at A- l. the self-duality point.eve ins latticeassmall as 3x3x3. Thaeslsowe find metastable state by using as an importance function awavefunctionwiththeadaordisordasuitedtothe aha phase. _ Another quantity that we have computed is the expects. tion value ofthe B field; more precisely the quantity IV-(dl 1-cosB(p)l¢) . (4.1) N74 Notethat Wisnotindcpendentoftheesegypepla- quette £0 /N,. tine: a 50 W" 3A IN, ] ' Howeve, we calculate Wdirectly from Eq. (4.1). not by using Eq. (4.2). The computed value of Ware shown as a function of coupling constant A in Fig. 4. for the three- dimensional theory. Thee the curve show the perturba- tion expansions of the expectation value of l-cothp) for hrge and small A. The points are variational and Monte Carlo etimate; the Monte Carlo points are computed from the mixed expectation value, i.e., Eq. (2.16). for the two importance functions. Figure 4(a) is for the harmonic trial function d2. Note that the ordinary expectation value in the variational state ‘2 agree with the large-A perturbation curve. but differs from the small-A curve. as expected. The GFMC method compute the correction to the variational etimate; at (4.2) SLOPE FIG. 4. (a) The expectation value of l-coaB for the three- dimensional theory. calculated from the trial function ‘3. The curve are perturbation expansions. The triangle (Al are for the simple expectation value in 6;; the crosse (x) are Monte Carlo calculations of the mixed expectation value. Eq. (2.l6). (bl The expectation value of l-cosB calculated from the trial func- tion“. Thecurveandpointshavethesamemeaningasinta). DAVIDWJIEYSANDDANIELLSTUMP 135 arnall A, say, A510, the corrected value are consistent with small-A peturbation theory. Figure 44b) is for the uncorrelated trial function in. I-Iee the expectation value in 1:. agree with the small-A perturbation expansion. but deviate from the large-A ex- pansion. In this case the correction computed by the GFMC method is not large enough to bring the reult into agreenent with the perturbation expansion at large A. As before we interpret this failure as a consequence of the metastability of the uncorrelated state with repect to the GFMC iteation, and claim it as evidence of the phase transition. Wehavenotincludederorbarsonthepointsonthee graphs. Thee reults involve aveage me 1000 itea- tions for ensenble of approximately 1m configurations. In all graphs except Figs. 4(a) and 4(b) the standard devia- tion is small compared to the size of the point plotted on the graph. To check whether the standard deviation is a reasonable measure of the eror, we verified that averaging ove half as many measurements increased the standard deviation by about ‘5. In Figs. 4m and 4(b) the standard deviations were somewhat large. but still comparable to the size of the point plotted on the graph. V. SUMMARY We have applied the Green‘s-function Monte Carlo (GMFC) method to the compact Ut ll lattice gauge theory in three and two dimensions on small lattice, 3X3x3 and 5x5. The GFMC importance-sampling technique was implemented with two trial wave functions: the un- correlated trial function sh. which reemble the strong- coupling eigenfunction; and the harmonic wave function ‘2. which derive from the weak-coupling eigenfunction but is also quite accurate at strong coupling as well. In the three-dimensional theory the vacuum energy per plaquette varie continuously with A, but undergoe a rathe sharp me from small-A dependence to largeA dcpenderce. around A~ 1.3. In the small-A region, whee there is little correlation between 8 fields on diffeent pla- quette. the two importance functions yield approximately equal value of the etegy. But in the large-A region. whee thee are long-range correlations between plaquette as decribed by the trial fumtion ‘2. the uncorrelated im- portance function it. yields value inconsistent with the variational bound placed by 5;. We interpret this as me testability of the disordered state. and as evidence of the phase transition of the UH) lattice gauge theory. In the two-dimensional theory. in contrast. the vacuum energy pe plaquette varie slowly with A. and the two im- portance functions yield equal enegie for all value of A. Weinterpretthisasanindication thattheeisnophase transition in the two-dimensional theory. We have used the tem “metastable” to decribe the false ground state found by the Monte Carlo calculation when using an importance function appropriate to the disordered phase in a region of coupling constant whee the true ground state has correlations decribed by the harmonic wave function. This choice of words may be misleading in that the false state may we converge to thetruestateiftheensenblesizeistoosrnall;thefalse 2_§ APPLICATION OF THE OREEN‘S-FUNCTION MONTE CARID . . . state is the: actually stable. This property is also seen in quantum many-body problems with phase transitions, such as solid to liquid helium. We have not attenpted to study the convegence of the metastable state by increas- ing the ensenble size. It is possible that the minimum size necesary to allow the convegence to occur is so large that the calculations are not feasible. All that can be said theoretically is that the iteation is stable only for the true ground state if the ensemble is large enough. Of course this is only an issue in systems with a phase transition. for which thee are two qualitatively diffeent low-enegy state. Our calculations wee retricted to mall lattice. Cal. culations for large lattice are certainly feasible; the only limitation is compute time. GFMC calculations have been done with several hundred quantum variable; for comparison. a 3x3x3 lattice gauge theory has Bl link variable. We believe that the reults of calculations on large lattice would be vey similar to those decribed above. In particular the vacuum energy pe plaquette does not depend very much on the lattice size. We have seen two indications of this. First. the perturbation expansions 136 1375 areindependentoflatticesizeformall A.andonlyvey weakly dependait for large A. as indicated by (3.21). Second. we carried out the ordinary variational calcula- tions for lattice of different size, and found only a vey mall lattice-size dependence. Thee U(1) lattice-gauge-theory calculations wee done in a special way. by formulating the problem in the space of configurations of the plaquette variable I: (p) defined in Eq. (3.4). That formulation leds to an equation that is especially simple to iteate by the GFMC method. It is our impresion that thee doe not exist a similar special formulation of the SU(2) lattice gauge theory. Theefore we intend to apply the GFMC method to that theory by anspproachmorealongthelinedecribed in Sec. IIA. ACKNOWLEDGMENTS We are pleased to thank the following people for discus- sions about this recarch: M. Creutz, T. DeGrand, J. Negele. S. Koonin. J. D. Stack, M. H. Kalos. and D. Ceperley. This recarch has been supported by the Na- tional Science Foundation and Michigan State Univesity. 'K. G. Wilson. Phys. Rev. D 1.0. 2445 (1974). 3). Kogut and 1.. Susskind, Phys. Rev. D 1.1. 395 (1975). 31. Kogul. D. K. Sinclair. and L. Susskind. Nucl. Phys. 3.1.11. 199 (1976); T. Banks. J. Kogut. and I.. Susskind. Phys. Rev. D n. 1043 (1976); A. Carroll. 1. K00“. D. K. Sinclair. and L. Susskind, ibid. 11. 2270 (I976). ‘5. D. Drell, H. R. Quinn. 3. Svetitsky. and M. Weinstein, Phys. Re. D 12. 619 (1979); D. Horn and M. Weinstein. ibid. 25. 333] (1982). SM. Creutz, L. Jacobs. and C. Rebbi, Phys. Rev. Lett. 12. 1390 (I979). ‘M. Creutz. Phys. Rev. Lett. 11. 553 (1979); Phys. Rev. D 2.1. 2308 (1980). 7A. M. Polyakov. Phys. Lett. 123. 79 (1977); Nucl. Phys. 3129. 429 (I977). '1'. Banks. R. Myerson. and I. Kogut. Nucl. Phys. 3122. 493 (I977). 5% also A. Guth. Phys. Rev. D 2.1. 2291 (I980); M. Gopfert and G. Mack. Commun. Math. Phys. 32. 545 (I982). ’D. R. Stump, Phys. Rev. D 21. 972 (1981). |oM. Creutz and G. Bhanot. Phys. Rev. D 21. 2892 (1980); B. Lauth and M. Nauenberg. Phys. Lett. 253. 63 (1980). "M. H. Kalos. Phys. Rev. 123, 1791 (1962); Phys. Rev. A z, 250(1970l. ”M. H. Kalos. D. Levesque. and I... Verlet, Phys. ReV. A 2. 2m (1974); D. M. Ceperley, G. v. Chete. and M. H. Kalos. Phys. Rev. B 11. 1070 (1978); P. A. Whitlock. D. M. Ceperley, G. V. Chete. and M. H. Kalos. ibid. 12. 5598 (1979); P. A. Whitlock and M. H. K31“. I. Cunp. Phys. 39. 361 (I979). ”D. M. Ceperley and M. H. Kale. in Topics in Current Physics, edited by K. Diode (Springe. Belin. 1979). Vol 7. This refeence is a comprehensive review of applications of the Mortte Carlo method to quantum systems. I‘Our approach is patterned afte the early GFMC method decribed in Ref. 11. but with importance sampling. l"In the calculations decribed in this paper this normalization constant was recalculated exactly after each Monte Carlo itestion. By contrast. in some calculations. e.g.. those of Ref. 12. the normalization is taken into account only in an approx- imate way. _ "I. Kogut. Rev. Mod. Phys. 51. 659 (1979). "U. I'Ieller. Phys. Rev. D 21. 2357 (1981). In this pape s varia- tionalwavefunctionoftheformoffiq. (3.13)isusedins study of the two-(space) dimensional U(l) gauge theory. "We later learned of the remarkable advance made by Horn and Weinstein (the second refecnce of footnote 4) for analytic evaluation of this quantity. Our variational Monte Carlo cal- culation. the reults of which are shown in Fig. 1. amounts to a numerical evaluation of the energy expectation value. which in their approach is computed by introduction of a ‘partition function" with a sum we state that projects onto the gauge~invariant space. Because we use a function of the pla- quette variable n(p). this projection 'e not explicitly used by In. although it may be implicit. l’I‘hat the phase transition of the U(1) gauge model in three di- mensionsisofsecondorderwasfintsuggetedbythecalcula- tions of B. Lauth and M. Nauenberg. Ref. 10. 137 In the preceding reprinted article graphs of the quantity <1-cos B(p)> versus A were not given for the two dimensional theory. For the sake of completeness, these are shown in Figures Bl and 32. Figure Bl is for the harmonic trial function oz and Figure 32 is for the uncorrelated trial function wl. The circles (0) are variational estimates and the crosses (+) are GFMC results. The solid curves are perturbation expansions. Notice that although the variational estimates using the two wave functions are quite different at large A, the two sets of GFMC results are reasonably consistent with each other. This is in contrast to the three dimensional theory where the wave function ll acted very poorly as an importance function for large A. This difference is due to the fact that there is a phase transition in the three dimensional theory whereas in two dimensions there is no phase transition. 138 1.0 C3 00 0.6 <1 — cos B(p)) C3 F‘ Figure 81: GFMC estimate of the expectation value of l-cosB(p) for the two dimensional U(1) theory using the harmonic wave function ¢2 for importance sampling. 139 1.0 0.8 <1—cos B(p)) C3 E» 0.4 Figure 32: GFMC estimate of the expectation value of l-cosB(P) for the two dimensional U(1) theory using the uncorrelated wave function *1 for importance sampling. 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