Density Matrix Renormalization Group: a New Approach to Lattice Gauge Theory

DENSITY MATRIX RENORMALIZATION GROUP: A NEW APPROACH TO LATTICE GAUGE THEORY

Tim Byrnes

A thesis submitted in satisfaction of the requirements for the degree of

Doctor of Philosophy

in the Faculty of Science

THE UNIVERSITY OF NEW SOUTH WALES UNSW 2 0 OCT 2003 LIBRARY ABSTRACT

Two lattice models are studied using the recently developed Density Matrix Renor malization Group (DMRG) method. First, we study the massive Schwinger model, equivalent to quantum electrodynamics in one space and one time dimension, in a

Hamiltonian staggered lattice formulation. We investigate in particular the point

0 = 1r, where a second order phase transition is known to occur. We find using finite size scaling analysis of DMRG data that the Schwinger model at 0 = 1r lies in the same universality class as the 2D classical Ising model. We confirm Coleman's picture of "half-asymptotic" particles, and locate the position of the critical point to good accuracy. We also find evidence for a possible dual symmetry and dual order parameter in the model. We also study the model at the point 0 = 0, and obtain estimates for the "vector" and "scalar" mass gaps. We significantly improve the accuracy of existing results for these quantities, in many cases by one or two orders of magnitude. A non-relativistic expansion of the "vector" mass gap is per formed to third order, correcting an error in a previous work. We find for both the

0 = 0 and 0 = 1r cases that DMRG is an extremely effective tool in studying the Schwinger model. We conclude that DMRG is an excellent algorithm for the study

(l+l)D lattice gauge theories, however in higher dimensions the method loses its comparative advantage, although many investigations are currently in progress to solve this problem.

The second model we examine is the Heisenberg chain with a single weak link.

This model is equivalent, via a Jordan-Wigner transformation to a ring of spinless electrons with short range interactions. The weak link then acts as a single defect in the chain. We examine this in the context of persistent currents in the presence

11 of a magnetic flux through the ring. We use DMRG to calculate the spin stiffness and correlation across the weak link. Our results show that the stiffness, and hence the current, renormalizes to zero in the presence of any weak link, agreeing with the renormalization group predictions of Kane and Fisher. The correlation does not renormalize, and approaches finite values for all weak link strengths, except for the open chain, due to the "local" nature of this quantity.

iii To my parents ACKNOWLEDGEMENTS

Firstly I would like to thank my supervisor, Associate Professor Chris Hamer, without whom I would have never been able to complete this work. He has provided me with consistent advice, support, insight and guidance throughout my PhD. I would particularly like to thank him for the financial support during the last few months of my PhD, which has made the writing of this thesis a much less stressful task.

Thanks also to Dr. Robert Bursill who taught me the DMRG method that was used to obtain the results for the bulk of this thesis. His patience and careful explanations were much appreciated particularly in the first year of my PhD, which laid the foundations for the rest of my time at UNSW.

I must also thank Dr. Hans-Peter Eckle and Dr. Anders Sandvik who were involved in the weak link Heisenberg chain project, and Dr. Zheng Weihong who I worked with on the Schwinger model project. It was a truly rewarding experience to collaborate with all these people.

The calculations on this thesis were performed primarily on the APAC national supercomputer facility, located at the Australian National University, Canberra.

Some calculations were also performed on the Napier machine at the New South

Wales Centre for Parallel Computing, and on the Fujitsu VPP machines at the

Australian National University Supercomputer Facility (ANUSF). I thank all these institutions for their facilities.

Thanks also to my family, in particular my parents, who have always been supportive of everything I have done throughout my studies.

Finally, the generations of Room 57 people who I shared an office during my

PhD has made my time here far more enjoyable. These people include: Doug,

Scott, Judy, Maria, Pradeep, Mushe, Jacinda, Elizabeth, Michael, and Julian.

V Special thanks to Pradeep, Mushe and Julian for the cricket, and Judy, Maria and

Jacinda for the entertaining conversation.

Vl Contents

List of Figures xi

List of Tables xvi

1 Introduction 1

2 The Massless and Massive Schwinger Model 13 2.1 Preliminaries ... 14 2.1.1 Definitions . 14 2.1.2 Electrodynamics in (l+l)D 15 2.2 Weak Coupling Approximation 17 2.3 The Massless Schwinger Model 21 2.4 The Massive Schwinger Model 24 2.4.1 Bosonization ...... 24 2.4.2 Charge shielding with mass 27 2.5 The Background Field 28 2.5.1 The Origin of 0 28

2.5.2 Why 0 = 1r is special 31 2.5.3 Phase Diagram 37

3 Lattice Gauge Theory 38 3.1 Abelian Lattice Gauge Theory . 38 3.2 Hamiltonian Formulation . . . . 41 3.3 Lattice Formulation for Fermions 44

Vll 3.3.1 The Schwinger Model Hamiltonian 44 3.3.2 Extracting the Continuum Limit 48 3.4 The Lattice Strong Coupling Limit 49 3.4.1 Background Field 0 = 0 50

3.4.2 Background Field 0 = 1r 51 3.5 Other Useful Formulae 53 3.5.1 Gauss' Law . . 53 3.5.2 Order Parameters . 54

4 Finite Size Scaling 57 4.1 Modifications of Critical Behaviour in Finite Systems 58 4.2 The Finite Size Scaling Hypothesis 61 4.3 Phenomenological Renormalization 63 4.4 Quantum Hamiltonian Limit .... 65 4.5 An Example - the Transverse Ising Model 67 4.5.1 Energy Spectrum and Order Parameters 67 4.5.2 Finite Size Behaviour ...... 72 4.5.3 The Self-duality of the Transverse Ising Model 73

5 The Density Matrix Renormalization Group Method 79 5.1 The Numerical Renormalization Group Procedure 80 5.2 Problems with the Numerical RG Procedure 83 5.3 The Infinite Lattice DMRG Method ..... 88

6 Results for the Massive Schwinger Model at 0 = 1r 92 6.1 Implementation of the DMRG 92 6.2 Convergence Tests ...... 97 6.3 Analysis of Critical Behaviour 101 6.3.1 Position of the Critical Point . 101 6.3.2 Critical Indices ...... 104 6.4 Mass Gaps and Order Parameters 107

6.4.1 Two Particle Gap D..2 /g 107 6.4.2 Loop energies D..o/ g . 112

Vlll 6.4.3 1-particle gap D.i/ g 114

6.4.4 Order Parameters f 5 , f°' 117 6.5 Conclusion ...... 119

7 Results for the Massive Schwinger Model at 0 = 0 124 7.1 Non-relativistic Series ...... 124 7.1.1 First Order Schrodinger Equation 126 7.1.2 Second Order Schrodinger Equation . 127 7.2 DMRG Results . . 129 7.2.1 Vector Gap 129 7.2.2 Scalar Gap 134 7.3 Conclusion ..... 136

8 The Persistent Current 139 8.1 The Aharonov-Bohm Effect ...... 139 8.2 Persistent Current Properties for Non-interacting Electrons . 143 8.2.1 No Disorder . . 144 8.2.2 Single Scatterer 145 8.3 The Effect of Electron-Electron Interactions 148 8.3.1 No Impurity .. 148 8.3.2 Single Impurity 151

9 Results for Heisenberg Chain with a Weak Link 155 9.1 Implementation of the DMRG 155 9.2 Quantities of Interest 157 9.3 Numerical Results .. 158 9.3.1 Convergence Tests 158 9.3.2 Surface Energy 160 9.3.3 Spin Stiffness . 163 9.3.4 Spin-Spin Correlation across the weak link 165 9.4 Conclusion . 168

10 Conclusion 170

IX References 174

A Weak Coupling Expansion of the Massive Schwinger Model 183 A.1 Dirac Fields in (1 + 1)D ...... 184 A.2 Evaluation of Matrix Elements . . 185 A.2.1 The Free Hamiltonian Hf}- 185 A.2.2 The Interaction Hamiltonian H!nt . 187 A.2.3 The Interaction Hamiltonian H[nt . 191

B The Transfer Matrix 194

C Finite Size Behaviour of the Transverse Ising Model 197

X List of Figures

1.1 A conducting ring with a magnetic flux . The persistent current I may flow paramagnetically (as shown), or diamagnetically...... 8 1.2 An electron must scatter many times before going around a ring once. 10

2 .1 a) The electric field around an isolated charge g b) The electric field around a pair of charges ±g. 16 2.2 Six lowest order diagrams contributing to the weak coupling expansion Hamiltonian. (e) and (f) refer to interactions with the back ground field...... 20 2.3 Complete vacuum polarisation of the Schwinger model. An electric field E imposed by plates at either end of the universe is exactly

cancelled by a polarisation field EP 01 • • • • • • • • • • • • • • • • 23 2.4 A quark-antiquark pair in the presence of a background field F. . . 29 2.5 The cascade of pair production in reducing the electric field. . . . . 29 2.6 Confining potential of a quark-antiquark pair at various 0 for weak coupling. 31 2. 7 Electric field configurations for ground and single quark states with 0 = 1r and 0 = 0...... 32 2.8 The bosonic potential U(cp) for 0 = 1r and m/g > (m/g)c- 33 2.9 Ground state energy versus 0 for three values of m/ g. 34 2.10 A half asymptotic 1-particle state is equivalent to a kink state in bosonic variables . 35 2.11 Phase diagram for the massive Schwinger model. 0 lies in range [0, 2n] for convenience. The jagged line shows location of the first order phase transition. The circle denotes a second order phase transition. 37

Xl 3.1 Angular field variables defined on the links of a lattice...... 39 3.2 A gauge transformation of the site n...... 40 3.3 a) Spin configuration for the ground state with background field 0 = 0. Numbers between spins denote the electric field on the links. Here we have L( n) = 0. b) Configuration with one flipped spin. c) Configuration with two adjacent flipped spins. 51

4.1 Specific heat of the 2D square Ising model with periodic boundary conditions for various lattice sizes (Ferdinand and Fisher, 1969). The vertical line denotes the position of the critical point in the bulk limit...... 59 4.2 Spectrum of the transverse Ising model...... 70 4.3 Phase diagram for inverse coupling 1/>. and magnetic field h. The jagged line represents a first-order phase transition. The open circle represents a second order phase transition...... 71 4.4 Two dimensional Ising lattices and their duals. Solid circles denote original lattice spins. Open circles denote dual lattice points. . . . . 74 4.5 (a) The ground state is a perfectly ordered eigenstate of az. (b) The ground state is dominated by the ordered state, with isolated kink antikink pairs. The size of the kink-antikink pairs a is much smaller that the distance between the kink-antikink pairs d » a. (c) A large number of flipped spins result in a kink condensate. The size of the kink-antikink pairs become comparable with the separation d ~ a. 77

5.1 A schematic picture of a Numerical RG iteration. Blocks B1 and

B2 are augmented to form the block B', which are copied to form blocks B~ and B;...... 81 5.2 The ground state eigenvalues for two 8-site blocks (black circles) and a single 16-site block (white squares), reproduced from White and Noack (1992)...... 84

5.3 An example of the superblock method. Blocks B1 and B2 are aug mented to form B', and passed onto the next iteration. Block B3 serves as an environment block...... 85 5.4 An augmentation process in the standard DMRG method ...... 88

6.1 Augmentation process for the Schwinger model with OBC imposed. Heavy dots indicate the location of the lattice spins. n labels the sites on the lattice, while L( n) labels the electric fields on the links. 93 6.2 As for Figure 6.1 but with PBC...... 94

xii 6.3 Formation of a loop of electric flux. a) Initially there is no electric flux throughout the ring. b) A qq pair forms with electric flux L = 1 between them c) The pair annihilate to leaving an electric flux L = 1. 95 6.4 Three basis states with the same fermion configuration, but with different loops of electric field. The left and right ends of the line are connected by PBC. Taking the state labelled as (b) as the base configuration, we may obtain states (a) and (c) by adding or subtracting a unit flux respectively. In this example, we therefore have 95

6.5 Dependence of the estimated ground state energy density E0 /2N x on the density matrix truncation eigenvalues, for x = 100, m/ g = 0.3, 0 = 1r, using PBC...... 102

6.6 The "loop gap" 110 / g for various m/ g near to the critical region with X = 25...... 102 6. 7 Pseudocritical points for three couplings x = 100, 44.4, and 25. 103 6.8 Critical line in the m/ g versus 1/ vx = ga plane. Open circles are our present estimates, and squares are the previous results of Hamer et al. (Hamer et al., 1982), which are in good agreement. The dashed line is a quadratic fit to the data in ga. 105 6.9 'Logarithmic' ratio estimates of critical indices -1/v and -/3/v for lattice spacing 1/ vx = ga = 0.45. Quadratic fits in 1/N provide the bulk extrapolations. We estimate here 1/v = 1.00(2) and /3/v = 0.125(5)...... 106

6.10 Bulk extrapolations for the 2-particle gap 112 / g for m/ g = 0.0. Dashed lines are the fits to the data, according to (6.7) ...... 108 6.11 Continuum extrapolations for the 2-particle gap for m/g = 0.0 and 0 = 1r. Data sets obtained through separate VBS and fit extrapolations...... 110 6.12 Final estimates for gaps in the 0-particle, 1-particle and 2-particle sectors at 0 = 1r. Dashed lines are merely to guide the eye...... 112

6.13 Convergence of lattice data for the loop gap 110 / g for various coupling x at m/ g = 0.1. At N = 256, all curves converge to within errors of the points...... 113 6.14 The electric fields at either end of the chain in the presence of a single particle are g /2, -g /2 respectively...... 114

xiii 6.15 Basis states for the two-site Schwinger model in the staggered lattice formulation. We see that setting a background field of a) a = 1/2 and b) a= -1/2 do not produce equivalent basis states...... 115 6.16 Extrapolation to the bulk limit for the order parameter re:. = (L + a) 0 , for m/ g = 0.5 and various couplings x, and an applied field of a= 1/2...... 118 6.17 Order parameters ra = ((L+a))o, f 5 = (ii/;15'1/;/g) 0 near the critical region. Dashed lines are merely to guide the eye. 120

7.1 "Vector" gap binding energies Eif g at m/g = 0, for various lattice spacings...... 130 7.2 Continuum extrapolation of "vector" gap binding energies Ei/ g for m/g = 0...... 131 7.3 An example of our extrapolation procedure for the "vector" gaps

E 1 / g for m / g = 0. Circles, squares and triangles show linear, quadratic and cubic extrapolants respectively. Dashed lines show the upper and lower bounds for our final estimate. Here we estimate Eifg = 0.56419(4) ...... 131 7.4 Comparison of our results for the "vector" state binding energies Ei/ g with other works. Squares mark the results of Sriganesh et al. (2000) and triangles the results of Kroger and Scheu (1998). The results of Vary, Fields and Pirner (1996) and Adam (1996) were used for the expansion around m/ g = 0. The 1st and 3rd order non relativistic expansions of Section 7.1 are also shown for comparison. 133 7.5 A typical excitation spectrum for binding energies E / g for the scalar state. Here we choose x = 25 and m/g = 0.125. Each excitation is labelled with an index i. Here the vector state approaches E1 / g ~

0.584, while the scalar state approaches E2 / g ~ 1.269...... 135 7.6 Comparison of our results for the "scalar" state binding energies E2/ g with other works. Squares mark the results of Sriganesh et al. (2000) and triangles the results of Kroger and Scheu (1998).- The results of Adam (1996) were used for the expansion around m/ g = 0. The 1st and 3rd order non-relativistic expansions of Section 7.1 are also shown for comparison. 136

XIV 8.1 An Aharonov-Bohm ring with a central magnetic flux (and mag netic potential A). An electron enters from the left and has equal probability of entering the upper and lower channel. An interference effect is seen when the two paths recombine...... 140 8.2 A conducting ring with a magnetic flux threading the axis. 141 8.3 'Band' structure of a ring with varying magnetic flux . For an impurity free ring (dashed lines), the energy levels are determined by a sequence of parabolic potentials. For a ring with impurities (solid lines) gaps open at level crossings of the energy...... 143 8.4 The persistent current for a ring of non-interacting electrons with no impurities. Solid lines show systems with an odd number of electrons, dashed lines for systems with an even number of electrons. 146 8.5 The effect of 8-function impurity on the persistent currents present in a ring of non-interacting electrons. In the limit of strong impurity, the current reduces to a sine function...... 147 8.6 Renormalization group flows as a function of the electron coupling g and the transmission t. For repulsive interactions g < l, the conductance renormalizes to G = 0, while for attractive interactions g > l, the conductance renormalises to that of zero barrier height. 153

9.1 DMRG augmentation process for the Heisenbe~g chain with a single weak link J'...... 156

9.2 !:::..EN= (E0 (N; J') - E0 (N; J' = J))/ J as a function of J'/ J for the ring with periodic boundary conditions. The data are extrapolated using a polynomial fit to obtain the bulk limit...... 161 9.3 As for Fig. 9.2, but with anti-periodic boundary conditions. 161 9.4 The stiffness factor Ps as a function of 1/N, for lattice sizes N = 4 to 64...... 163 9.5 The spin stiffness obtained by Laflorencie, Capponi and S0rensen (2001) by Bethe ansatz and renormalization group methods. . . . . 165 9.6 The same data as Figure 9.4 but plotted on a log-log scale. 166 9. 7 Stiffness Ps obtained through SSE Quantum Monte Carlo methods, courtesy of A. W. Sandvik. x = J' / J in the figure...... 166 9.8 The correlation (SNSi) across the weak link for various J' / J for the periodic chain. 168

C.1 (a) Contour of integration for (C.14). Crosses indicate non-analytic

points given in (C.13). (b) The deformed contour C2...... •. 199

xv List of Tables

4.1 Bulk and finite-size dependences of various thermodynamic quantities...... 63 4.2 Equivalences between statistical mechanics and field theory. 66

6.1 Comparison of exact finite-size ground state energies with DMRG estimates using PBC, with 0 = 1r, mfg= 0.2, x = 100 and Lmax = 5. DMRG results use moMRG = 930 states in a single block. The dif ference t5E8MRG = E8MRG - Eixact between the DMRG and the exact results are quoted as "round off" if they agree to within machine prec1s1on,. . wh' 1c h 1s. rv 1 . 0 x 10-11 ...... 97 6.2 Comparison of ground state energies and two order parameters be tween exact diagonalisation and DMRG with OBC, with 0 = 1r, mfg -: 0.3, x = 100. DMRG results use moMRG = 400 states in a sin gle block. The difference t5E8MRG is defined in the caption for Table 6.1, and similarly t5r DMRG = r DMRG - r Exact• The difference between the DMRG and the exact results are quoted as "round off" if they agree to within machine precision, which is t5E8MRG rv 1.0 x 10-11 . For order parameters, the round-off level is rv 1.0 x 10-6 . . . . • • 98 6.3 Comparison of DMRG data to strong coupling series estimates (Hamer et al., 1997), for x = 1, mfg= 0.5, 0 = 1r, using PBC and Lmax = 5. Or- der parameter estimates for the strong coupling series are supplied courtesy of Z. Weihong...... 98 6.4 Convergence with Lmax for a sample calculation with x = 100, mfg = 0.3, 0 = 1r, N = 20 using PBC and calculated using exact diagonal isation...... 99 6.5 DMRG estimates of the ground state energy density E0 f2Nx, the "2-particle" gap !:!.. 2f g,. and two order parameters, the mean field file= ((£ + a))o and axial density f 5 = (ii/;151/Jf g) 0 as functions of moMRG, the number of basis states retained per block. These results are for PBC, at x = 100, mfg= 0.3, 0 = 1r, and N = 256 sites. 100 6.6 As for Table 6.5, with OBC...... 101

XVI 6.7 Estimates for critical points (m/g)c(x) and critical exponents 1/v and (3 / v as functions of the lattice spacing parameter 1/ Ji = ga. . 107 6.8 VBS extrapolations for the 2-particle gap with m/g = 0.0, x = 16.

The two tables are for the VBS parameters a) av8 s = 1 and b)

av8 s = 0. Here our extrapolation is !::,.2/ g = 0.511(1)...... 110 6.9 Estimates for critical points and the critical exponent 1/v, using the I-particle gap !::,.if g, as a function of 1/ Ji= ga...... 117 6.10 Our results for the loop energy !::,.of g, I-particle gap !::,.if g, 2-particle gap !::,.2/ g at background field 0 = 1r. We also quote our results for the order parameters r 0 = ((L + a))o and f 5 = (i{ry57/J/g) 0 • 120 7.1 Results of numerical integration for various operators. 128 7.2 Comparison of bound-state energies for the "vector" state Ei/ g with previous works. The results of Sriganesh et al. (2000) and Crewther and Hamer (1980) were obtained through finite-lattice studies, and Kroger and Scheu (1998) used light-cone methods to obtain their results. We also show our 3rd order non-relativistic calculation of Section 7.1.2 for comparison...... 132 7.3 Comparison of bound-state energies for the scalar state E2 / g with previous works. We display here the results of Sriganesh et al. (2000), Crewther and Hamer (1980), Eller et al. (1987), Mo and Perry (1993), and Kroger and Scheu (1998)...... 137

9.1 Comparison of exact finite-size ground state energies with DMRG estimates, where the difference r5EgMRG = (EgMRG - E~xact)/ J. The difference between the DMRG and the exact results are quoted as "round off" if they agree to within machine precision, which is rv 1.0 x 10-11 . The Hilbert space size is also quoted for the two calculations...... 159 9.2 Comparison of exact finite-size ground state energies obtained by Bethe ansatz with DMRG estimates for J' / J = 1, where r5EgMRa = (£gMRG _ E~xact)/ J...... 159 9.3 DMRG estimates of the ground state energy E0 / J and correlation (S1vSi) for N = 64 sites at J' / J = 0.5 as a function of m, the number of states retained per block for periodic and anti-periodic boundary conditions imposed...... 160 9.4 A summary of the technical parameters used for the calculation. 160

xvu Chapter 1

Introduction

Strongly correlated electron systems have attracted a large amount of interest in recent times (Edwards et al., 2000). This follows from the discovery of a large class of materials where the traditional Fermi-liquid behaviour breaks down. Examples of such materials include high temperature (cuprate) superconductors, quantum

Hall systems, magnetic systems, and systems where the electron degree of freedom is effectively one-dimensional. The common element between these materials is that the electrons in these materials do not obey the quasi-free particles description of

Fermi-liquid theory. In a Fermi-liquid theory, electrons are treated as essentially free particles, and any interaction is taken into account through a renormalization of the quasi-particles. In strongly correlated systems the picture is very different. The electron behaviour now cannot be described in this quasi-free particle framework, due to the strong correlations between the electrons. These correlations may arise due to several reasons, including the presence of strong interactions, or a reduced dimensionality in the freedom of the electrons, or both.

Several mathematical models have been put forward to attempt to explain the phenomena that these materials exhibit (for high temperature superconductivity,

1 INTRODUCTION 2 typical models are the Hubbard, t-J, and Heisenberg models). Unfortunately, analytical solutions to many of these models are generally few and far between, and are often restricted to special cases. Hence various approximate schemes must be applied, both analytically and numerically. The traditional method in such a situation is many-body perturbation theory, however this is inappropriate due to the strong nature of the interactions. Another approach, that of mean field theory, also fails, as it is unable to take into account the subtle correlations present between the electrons.

The approach in recent times has been dominated by various non-perturbative methods, in particular numerical algorithms, spurred on by the availability of high speed computers. The main forms of numerical algorithm used prior to 10 years ago were quantum Monte Carlo and exact diagonalisation. However each of these methods have their own drawbacks. In Monte Carlo algorithms involving fermions, the infamous "minus-sign problem" arises due to the anti-commutation of fermionic operators. Exact diagonalisation is limited to relatively small lattice sizes, due to the exponential growth of the size of the Hilbert space of states. Therefore some opportunity existed at the time for a new kind of algorithm that could overcome these shortcomings.

Steven White invented exactly such an algorithm, the Density Matrix Renor malization Group (DMRG) method (White, 1992; White, 1993). This was based on improvements upon the work of Wilson (1975), who used a numerical version of the renormalization group to examine the Kondo problem. While successful for the Kondo problem, when applied to other models the numerical renormalization group (NRG) procedure was found to be rather unreliable (see White and Noack

(1992) and references therein). The work of White and Noack (1992) addressed these issues and identified the problems with the method. In short, it was found INTRODUCTION 3 there was a problem with the boundary conditions of the algorithm, which resulted in a set of basis states less than ideal for representing the system. A solution of this problem, plus the introduction of a density matrix gave rise to the DMRG method in its current form today (White, 1992; White, 1993). We will discuss the DMRG method in more detail in Chapter 5. One of the first applications of

DMRG was a study of the S = 1 Heisenberg chain, where quantities such as the ground state energy density, correlation length and the Haldane gap were obtained.

Unprecedented accuracies were obtained for all quantities calculated, for example the ground state energy density was determined to better than 1 part in 10 11 .

The application of DMRG these days is varied, for a review on its applica tions see Hallberg (1999) or Gehring, Bursill and Xiang (1997). The early ap plications of the method were for examining short-range one-dimensional models

(White and Huse, 1993; Moukouri and Caron, 1995; Noack et al., 1994; White et al.,

1994; Schollwock and Jolicoeur, 1995), but were quickly extended to models with dimerization and frustration (Bursill et al., 1994; Chitra et al., 1995). In ad dition, systems with disorder (Schmitteckert and Eckern, 1996), ladder systems

(Azzouz et al., 1994; Weihong et al., 2001), and phononic systems (Zhang et al.,

1998) have also been considered. Two-dimensional systems (Liang and Pang,

1995; White, 1996; Xiang, 1996; White and Scalapino, 1997) have also been consid ered, however generally the performance of the DMRG is not as spectacular as in

1D. The accuracies of current 2D calculations are on par with Monte Carlo results, hence there is no such calculational advantage as is present in 1D. However a large amount of effort is being expended currently to improve the algorithm to make it more suitable for 2D calculations (Xiang et al., 2001; McCulloch et al., 2001).

Other applications include 2D classical systems (Nishino, 1995), dynamical properties (Hallberg, 1995), finite temperature properties (Bursill et al., 1996), INTRODUCTION 4 and dynamical correlation functions (Kuhner and White, 1999). A particularly interesting study of a simple case of asymptotic freedom was considered by Martfn

Delgado and Sierra (1999). These authors considered a single particle in the pres ence of a two-dimensional delta function potential, using an algorithm similar to momentum-space DMRG. The authors obtain very competitive results compared to existing similarity renormalization group and discrete light-cone quantisation methods. This work represents the first attempt to apply DMRG to a model in high energy physics; however it can be considered to be a rather simplified toy model, as no field theoretical dynamics are present in the model, as is the case with models of interest such as QCD.

In this thesis we apply DMRG to a fully field theoretical model, albeit the sim plest possible example, the massive Schwinger model, identical to spinless quantum electrodynamics in one space and one time dimension. The original version of the model, with massless fermions m = 0, was solved exactly by Schwinger (1962; 1963), through Green's function methods. Later, using an operator approach, Lowenstein and Swieca (1971) discovered that this solution was incomplete, although many of the main features of the model remained intact.

One of the main features of the exact solution is that the spectrum consists entirely of non-interacting bosons of mass g / .,fir, where g is the charge of the one dimensional electron1 . An alternative, and more physical derivation of the exact result was realised by Coleman (1975). Using bosonization methods (Coleman,

1975; Mandelstam, 1975) he was able to transform the Hamiltonian into a theory 1 We also occasionally call the fundamental fermion of the model a "quark", although these are completely unrelated to the particles of QCD. The true nature of these particles is that they are spinless electrons in one dimension. This does, however, follow the language of the literature. INTRODUCTION 5 of massive free bosons

(1.1) where

We may immediately observe an interesting feature of this result, namely the lack of fermion excitations in the spectrum. The original fermions of the theory never occur in the excitation spectrum, so in a sense we have a form of "confine ment". The mechanism of confinement originates from a charge shielding effect, where the vacuum state of the Schwinger model is infinitely polarisable. Therefore any electric field that may be created is neutralised to zero via dynamical fluctu ations in the vacuum. Any isolated charge is instantaneously shielded, and never appears in the spectrum. Another interesting phenomenon is that chiral symmetry is broken in the model, which arises due to an axial anomaly

.,_,,_ gE aµ,]5 -- (1.2) 7r where Jt is the chiral current and E is the electric field (Brown, 1963). In fact it is known that chiral transformations transport the model across a 0-vacuum

(Coleman et al., 1975), which is the nature of the ground state. The vacuum is therefore infinitely degenerate, labelled by an angle 0, which runs from O to

21r. An important intuitive development was the identification of 0 as a constant background electric field (Coleman, 1976). For the massless model the spectrum is completely independent of 0, and may be treated as a free parameter. However other quantities such as the chiral condensate (i/J'I/J) are 0 dependent. These features will be discussed in more detail in Chapter 2. INTRODUCTION 6

Many of these properties, such as confinement, chiral symmetry breaking, and a topological 0-vacuum are present in QCD, hence the model has been used in past decades both as a testing ground for various numerical methods, and also studied directly in order to gain some insight into QCD itself. The original motivation for studying the model was to illustrate the possibility that the gauge bosons of a gauge theory could acquire a non-zero mass, through a dynamical Higgs mechanism

(Schwinger, 1962). This was put forward as a possible explanation of the non observation of the strong force gauge field, and later the non-observation of quarks

(Casher et al., 1973; Casher et al., 1974). These theories were later superseded by the discovery of asymptotic freedom (Politzer, 1973; Gross and Wilczek, 1973).

Later, the massive version of the model was used as a first test for the applicability of Hamiltonian lattice gauge theory (Kogut and Susskind, 1975a), where it was shown that the method could obtain good results in a non-perturbative regime.

It was first demonstrated through this model that a continuous chiral symmetry could be recovered from the discrete version on a Kogut-Susskind lattice gauge theory. The model also has interest in the context of bosonization ( Coleman et al.,

1975), as both the massive and massless versions of the model may be bosonized.

More recently Creutz (1995a; 1995b) has used this model to study the effects of the CF-violating parameter 0 in QCD.

The version of the model with massive fermions is no longer exactly solvable, hence various approximate schemes have been employed in the past. Such meth ods include mass perturbation theory (Coleman et al., 1975; Vary et al., 1996;

Adam, 1996; Harada et al., 1998), Hamiltonian lattice series methods (Banks et al.,

1976; Carroll et al., 1976; Berruto et al., 1998; Hamer et al., 1982; Hamer et al.,

1997), non-relativistic weak coupling expansions (Coleman, 1976; Hamer, 1977; Sri ganesh et al., 2000), "finite-lattice" methods (Crewther and Hamer, 1980; Irving INTRODUCTION 7 and Thomas, 1983), and exact diagonalisation (Sriganesh et al., 2000). Other methods include Monte Carlo calculations (Martin and Otto, 1982; Schiller and

Ranft, 1983; Carson and Kenway, 1986; Baillie, 1987; Azcoiti et al., 1994), and various light-front approaches including those using an infinite momentum frame

(Bergknoff, 1977), discrete light-cone quantisation (DLCQ) (Eller et al., 1987), and a Tamm-Dancoff approach (Mo and Perry, 1993). More recent studies have used techniques such as "contractor renormalization group" methods (Melnikov and We instein, 2000), coupled-cluster expansions (Fang et al., 2001), and the "fast-moving frame" technique (Kroger and Scheu, 1998).

Most of these studies have been focussed on the case with 0 = 0, i.e. zero background field. It is known however, that the physics of the model is distinctly different for the special case of vacuum angle 0 = 1r ( Coleman, 1976). At this point a second order phase transition is known to occur in the dimensionless variable m/ g.

Above the critical point m/g > (m/g)c, a spontaneous symmetry breaking phase is entered, while below there is expected a unique vacuum. This is in contrast with

0 =/:- 1r where there is no phase transition for any m/g. A particularly interesting phenomenon of "half-asymptotic" particles also occurs at 0 = 1r. Normally in the model, excitations corresponding to qij_ mesons are confined such that singly charged states cannot appear in the spectrum. However at 0 = 1r it is possibly to deconfine these particles, such that singly charged states are possible. In this regime however, the deconfined q and if. must always maintain the same ordering. Due to this special property, these particles are described as being "half-asymptotic". This special type of excitation only occurs above the critical point m/g > (m/ g )c- We will discuss many of these features in more detail in Chapter 2.

There have been very few studies of the Schwinger model at 0 = 1r. The first in-depth treatment was carried out by Hamer et al. (1982), who used finite-lattice INTRODUCTION 8

Figure 1.1: A conducting ring with a magnetic flux . The persistent current I may flow paramagnetically (as shown), or diamagnetically.

techniques to calculate the ground state energy, string tension as a function of 0 and m/g. They located the phase transition to be (m/g)c = 0.325(20) and correlation index v = 0.9(1). Schiller and Ranft (Schiller and Ranft, 1983) used Monte Carlo techniques to locate the critical point at (m/g)c = 0.31(1). In this thesis we apply the DMRG technique to the massive Schwinger model, and in particular concentrate our attention to the less studied 0 = 1r case. Our aims will be to study in some detail the model in the region of the critical point, and obtain results for the critical point, critical indices, and the mass gap spectrum. In particular we will attempt to locate the spectrum of the "half-asymptotic" particle, which has not been directly performed to date, as far as we are aware. We also apply our algorithm to the 0 = 0 case, and obtain some improved estimates for the "vector" and "scalar" states. The primary aim of this will be to demonstrate the power of the DMRG algorithm compared to other related methods, such as exact diagonalisation. We also perform some analytic calculations around the non relativistic (large m/ g) limit, and compare this to our numerical data. We apply DMRG to a second model in this thesis, namely the Heisenberg chain with a single weak link. Although this particular model has been studied by many authors before, we examine this model in a new context, relevant to the phenomena of "persistent currents". Persistent currents are electrical currents that flow in rings of conducting material under the influence of an Ahanorov-Bohm flux INTRODUCTION 9 penetrating the ring (see Figure 1.1). As the name suggests, the currents flow as long as the magnetic field is present through the ring without decay, i.e. they are truly "persistent". This is a purely quantum mechanical effect, relying on the coherence of the electron wavefunction throughout the sample. It is related to the

Ahanorov-Bohm effect itself, as one finds that varying the magnetic flux produces oscillations in the persistent current, with period of a flux quantum 0 = he/ e. The theoretical existence of such persistent currents was known for many years, but it was only more recently realised that these currents could be experimentally realised (see Imry 1997 and references therein). Consider the schematic picture of some experimentally produced ring in Figure 1.2. In a real ring, for an elec tron to travel once around its perimeter, it must scatter many times. One would expect that such multiple scattering events would destroy any coherence of the electron wavefunction, and therefore destroy any experimentally measurable per sistent current. Furthermore, our diagram is already an idealisation, as in any real experiment it is extremely difficult to prepare any sample with absolutely no defects (e.g. impurities, or in the shape of the ring itself), which will also add to the loss of coherence. It was therefore widely expected that any experimental measurement of persistent currents was impossible. This conclusion is now known to be incorrect, as it was realised that there is a distinction between elastic and inelastic scattering in terms of phase coherence (Imry, 1986). Elastic scattering results from a static potential, and thus well-defined phases can be defined for the scattering event. In inelastic scattering any phase information is lost, which may arise typically from scattering with phonon excitations, or inelastic interactions with impurities in the sample. Therefore if one can experimentally produce small enough rings of circumference L, provided that the mean free path l(T) (where T is the temperature) between inelastic collisions is less than the sample size, there INTRODUCTION 10

Figure 1.2: An electron must scatter many times before going around a ring once.

is a possibility of observing such currents. If we denote a the microscopic length scale (e.g. the lattice spacing), then the regime where such currents are observable is in the range a « L '.S l(T), which has the name "mesoscopic". Many of the early theoretical advances in the study of persistent currents arose in connection to superconducting phenomena (Byers and Yang, 1961; Schick,

1967; Gunther and Imry, 1969; Kulik, 1970; Bloch, 1970). Byers and Yang (1961) made the important connection between a magnetic flux through a ring and twisted boundary conditions, while attempting to understand the Meissner effect. Kulik

(1970) was one of the first to derive an expression for the persistent current in a sim ple system, while investigating flux quantisation in normal metal rings. Many years later, Biittiker published a series of papers (Biittiker et al., 1983; Biittiker et al.,

1984; Landauer and Biittiker, 1985; Biittiker, 1985) investigating the effects of dis order in free (i.e. non-interacting) electron rings. An important conclusion was that the spectrum in the presence of a magnetic flux should look similar to that of Bloch electron bands in the presence of disorder ( this will be discussed more in

Chapter 8). Cheung et al. (1988) also examined free electron rings in the presence of disorder and temperature.

Around this time (rv 1985), through technological advances it became possible INTRODUCTION 11 to experimentally produce rings small enough such that many of the theoretical predictions could be for the first time compared to experiment. The first exper imental measurement was performed by Levy et al. (1990), who fabricated 107 copper rings, approximately 0.5µm in diameter. Shortly after, Chandrasekhar et al. (1991) produced a single gold ring of somewhat larger size, and performed similar measurements, while a third group produced single GaAs rings in the bal listic regime. All three groups were able to give a positive result for the existence of persistent currents. However, quantitative agreement between the theory and experiment was lacking. The first two groups gave much larger magnitudes of per sistent currents than theoretical models, while the first group measured periodic oscillations of the current with magnetic flux with period <1!0 /2, in discrepancy with theories which predict period if!0 oscillations. Many explanations for these discrep ancies were proposed, which will not be discussed here, see for example Imry (1997) for a discussion on this subject.

To account for the larger magnitude of the current, and to provide a more real istic model of electrons in the rings, many investigations were prompted involving interacting electron systems (e.g. the Wigner crystal, the Luttinger liquid, the

Hubbard model and the Heisenberg model - see Zvyagin and Krive (1995) for a review). The Heisenberg model, is not directly a model involving electron trans port, but can be transformed via a Jordan-Wigner transformation into a model of spinless fermions, with short-range interactions. It is this model that will be investigated in this thesis, with the additional presence of a weak link - a simple example of a single defect in the ring. The Hamiltonian is therefore in standard notation

(1.3) INTRODUCTION 12 where J' / J lies in the range [O, l]. We again use DMRG to obtain our results.

Instead of calculating the persistent current directly, we calculate a related quantity, the spin stiffness Ps, which will be non-zero in the presence of a persistent current.

This will be discussed in more detail in Chapter 8.

This thesis is organised as follows. We first start in Chapter 2 with an overview of the physics of the massless and massive Schwinger models, with particular em phasis on the background field 0 = 1r case. We explain here the origin of the phase transition, and the existence of the half-asymptotic particles introduced previously.

As the bulk of our results are obtained using the lattice formulation of the model, we devote Chapter 3 to this subject, including the lattice strong coupling limit, which is useful for some intuitive understanding of the states calculated. In order to extract our results from the raw lattice data, we rely on finite size scaling the ory, which is reviewed in Chapter 4. Our primary numerical technique of DMRG is reviewed in Chapter 5. Finally the results of our calculations for the Schwinger model are displayed in Chapter 6, which exclusively shows results for the 0 = 1r case, which is our main interest. Chapter 7 shows results for the Schwinger model at 0 = 0, which includes both DMRG results and a non-relativistic expansion for the "vector" and "scalar" states. We then turn to the second part of this thesis, examining the persistent current effects in the weak link Heisenberg model. We give brief review of the model in Chapter 8, and present our DMRG results in

Chapter 9. We summarise our findings in Chapter 10.

Throughout this thesis we use units such that n = c 1, unless otherwise specified. Chapter 2

The Massless and Massive

Schwinger Model

In this chapter we give an overview of the known aspects of the Schwinger model.

After some preliminaries in Section 2.1, we illustrate some of the main features of the model using a weak coupling approximation in Section 2.2. In Sections 2.3 and

2.4 we review the massless and massive models briefly. The extra parameter 0 is examined in Section 2.5, which will be shown to be equivalent to a background electric field. The main emphasis of this chapter will be placed on Section 2.5, as our primary results are concerned with the case with background field 0 = 71'. For a more thorough review of the physics of the massless and massive Schwinger models, see Abdalla, Abdalla and Rothe (1991).

13 THE MASSLESS AND MASSIVE SCHWINGER MODEL 14

2 .1 Preliminaries

2.1.1 Definitions

The massive Schwinger model is nothing more than quantum electrodynamics in ( 1 + 1) D. The Lagrangian is

(2.1) where (2.2)

Here the Lorentz indices run from µ, v = 0 to 1. Gamma matrices are defined in the same way as in (3 + l)D:

(2.3) note that these are 2 x 2 matrices, as there is no spin in one spatial dimension. The equations of motion may be written using Euler-Lagrange equations,

0 (2.4)

(2.5) where f = ifryv'l/J. The fields are quantised by imposing anti-commutation relations on the Fermi fields

{ 'l/J!(x, t), 'l/Jf3(Y, t)} 8(x - y)6af3 (2.6)

{ 'l/Ja(x, t), 'l/J{3(Y, t)} { 'l/J!(x, t), 'l/J1(Y, t)} = 0, (2.7) THE MASSLESS AND MASSIVE SCHWINGER MODEL 15 and commutation relations on the gauge fields

[E(x, t), A1(y, t)] = -io(x - y), (2.8) where the electric field is defined

E = pOI = -FIO. (2.9)

2.1.2 Electrodynamics in (l+l)D

Equation (2.5) may be expanded to give Maxwell's equations,

8E ox gp (2.10) 8E at gJ. (2.11)

The first of these is equivalent to Gauss' Law. There are only two equations of motion, in contrast to four in (3+1)D, as magnetic fields are not possible in one spatial dimension. This is due to the lack of transverse directions available in

(l+l)D. Photons also do not exist in (l+l)D for the same reason. This may also be seen by noting that Maxwell's equations (2.10) and (2.11) do not yield wave solutions with no sources p = 0, j = 0.

Integrating (2.10) between two arbitrary points x1 and x2 we have

(2.12)

The Qenc is the total charge enclosed in the region of integration. The electric field around an isolated charge g is therefore g /2 to the left of the charge, and g /2 to its right (see Figure 2.la). Note that the electric field does not diminish with the THE MASSLESS AND MASSIVE SCHWINGER MODEL 16

a) b)

• • • E=g/2 0 E=g/2 E=O 0 E=g 0 E=O

Figure 2.1: a) The electric field around an isolated charge g b) The electric field around a pair of charges ±g.

distance from the charge, and persists to infinity. The electrostatic energy per unit length of such a configuration is

1 u = -E2 2 = g2/8 . (2.13)

Therefore compared to the ground state where E = 0 everywhere, the excitation energy of such a singly charged state is infinite.

If we now consider two oppositely charged particles, as shown in Figure 2.lb, an electric field is present only between the two charges, hence the excitation energy is finite. We can immediately conclude that physical states must be uncharged states. Separating the charges a distance x requires an energy

(2.14) which is a linear confining potential. This same confining potential will be seen in the following section where we examine the Schwinger model in a weak coupling approximation. THE MASSLESS AND MASSIVE SCHWINGER MODEL 17

2.2 Weak Coupling Approximation

We saw in the previous section that charges are confined in classical electro dynamics. Let us now rederive this relationship, using the Schwinger model La grangian (2.1). In the limit g = 0, the Fermi fields in the Lagrangian (2.1) decouple from the gauge fields, hence the theory reduces to a theory of free particles. We now expand around this point, in the limit m » g, following the work of Coleman (1976). This will be presented in some detail, as it forms the necessary background for our non-relativistic expansion in Section 7.1.

The first step is to write the Hamiltonian of the theory. To do this we must impose a gauge, the convenient one for our current purposes is

(2.15) which is the two dimensional version of the Coulomb gauge. In this gauge we have

2A - · _ 0 /,ta/, 81 o - -gJo - -g'I-' '!-', (2.16) which follows from (2.5). Integrating (2.16) once we get

(2.17) where F is a constant of integration, which may be interpreted as a constant background electric field. We shall see later that this is in fact a very significant parameter. The potential is written

Ao= -g812jo - Fx - G, (2.18) THE MASSLESS AND MASSIVE SCHWINGER MODEL 18 where G is another constant of integration.

The Hamiltonian density is defined as

H(x) = 1r(x)~(x) - £('1/J,8µ'1/J) (2.19) where the canonically conjugate field is

( ) - f),C - ·.1,t 7r X = -. - 1,'f'' (2.20) 8'lj) and the dot on the 'ljJ denotes the time derivative. This gives

(2.21)

We now make a restriction on the charge density such that

Jj 0 (x)dx = J'lj)t'lj)dx = 0, (2.22) according to the considerations of Section 2.1.2, where it was seen that physical states only correspond to those with charge zero.

The Hamiltonian is obtained as

(2.23) where the third and fourth terms in (2.21) combine under an integration by parts.

We may substitute (2.17) into the second integral above, to obtain an expression purely in terms of matter fields. We therefore see that the electric field E is not an independent degree of freedom, but is determined purely by the fermion THE MASSLESS AND MASSIVE SCHWINGER MODEL 19 configuration. We then have

9 H = Jdxi/J(-i,181 + m)'lj;- : Jdxdyjo(x)jo(Y)lx-yl - F9 Jdxxj 0 (x), (2.24) where every appearance of 812 was handled by substituting the Coulomb's Green's function, defined as the solution of 8fG(x, x') = o(x - x'), and integrating. In one dimension this is 1 G(x, x') = 2ix - x'I- (2.25)

We may now use this Hamiltonian to expand around the point 9 = 0. Our calculations are restricted to the two-particle subspace, and we only perform first order calculations. The second and third terms in (2.24) are treated as perturbative terms, and give rise to the diagrams shown in Figure 2.2. The working for these calc~lations is fairly tedious, and is relegated to Appendix A.

Evaluating the matrix elements from each diagram, and recasting them back into Hamiltonian form yields the effective Hamiltonian

(2.26) where

HR (p2 m2)1/2 0 2 + (2.27) HR -g2[ lxl+-lxl-+-lxl-p p m ml (2.28) 1 4 Ep EP EP Ep 92 1 HR2 --- (2.29) 7r Ep 92 m m HR (2.30) 3 E2o(x) E2 4 p p HR 4 -9Fx, (2.31) THE MASSLESS AND MASSIVE SCHWINGER MODEL 20

(a) (b)

(c) (d) > ~,-< (e) (f)

•

Figure 2.2: Six lowest order diagrams contributing to the weak coupling expansion Hamilto nian. (e) and (f) refer to interactions with the background field.

and Ep = (p2 + m2)112. Hfj is simply the free energy of two particles. We now wish to do a semiclassical analysis on this Hamiltonian. To do this, we throw away all terms in lowest order in fi. As we have set fi = 1, some dimensional analysis is needed, specifically if we note that

(2.32) then we find to zeroth order in fi we have (Coleman, 1976)

2 2F HR ~ 2(p2 + m2)1/2 + L(lxl - -x) + O(fi). (2.33) 2 g

We can interpret the above Hamiltonian in a very simple way. The first term is of course the total energy of two free particles. We saw in equation (2.14) that the THE MASSLESS AND MASSIVE SCHWINGER MODEL 21 energy due to the separation of two charges was V(x) = g2 lxl/2, precisely giving the second term. In the presence of a constant background electric field, we would expect that an energy Fg would be gained by moving in the electric field. This is exactly the third term. We therefore see that in this weak coupling approximation the dominant effects are the same as the classical effects discussed in Section 2.1.2.

A second, more important result is that the background field F is a physically significant parameter. This will be of primary importance in Section 2.5.

2.3 The Massless Schwinger Model

We now review some of the significant results of the massless version of the model. As discussed in the introduction, the massless Schwinger model can be exactly solved (Schwinger, 1963). To do this, Schwinger realised that the current must be defined according to the limit of a point-separated current

(2.34)

where E is some spacelike vector. This definition is necessary to ensure gauge invariance, and the current operator is non-singular. A crucial part of the derivation is evaluating the correct form of the chiral current, which in (l+l)D can be written

(2.35) where the Levi-Civita symbol is

(2.36) THE MASSLESS AND MASSIVE SCHWINGER MODEL 22

It turns out (Brown, 1963), from the definition of the current (2.34) the divergence of the chiral current is

g ·/J, ·I -o aµ,]5 = aOJ + aI] -E(x, t). (2.37) 7r

We see that the chiral current is not conserved, despite the chiral symmetry of the Lagrangian (2.1). This model therefore contains a chiral anomaly. Combining

(2.37) and the equation for current conservation

·µ, 0 1 aµ,J = aoJ · + aI] · = o ' (2.38) and Maxwell's equations (2.5) we obtain

(-a 2 +~)jµ,=0. (2.39)

We see that the current jµ, satisfies a Klein-Gordon equation. The spectrum of the theory is therefore populated by non-interacting bosons of mass g / Y1f· One can interpret this as the gauge field acquiring a mass through a dynamical Higgs mechanism. This was the original motivation of introducing the Schwinger model, which was to show the possibility of a non-zero mass gauge field.

Schwinger also found that the vacuum of the Schwinger model has the property of complete charge shielding (Schwinger, 1963). Inserting an arbitrary external charge Q induces an equal and opposite charge -Q, cancelling the original charge.

We may understand this in analogy with a classical dielectric medium. In a normal dielectric medium, an applied electric field induces a polarisation field in the oppo site direction, thereby weakening the field. In the vacuum state of the Schwinger model, a similar effect occurs, except that here the polarisation field is strong THE MASSLESS AND MASSIVE SCHWINGER MODEL 23 n ~ 00000000~

Figure 2.3: Complete vacuum polarisation of the Schwinger model. An electric field E imposed by plates at either end of the universe is exactly cancelled by a polarisation field Epo1·

enough to completely cancel the original field (see Figure 2.3). The net field is therefore zero at all points, and we have a complete polarisation of the vacuum.

This is another example of the confinement in the model, as bare charges are sup pressed by the action of the vacuum (Casher et al., 1973; Casher et al., 1974).

It was first realised by Lowenstein and Swieca that the vacuum state of the

Schwinger model was infinitely degenerate, and could be described by a 0-vacuum

(Lowenstein and Swieca, 1971). This followed from the discovery that Schwinger's original solution was only valid in a subspace of the full Fock space of the theory.

Specifically, if we define distinct chiral charge sectors as

(2.40)

where Q5 = J 'I/Jt'Y5'!/Jdx, then Schwinger's solution corresponds to restricting the Fock space to n = 0 only (Kogut and Susskind, 1975 b). It turns out (Lowenstein and Swieca, 1971) that this violates the cluster decomposition property for Wight man functions (Roman, 1969), and the space of states should be more correctly chosen as

(2.41) n THE MASSLESS AND MASSIVE SCHWINGER MODEL 24

According to this definition, chiral rotations of angle 00 transform the above from

I0) - I0 + 00). Since the Lagrangian contains no terms that act on the chiral space of states, the choice of 0 is arbitrary and is therefore a free parameter of the theory.

The physical states of the theory then lie in a tensor product of the states above, with the Fock space of states generated by the Lagrangian in a single chiral sector.

We will see later a more physical interpretation of this parameter in Section 2.5.1.

2.4 The Massive Schwinger Model

2.4.1 Bosonization

We now turn to studying the properties of the Schwinger model with a non-zero fermion mass. The first to do such a study were Coleman, Jackiw and Susskind

(1975), using bosonization methods (Coleman, 1975). A summary of correspon dences is (Mandelstam, 1975)

:'l/;'1/;: - -cµNµ cos(2./i) (2.42) : i'l/;'"'(s'I/; : - -cµNµ sin(2./i) (2.43) : i/;1µ'1/; : - ~Eµv8µq> (2.44) : ii/;8µ'"'(µ'1/; : - 18µ8µ (2.45) where µ = g / vii and c = e'Y /21r, and 'Y = 0.577 4 ... is Euler's constant. Colons denote normal ordering with respect to Fermi fields, and Nµ denotes normal order ing with respect toµ. Using this the Hamiltonian density may be written in Bose form (Coleman, 1976),

(2.46) THE MASSLESS AND MASSIVE SCHWINGER MODEL 25 where 0 is the same parameter introduced in Section 2.3. Let us look at this Hamil tonian in various limits. With zero fermion mass m = 0, we explicitly have a theory of free bosons of mass µ = g / .Jii, in agreement with (2.39). In particular note that the dependence in 0 completely drops out in this limit, making the Hamiltonian invariant for different 0. The ground state is therefore also independent of 0, and is infinitely degenerate. For a non-zero fermion mass m, an extra interaction term is included in the Hamiltonian, which introduces interactions between the bosons. In contrast to the massless case, here 0 does become a physically relevant parameter, i.e. the spectrum of states will depend explicitly on 0. Let us now write the ground state energy dependence on 0 ( Coleman et al.,

1975), using semi-classical methods. For a small fermion mass m, the minimum of the Hamiltonian (2.46) lies at cp = 0. The ground state energy density is therefore

E0 (0) ~ -cmµ cos 0. (2.47)

The "string tension", or the energy shift per unit length relative to 0 = 0 is therefore

T(0) = cmµ(l - cos 0). (2.48)

Thus the degeneracy in 0 of the vacuum which we had for the massless case is broken by the fermion mass.

We may also derive an explicit value for the chiral symmetry breaking through the Feynman-Hellman theorem for zero fermion mass, m = 0 (Hamer et al., 1982)

(2.49) (7/J'I/J )o 8~ Eo(m, 0) lm=O -cµcos0. (2.50) THE MASSLESS AND MASSIVE SCHWINGER MODEL 26

Corrections of order"' (m/g) result for a finite fermion mass (Adam, 1995).

Mass perturbation theory may be used to obtain corrections to the Schwinger boson mass for small fermion mass. A second order calculation for the first excited state at 0 = 0 (the "vector" mass gap) has been calculated and yields (Vary et al., 1996; Adam, 1996)

2 gM 1 = y'1r1 + 1.781 (m)g + 0.1907 (m)g + .... (2.51)

The second excited state, corresponding to a bound state of two Schwinger bosons

(the "scalar" gap) was calculated by Adam (1996) to be

2 M2 2 m m g = y'1r + 3.562 ( g ) - 13.512 ( g ) + .... (2.52)

In the opposite limit of large mass m/g ---+ oo, Hamer (1977) using a non relativistic expansion has calculated the masses of the same two states. For the

"vector" state he obtains

E1 ( g ) 1/3 g = 0.642 m + ... , (2.53) while for the "scalar" state

E2 ( g) 1/3 g = l.473 m + ... , (2.54) where the above results are for the "binding energies" of the vector and scalar state, and do not include the 2m of energy required to excite the qij_ pair. The mass gaps are therefore given by M E 2m -=-+-. (2.55) g g g THE MASSLESS AND MASSIVE SCHWINGER MODEL 27

Sriganesh et al. (2000) has extended this to third order, who found

13 13 0.642 (!)1 - ¼(!) - 0.252 (!)5 (2.56)

113 513 1.473 (!) - ¼(!) + 0.108 (!) . (2.57)

However, on a re-examination of their results, we found an error in their calculations and hence a recalculation will be presented in Section 7 .1 displaying the correct result.

2.4.2 Charge shielding with mass

To study the effect of charge shielding for the massive case, Coleman studied the effect of two widely separated external charges ±Q, using mass perturbation theory (Coleman et al., 1975). Using the bosonic form of the theory he obtained the Hamiltonian for the region between the charges

(2.58)

This is identical to the Hamiltonian (2.46), up to a redefinition of the constant term

0. We may thus immediately write down the ground state energy using (2.47). We obtain 21rQ E 0 (0) ~ -cmµcos(0 - -). (2.59) g

We see that for arbitrary Q there will be some shift in the ground state energy, as one would expect. This indicates the presence of a long-range force between the separated charges. However, if Q is an integer multiple of g, we see that there is no shift in the ground state energy, and hence zero force between the charges. This indicates that there is complete charge shielding in this case. This is distinct from THE MASSLESS AND MASSIVE SCHWINGER MODEL 28 the behaviour of the massless case, where complete charge shielding took place for all charges Q. We will see a more physical reason for these two regimes of behaviour in the next section.

2.5 The Background Field

2.5.1 The Origin of 0

We saw that in (2.33) that the constant electric background field F was a physically significant parameter. Let us take a moment to discuss the dynamics of such a field, using simple classical considerations. Consider some source of the background electric field, placed at the ends of the one-dimensional universe, which we denote schematically by condenser plates in Figure 2.4. Due to the infinite electrostatic energy (2.60) of the electric field, it will be energetically favourable for the vacuum to produce a quark-antiquark pair, which will lower the field between them to F - g, according to Gauss' law (2.12). The charges then travel to the condenser plates, lowering the energy by

!:lU 211 dx(F-g) 2 - 211 dxF 2 (2.61) 1 2L(-2gF + g2), (2.62) where L is the distance between the condenser plates, assumed to be large. For this process to be energetically favourable we require that !:lU < 0, which in turn means IFI > 0.5g. THE MASSLESS AND MASSIVE SCHWINGER MODEL 29

-L- ~ ~·· -01-----101----- ······ F F-g F

Figure 2.4: A quark-antiquark pair in the presence of a background field F.

F=2.2g --of---0-,___- F=2.2g F=l.2g F=2.2g -----{o\--___,0,___- F=l.2g F=0.2g F=l.2g

F=0.2g

Figure 2.5: The cascade of pair production in reducing the electric field.

To understand the significance of the point JFI = 0.5g, let us consider a simple example, shown in Figure 2.5. We start with an electric field of F = 2.2g between the two condenser plates. The vacuum will produce one pair, and then another, to reduce to field in the sequence F = 2.2g -+ 1.2g -+ 0.2g. However, producing another pair does not help to reduce the energy, as Gauss' law can only reduce the field in units of g. Therefore pairs will be produced until the background field lies in the range -0.5g :S F :S 0.5g. We can now see what is special about the point JFI = 0.5g. Any field with IFI > 0.5g can reduce its energy by producing pairs, but a state with IFI < 0.5g cannot, due to the nature of Gauss' law. We also conclude that the physics of the model must be periodic in units of g, as any state with IFI > 0.5g can always reduce its field by producing a number of pairs. Coleman (1976) has argued that this background field is in fact identical to the THE MASSLESS AND MASSIVE SCHWINGER MODEL 30 parameter 0, introduced in Section 2.3, if one makes the identification

0 = 21rF_ (2.63) g

Background fields in the range IFI :S 0.5g are mapped to the interval 0 = [-1r, 1r]. Another way of seeing this equivalence is to note that the Bosonic field is pro portional to the electric field g E = --vn (2.64) which may be seen by combining (2.17) with (2.44). Using the result that chiral transformations transform the bosonic field by a constant (Kogut and Susskind,

1975b)

(2.65) we can conclude that a chiral transformation (2.64) simply shifts the background electric field by a constant. As chiral rotations transport us around the 0-vacuum, any vacuum angle 0 is then equivalent to a constant background electric field.

We may now ask why such a background field does not occur in our three dimensional world. For concreteness, let us revisit the case of two condenser plates placed at either end of the universe, but this time in three dimensions. Again, the vacuum would respond by creating pairs, which will travel out to the condenser plates to neutralise the charges. Unlike the one-dimensional case however, Gauss' law does not restrict the electric field to change in steps of g. In three dimensions it is possible to have some configuration of charge to neutralise any arbitrary charge on the plates. Therefore even if such condenser plates did exist, we would never be aware of them. We note that even in the one-dimensional case, a fractional charge is necessary on the plates to produce a field F < g /2. We do not give an explanation of where such charges could come from, hence one may argue the whole THE MASSLESS AND MASSIVE SCHWINGER MODEL 31

0=0 0 = rr12 0=7t

V(x) V(x) V(x)

X X X

Figure 2.6: Confining potential of a quark-antiquark pair at various 0 for weak coupling.

situation is rather hypothetical. However a field of exactly F = g /2 is possible by placing a single charge on one of the plates. This in fact corresponds to 0 = 1r, which shall be examined in the next section.

2.5.2 Why 0 = 1r is special

Let us again return to the weak-coupling Hamiltonian (2.33) discussed previ ously. The potential term is

g2 0 V(x, 0) = 2 (1xl - :;;:x). (2.66) where we have rewritten the background field F using (2.63). We see that for

0 = 1r, the potential "tips" to its side, so that one side is flat along the axis (see Figure 2.6). This implies that if the quark is to the left of the antiquark, there is a confining potential, but on the other side, there is no force. Coleman called such particles "half-asymptotic" particles, as they are deconfined particles as long as they maintain the correct ordering, but become confined when they cross over.

For 0-=/=- 1r, there is an (asymmetric) long-range confining potential, hence no half asymptotic particles exist.

We must not forget that (2.66) was derived for weak coupling only, and fur- THE MASSLESS AND MASSIVE SCHWINGER MODEL 32

0=7t 0=0

g/2 0 Ground

I-particle

Figure 2. 7: Electric field configurations for ground and single quark states with 0 = 1r and 0 = 0.

thermore is only the leading term in h. The long-range forces however, can only originate from the electrostatic potential, which is included. Hence in the weak cou pling regime we expect true deconfinement. We shall see that for strong coupling these results do not hold.

It is also easy to see that unlike the case for 0 =J. 1T', one-particle deconfined states exist for 0 = 1!'. We may consider this from a classical viewpoint, shown in

Figure 2.7. We have already seen from the discussion in Section 2.1.2 that charged states are not possible for 0 = 0. The situation here is different, as shown in the figure, as the electric field is E = g/2 on the left of the charge and E = -g/2 on the right. The electrostatic energy is therefore no higher than that of the ground state, and hence charged states are possible.

All our arguments so far were in the weak-coupling limit. Let us now turn to the strong-coupling limit, where the suitable form of the theory is written using the bosonized Hamiltonian (2.46). For 0 = 1T' we have

(2.67) where

(2.68) THE MASSLESS AND MASSIVE SCHWINGER MODEL 33

U(ct,)4

-1 ~--_.______...__ __ _.______._ ___...... __ ___,

-3 -2 -1 0 3

Figure 2.8: The bosonic potential U() for 0 = 1r and m/g > (m/g)c,

In the limit m = 0, there is only a single ground state, which in the semiclassical approximation is

suffers spontaneous breakdown. The expectation value of () will have a non-zero value in this regime.

Let us now attempt to interpret the nature of this doubly degenerate state. It is easiest to do this in the non-relativistic limit, m/ g - oo. In this limit exciting pairs out of the vacuum costs a large amount of energy, hence the ground state has no quantum fluctuations. The energy of the ground state then comes only from the electrostatic energy term u = ½E 2 . In the presence of an applied electric field

F, the ground state has an energy density

g2 2 E0 (0) = - 2 0, m/g - oo (2.69) 81r THE MASSLESS AND MASSIVE SCHWINGER MODEL 34

0.14 rr------r---..,.,..----....-----,.,.,....------r-----n m/g= co - m/g = 0.1 0.12 m/g = 0 ------

0.1

0.08 Eo(e )/92

0.06

0.04

0.02 o======~ -3 1t -2 1t -Tt 0 1t 2Tt 3Tt e

Figure 2.9: Ground state energy versus 0 for three values of m/g.

which holds for 0 = [-1r, 1r]. We have already seen from considerations of Section 2.5.1 that this must be periodic in 0 with period 21r. A plot of this is shown in

Figure 2.9. We see that the curves for the non-relativistic case m/g = oo intersect at 0 = ±1r, ±21r, ±31r . . . . At each of these points there is a doubly degenerate ground state, which arises because there are two values of the background electric field with the same energy, namely F = ±0.5g. We can now see what the double well of Figure 2.8 represents. Each well corresponds to one of the vacuum states with F = ±0.5g. Indeed we may have deduced the same conclusion from relation (2.64), from which we see that a non-zero expectation value of (cp) gives a non-zero average (E), which corresponds to a background field.

Armed with this interpretation, we may now interpret the "half-asymptotic" particles in terms of the scalar field cp. If we look at the electric field configuration around a 1-particle state at 0 = 1T in Figure 2.7, we see that the electric field across the particle swaps between the two electric field values F = 1/2 --+ F = -1/2. THE MASSLESS AND MASSIVE SCHWINGER MODEL 35

Figure 2.10: A half asymptotic 1-particle state is equivalent to a kink state in bosonic variables

In terms of the scalar field , using the correspondence (2.64), this is equivalent to a solution which passes from 1> = -1>0 -----+ o (see Figure 2.10), at the location of the particle. These are soliton solutions, as they are time-independent classical solutions of the field equations. An increasing solution may be called a "kink" state, and a decreasing solution may be called an "antikink" state. Note that for multi-kink states, the kinks and antikinks must alternate, in the same way that half-asymptotic particles do.

Let us summarise what we have found so far. For weak coupling (m / g large):

• Half-asymptotic particles should exist.

• There are two vacua.

• The symmetry 1> -----+ -1> suffers spontaneous breakdown.

Only the other hand for strong coupling (m/g small):

• Half-asymptotic particles do not exist. THE MASSLESS AND MASSIVE SCHWINGER MODEL 36

• There is only one vacuum.

• The symmetry cp--+ -cp is unbroken.

Coleman (1976) predicted that this change of behaviour points to the existence of a phase transition somewhere between strong and weak coupling at (m / g )c- The critical point is signaled by the breaking of the degeneracy of the ground states corresponding to F = ±0.5g. Another way to see the transition is to measure the expectation value of (cp), which should be zero for strong coupling and non-zero for weak coupling. Alternatively, using the relation (2.64) we can measure

(2.70) which should also have the same behaviour. An expression for another order pa rameter was suggested by Creutz (1995a). According to bosonization rule (2.43), a non-zero expectation value of (cl>) will in turn give a non-zero value of

(2.71)

In our calculations we will calculate both order parameters to signal the location of the phase transition.

Another effect of the phase transition may also be seen by examining the de pendence of the ground state energy on 0. In Figure 2.9 we plot the ground state energy for a small mass (m/ g = 0.1), using equation (2.48), and also the case m/ g = 0 where there is no energy shift due to 0. We see that somewhere between strong and weak coupling the point at 0 = 1r undergoes a transition from a zero gradient to a cusp. The point at which this occurs signals the presence of a phase transition (Hamer et al., 1982). THE MASSLESS AND MASSIVE SCHWINGER MODEL 37

2n.-----.------~

0 n

< > =- 0

o~----~------' (mfg) 00 0 C mfg

Figure 2.11: Phase diagram for the massive Schwinger model. 0 lies in range [O, 21r] for con venience. The jagged line shows location of the first order phase transition. The circle denotes a second order phase transition.

2.5.3 Phase Diagram

Let us now construct the phase diagram for the massive Schwinger model. We have already discussed that a phase transition should exist at some (m/g)c with

0 = 1r. It is also easy to see that there will be a first order phase transition line for m/g > (m/g)c- For 0 = 1r - E, where Eis a small parameter, the two wells are no longer degenerate, hence there is no longer any spontaneous symmetry breaking.

Here the left well of Figure 2.8 becomes the ground state, and we have (q;) = -q;0 .

For 0 = 1r + E, a similar argument results in (

Figure 2.11. The most accurate value for the critical point previous to this study was obtained by Hamer et al. (1982), who obtained a value of (m/ g)c = 0.325(20), and Schiller and Ranft (1983) who obtained (m/g)c = 0.31(1). Our aim in this thesis will be to investigate this critical region in more detail. Chapter 3

Lattice Gauge Theory

To study the Schwinger model using DMRG, we must construct the equivalent theory on a lattice, which will be the subject of this chapter. As pure gauge electrodynamics is equivalent to a continuous Abelian lattice gauge theory, we will consider this first in Section 3.1. Section 3.2 discusses the Hamiltonian limit of the Abelian theory, and Section 3.3 introduces fermions into the lattice theory.

Finally Section 3.4 discusses the lattice strong coupling limit of the Schwinger model

Hamiltonian, from which we will see some of the basic features of the spectrum for

0 = 0 and 0 = 1r. Section 3.5 examines the lattice formulation of Gauss' law and the order parameters that we will calculate.

3.1 Abelian Lattice Gauge Theory

An Abelian lattice gauge theory (Wilson, 1975) is defined is such a way that the field variables on a lattice obey a local gauge symmetry. The discussion here follows that presented by Kogut (1979). One can formulate such a theory by defining an angular variable on each link of the lattice 0µ(n) (see Figure 3.1). The labels µ

38 LATTICE GAUGE THEORY 39

n + v 0 -µ( n + µ + V ) n + µ + v

e_Jn +v) 0v( n+µ)

n n+µ

Figure 3.1: Angular field variables defined on the links of a lattice.

denotes the direction of the link, while n labels the site1. We may imagine that the angle 0 refers to the direction of a planar spin located on the link. It should be clear that the link 0µ(n) can be also labelled as 0_µ(n + µ). These are related by

0_µ(n + µ) = -0µ(n), (3.1) for reasons which will be apparent later.

Our aim is to define an action S which is invariant under gauge rotations. A gauge rotation is defined as

0µ(n) -+ 0µ(n) - x(n), (3.2) where the x( n) is an arbitrary function. This transformation is applied to every link connected to the site n. According to the definition (3.1), the same gauge transformation will affect links in the reverse direction by

0µ(n - µ) -+ 0µ(n - µ) + x(n). (3.3)

1 We trust the index ( n) will differentiate 0µ, ( n) from the background field parameter 0, of Section 2.5.1 LATTICE GAUGE THEORY 40

0v(n) 0v(n) -x(n)

0µ(n - µ) 0µ(n) 0µ(n-µ)+x(n) 0µ(n)-x(n) ~

0v(n-v)+X(n)

Figure 3.2: A gauge transformation of the site n.

This is sketched in Figure 3.2.

We may start to see a glimpse of how this may be related to electrodynamics by considering the total effect of two separate transformations on sites n and n + µ. We see that the link between the sites transforms as

(3.4) where .6.µx(n) = x(n + µ) - x(n). This is analogous to the gauge transformation familiar in electrodynamics gAµ - gAµ +BµX, where g is the elementary charge. The action must be constructed out of quantities invariant under gauge trans formations. One such object is the discrete curl

.6.µ0v(n) - .6.v0µ(n) (3.5)

0µ(n) + 0v(n + µ) + 0_µ(n + µ + v) + 0_v(n + v). (3.6)

It is easily verified that under gauge transformation (3.2), 0µv(n) is invariant. The obvious analogy in electrodynamics for 0µv(n) is Fµv, which is also invariant under gauge transformations. LATTICE GAUGE THEORY 41

Now we can define a gauge invariant action

s = 1 I: [1 - cos eµ11 ( n) i . (3.7) n,µ,11

To see how this action relates to that of electrodynamics, let us expand the cosine for small 0µ 11 . To first order we obtain

(3.8)

where the summation was replaced by the integral :E --+ a-4 J d4x. If we now make the association

J = 1/2g2, 0µ(n) = agAµ(r) (3.9) we recover the conventional form of the action for electrodynamics

S = 41 / d4 xFµ 11 Fµ 11 . (3.10)

3.2 Hamiltonian Formulation

We now formulate the results of the previous section in a Hamiltonian frame work. Let us assume that we are in only (l+l)D, hence the only plaquettes that are present are time-like plaquettes. A change of notation is appropriate here, let us denote such a plaquette as D( n). The action is therefore

S = JJ L [1 - cos D (n)] . (3.11) n LATTICE GAUGE THEORY 42

We will be interested in a lattice with different time and space lattice spacings, hence we label the lattice spacings in the time direction as ar and those in the space direction as a. Let us impose the gauge condition 00 (n) = 0, such that the plaquette is

D(n) = 01(n + r) - 01(n), (3.12)

Our general procedure is to look at the action in the time-continuum limit ar -----+ 0.

Expanding the cosine in the action thus gives

1 . 1 - cos D(n) ~ 2a;0~(n), (3.13) where

01 (n) = 01 (n + T) - 01 (n) . (3.14) aT

The action is then

(3.15)

The combination ar I:n,. ensures that the time summation will be finite. By con structing a transfer matrix, we obtain an expression for the Hamiltonian (see Ap pendix B):

(3.16) where we now have operators satisfying

[L(n), 01 (n')] = -i6nn'· (3.17)

L( n) is the canonically conjugate operator to 01 ( n) and is therefore an angular momentum operator. Since 01 ( n) is periodic, it follows that the spectrum of L( n) LATTICE GAUGE THEORY 43 is discrete

Lll) = lll), l=0,±1,±2, .... (3.18)

It will be useful for later to note that the quantities e±iBi are raising and lowering operators in this space

(3.19) which directly follows from the commutation relation (3.17).

We may recall from continuum electrodynamics that the electric field E(r) and the vector potential A(r) are canonically conjugate variables

[E(x), A(x')] = i<5(x - x'). (3.20)

Using the association 01(n) = agA 1(x), <5nn' = <5(x - x')a, and (3.17), we see that

E(x) = gL(n), (3.21) hence L( n) is nothing other than the electric field on a link. An important point to note here is that due to the discrete spectrum of L( n), the electric flux on a link

E will also be discrete. This may seem like an unphysical restriction, until it is realised that exactly this discrete set of flux values are possible in the continuum formulation. So in fact it is a fortunate coincidence that the spectrum of the two match up exactly. Had this not been so, it would have been necessary to have a full range of values of 0(n), not just those lying in the range [O, 21r].

We may now write the full Hamiltonian in more familiar variables

(3.22) LATTICE GAUGE THEORY 44

Clearly this expression must reduce to ½J dxE2• For these to agree, we must have j3 = g2a. In the last chapter we showed that it is possible to incorporate a background field into the Schwinger model. If we imposed a background field of

F, we can generalise the above Hamiltonian trivially to obtain

1 H = 2a l)E(n) + F)2, (3.23) n which is our final expression for the Hamiltonian for gauge fields.

3.3 Lattice Formulation for Fermions

3.3.1 The Schwinger Model Hamiltonian

We now must write a lattice form for the fermionic degrees of freedom. The fermions obey the Dirac Hamiltonian

(3.24)

Since we are in ( 1 + 1) D, there are only two gamma matrices 'Yo, "(1 . The particular representation that will be used here is

'Yo= ( 1 0 ) (3.25) 0 -1

Each '1/J field is a two component spinor

'1/J = ( '1/Jupper ) (3.26)

'l/J1ower LATTICE GAUGE THEORY 45

Explicitly multiplying out the matrices gives

H - 0 0/,t OVJtower 0 0/,t OVJupper o/,t of, o/,t of, - -i'l-"upper ox - i'l-'lower ox + m'l-'upper'l-'upper - m'l-"lower'l-'lower· (3.27)

According to the "staggered" lattice formulation of Kogut and Susskind (1975a), we place the upper components on even lattice sites and lower components on odd sites

1Pupper +--+ cl>( n + 1) (3.28)

1P1ower +--+ c/J ( n) , (3.29)

where n is an odd integer2 . The staggered lattice formulation is particularly conve nient in one dimension as it completely eliminates the fermion doubling problem.

To see this, let us recall that in d-dimensional space-time we normally have 2d doublers (see e.g. Gupta, 1998). In d = 2 dimensions this is reduced by a factor of four, hence we have 22 / 4 = 1 fermion doublers. The staggered formulation also has the added benefit of preserving a discrete version of chiral symmetry, which has been useful in previous studies (Hamer et al., 1982).

The lattice field variables obey the usual anticommutation relations

{ cp(n), cp(n')} = 0. (3.30)

A lattice derivative is defined

81/} 1 - = - (cp(n + 1) - cp(n - 1)), (3.31) ox 2a 2The lattice fields

H= L - 2ia [(n + 2) - t(n + 1)(n) nE{odd} +t(n)(n + 1) - t(n)cp(n - 1)]

+mcpt(n + l)cp(n + 1) - mcpt(n)cp(n). (3.32)

We may remove the restriction on the summation by grouping together terms in pairs

H = L - 2ia [ qit (n)(n + 1) - qit (n + l)cp(n)] + m(-ltt (n)cp(n). (3.33) n

We must be careful that we obtain a locally gauge invariant Hamiltonian, so as not to spoil the hard work of the last two sections. A local gauge transformation is made by the replacement

1 Aµ(n) -+ Aµ(n) + -~µx(n). (3.34) ga

The last term in (3.33) is therefore gauge invariant, but the first term in the brackets is not. To make the first term gauge invariant, we recall an observation made by

Schwinger (1959) from the continuum theory. Schwinger worked with point split operators such as

(3.35) which are explicitly not gauge invariant. To fix this, Schwinger suggested that LATTICE GAUGE THEORY 47 instead one should use the operator

(3.36)

Following the same prescription, on a lattice the corresponding factors will be exp[-iagA1(n)] in the A0 = 0 gauge. Combining this with the result of the pre vious section for the kinetic energy term of the electric field (3.23), we obtain the

Schwinger model Hamiltonian

H - 2ia L [t(n)i0(n)(n + 1) - 4>t(n + l)e-i0(n)(n)] n g2a +m L(-ltt(n)(n) + 2 L(L(n) + a)2, (3.37) n n where we have dropped the 'l' on 01 and written

0(n) = -agA1(n), (3.38) and 0 F a= - = -. (3.39) 271" g

It is easily verified that the above Hamiltonian is fully gauge invariant, using the transformations (3.34).

In principle the derivation of the Schwinger model Hamiltonian is complete at this point. However, it is more convenient from a numerical point of view to use spin operators rather than the fermionic fields (n) (Banks et al., 1976). We can LATTICE GAUGE THEORY 48 transform to spin variables by using the well-known Jordan-Wigner transformation,

cp(n) (3.40) l

Under such a transformation we obtain

2 W=-2 H=W0 +xV, (3.42) ag where

n n V L [a+(n)eiB(n)a-(n + 1) + a-(n + l)e-iB(n)a+(n)], (3.43) n withµ= 2m/g2a and x = l/g2a2• We have introduced the operator W to work with a dimensionless Hamiltonian.

3.3.2 Extracting the Continuum Limit

We now discuss how one can recover the continuum results from the Hamil tonian (3.42). First note that a trivial rescaling is necessary, to return from the

Hamiltonian W, to the original Hamiltonian H. Second, as we are now on a lattice of N sites, with spacing a, we must perform a double extrapolation to the bulk limit N --+ oo and the continuum limit a--+ 0. In terms of the variables of (3.42), the continuum extrapolation is equivalent to x --+ oo. Therefore our prescription LATTICE GAUGE THEORY 49 for finding a particular gap is

.6.n En(N, x) - Eo(N, x) lim (3.44) g N-+oo,x-+oo 2ft where En(N, x) are lattice energy states at finite lattice spacing N and coupling x.

Note that the extrapolations must be carried out in the order specified, first N-+ oo, then x -+ oo. The reverse order of extrapolation gives in general nonsensical results since x -+ oo is a singular point of the lattice theory (Hamer et al., 1982).

Alternative extrapolation procedures have been advocated by Irving and Thomas

(1983), who perform the double extrapolation simultaneously. However we do not find these are necessary in our results, and will not be discussed here.

In the continuum limit x-+ oo, the hopping term V of the Hamiltonian (3.42) dominates, hence the ground state energy should reduce to that of the XY antifer romagnet Eo(N,x) 1 lim (3.45) N-+oo,x-+oo 2Nx

This serves as a check of the extrapolation procedure and should be true regardless of m/g.

3.4 The Lattice Strong Coupling Limit

Let us now examine this Hamiltonian in the lattice strong coupling limit x-+ 0.

In this limit the Hamiltonian is diagonal, therefore it should be possible to write down an exact expression for the ground state and excited states. This will help to give an intuitive picture of the states we shall calculate later. We consider two values of the background field, 0 = 0 and 0 = 1r which are of interest as discussed in Chapter 2. LATTICE GAUGE THEORY 50

3.4.1 Background Field 0 = 0

First let us write down the ground state. We can minimise the electric field in

W0 by setting this to zero everywhere, and the energy due to the spin configuration can be minimised by making sure each spin contributes -µ/2, i.e. (see also Figure

3.3a) n odd L(n) = 0, a'(n) = {: (3.46) n even or in terms of fermion operators,

0 n odd L(n) = 0, q}(n)(n) = { (3.47) 1 n even

We see that the states on odd sites are completely empty, while states on even sites are completely filled. We can interpret this completely filled set of states as the Dirac sea. Excitations on even sites correspond to holes in the Dirac sea, and have energy µ. Therefore quarks lie on the odd sites of the lattice, and antiquarks lie on the even sites of the lattice. In the spin formulation, a down spin on an odd site corresponds to a quark, and an up spin on an even site corresponds to an antiquark, therefore from any spin configuration we can deduce the corresponding quark configuration.

Now let us turn to the low-lying excitations above the ground state. A single spin flip costs energy µ, but induces an electric field everywhere to the right of the flipped site (see Figure 3.3b), which costs an infinite energy. Therefore the lowest energy excitations must be states with two flipped spins ( a quark-antiquark pair), with only one lattice separation to minimise the electric field energy (shown in Figure 3.3c). To maintain translational symmetry we must have superpositions LATTICE GAUGE THEORY 51

i=l i=2 i=3 i=4 i=5 i=6

a) 0 [ 0 ! 0 [ 0 ! 0 [ 0 ! 0

b) 0 [ 0 [ 1 [ 1 ! 1 t 1 ! 1

c) 0 I O I 1 ! 0 ! 0 [ 0 ! 0

Figure 3.3: a) Spin configuration for the ground state with background field 0 = 0. Numbers between spins denote the electric field on the links. Here we have L(n) = 0. b) Configuration with one flipped spin. c) Configuration with two adjacent flipped spins.

across the lattice, of which there are two choices: the "vector" state

Iv)= ~ L [q}(n)eie(n)(n + 1) + H.c.] IO) (3.48) n and the "scalar" state

Is)= ~ L [t(n)ei0(n)(n + 1) - H.c.] IO). (3.49) n

Each of these have an excitation energy 2µ + 1 in the strong coupling limit. Any finite x breaks the degeneracy of these states, with the "vector" state having the lower energy.

3.4.2 Background Field 0 = 1r

We have seen from the discussion of Section 2.5.2 that this is a special value of the background field, as there are two degenerate vacuum states for large m/g. In LATTICE GAUGE THEORY 52

the lattice strong coupling limit x ---+ 0 these are

n odd L(n) = ±1/2, (3.50) n even which we denote by I ± ½). These vacuum states are in fact only the true vacuum states strictly in the bulk limit N ---+ oo. On a finite lattice ( with periodic boundary conditions imposed) the ground states will be in fact

(3.51) and ID')= ~[I½) -1- ½)]. (3.52)

These states only become degenerate in the strong coupling limit x ---+ 0. For finite x and N, there is a gap between these states, which we call the "loop gap" .6.0 .

The subscript O refers to the fact that this is a 0-particle excitation.

We have also seen in Section 2.5.2 that for a = 1/2 a single particle state is possible. On a lattice such a state in the strong coupling limit has a form

N N 11) = Vi; ~ (n) IJ e-i0(j)I½), (3.53) j==n n == 1 {n odd} where we have assumed that the ground state is spontaneously broken and settles in the I½) state. The product of e-iO(j) is to account for the fact that the electric field is lowered by one unit everywhere to the right of the particle. We shall denote the gap associated by this state by .6.1. LATTICE GAUGE THEORY 53

The two-particle state is

N 12) = - 1- JN L n=l {n odd} (3.54) which will have a gap denoted by ~ 2 .

3.5 Other Useful Formulae

3.5.1 Gauss' Law

Only states satisfying the correct equations of motion can be considered when constructing a basis set diagonalising the Hamiltonian. Therefore it will be useful to write a form of Gauss' law on the lattice. In the continuum language Gauss' law

18 oE ox = g : 'Ip t 'Ip : ' (3.55) where : : denotes normal ordering of the operators. Once we make the same lattice associations made in (3.28) we obtain

L(n + 1) - L(n - 1) N [

1 L(n) - L(n -1) = q)(n)(n) - 2 [1 + (-lf]. (3.57)

We have associated the -1 present in (3.56) with even sites only to account for the

Dirac sea present on these sites.

As we saw in Section 2.2, we see that the electric field can simply be worked out from the fermion configuration. We do however need to now the electric field L(O) at the boundary of the lattice. The only true degree of freedom associated with the electric field is therefore the boundary term L(O). This field in fact corresponds to the background field a= F/g, since the action of Gauss' law is simply to add or subtract a unit flux across a particular lattice site. Therefore in any lattice calculation we may simply impose some boundary electric field to obtain results for various a.

3.5.2 Order Parameters

It was noted that in Section 2.5.3 that the quantity (ii/ry5'lj;) 0 is a convenient order parameter for the Schwinger model at 0 = 1r. In order to perform numerical calculations, we will need a lattice version of this quantity. If we follow the lattice transcription above, we obtain

r5 (i{ry5'l/J / g)o (3.58)

-if: (~(-1)" [(n + 1) - H.c]) (3.59) 0 where

(3.60) LATTICE GAUGE THEORY 55

The other order parameter mentioned in Section 2.5.3 was the average electric field order parameter (E). In the notation of this chapter we may write down trivially r• ~ ~ (~[L(n) +<>]) (3.61) 0

Both these quantities will be calculated in our lattice calculations.

There is one problem with calculating such quantities on a finite lattice, related to the fact that spontaneous symmetry breaking only occurs once we have reached the bulk limit N = oo. To illustrate this problem, let us again consider the lattice strong coupling limit again x - 0, and evaluate the expectation fC-1' given in (3.61).

In the true bulk limit, the ground state is given by (3.50), and we are in the spontaneously broken symmetry regime. The expectation value of f°' is

(3.62) hence we obtain a finite value, as desired. One the other hand, on a finite lattice, the ground state is given by (3.51), and the expectation value is

(3.63)

Since all our calculations are performed on a finite lattice, it appears that we will inevitably obtain a zero value for the expectation value. The remedy for this is well-known, however (Yang, 1952; Uzelac and Julien, 1981; Hamer et al., 1982).

Instead of calculating the matrix element (3.63), if we calculate the overlap matrix element between the states (3.51) and (3.52), we obtain

(01r 0 10') = 1/2, (3.64) LATTICE GAUGE THEORY 56 which is non-zero. For our calculation using periodic boundary conditions we use this approach to calculate the order parameters. For open boundary conditions, this approach is unnecessary as we fix the boundary conditions in such a way that only one of the states I ± ½) is present. Therefore the ground state will be "chosen" as one of the states IO) = I±½), and hence we may directly calculate the matrix element. Chapter 4

Finite Size Scaling

In Section 2.5.2 it was found that a second order phase transition should be present for the Schwinger model at 0 = 1r. One of our main aims in this thesis is to study the physics of the model in the vicinity of this critical point. In general, it is known that phase transitions only occur in the bulk limit N ----+ oo. Since our calculations must always be performed on a finite lattice, strictly we can never observe the phase transition directly. However, using finite size scaling theory (Fisher and

Barber, 1972) it is possible to extract quantities describing critical phenomena from finite lattice data. In particular we will use finite size scaling theory to obtain the location of the critical point (m/ g )c, and critical exponents. Sections 4.1 and 4.2 give a brief overview of general finite size scaling theory. Section 4.3 discusses phenomenological renormalization, where we introduce a practical application of finite size scaling theory, as formulated by Nightingale (1976). In Section 4.5 we review several aspects of the transverse Ising model, which will be useful later for comparison with the results of the Schwinger model at 0 = 1r.

57 FINITE SIZE SCALING 58

4.1 Modifications of Critical Behaviour in Finite

Systems

Although true phase transitions only occur in the thermodynamic limit, one would expect however, with increasing system size the properties of the system approach the critical bulk behaviour asymptotically. As an example of this, let us reproduce the results of Ferdinand and Fisher (1969), who studied the effect of a finite lattice on the critical behaviour for the 2D Ising model on a square N x N lattice. Their results are reproduced in Figure 4.1. We see that on a finite lattice the divergence of the specific heat is suppressed to increasing extents for smaller lattice sizes at the critical point. However, we also see that the position of the peak for various lattice sizes is modified depending on the size of the system. The position of the peak for a given lattice size is called the pseudocritical point Tc(N).

The pseudocritical temperature approaches the true critical temperature Tc in the limit N - oo. We may quantify this approach by defining the shift exponent A, according to

N-oo. (4.1)

The another significant feature shown in Figure 4.1 is that away from the critical region, all curves converge towards a single curve. Therefore away from the critical region finite size effects are relatively small, or equivalently, finite size effects only become significant in some finite range around the critical point. The temperature where the finite lattice curve departs significantly is called the rounding temperature T*(N). We see from the figure that the rounding temperature approaches the critical temperature Tc for larger lattice sizes. We may also define a rounding FINITE SIZE SCALING 59

2.5 r---.---....---.--..----r------.---,

2 32x32

16xl6 1.5

C Nk l

0,5

2 kT/J 3 4

Figure 4.1: Specific heat of the 2D square Ising model with periodic boundary conditions for various lattice sizes (Ferdinand and Fisher, 1969). The vertical line denotes the position of the critical point in the bulk limit. FINITE SIZE SCALING 60 exponent, according to

N-----+ oo. (4.2)

This behaviour is generally valid for systems that are finite in extent in all dimensions. We note here that it is also possible to consider semi-infinite systems, where the size of the system extends to infinity in some directions. In these systems it is possible to observe true divergencies in thermodynamic quantities, as opposed to the pseudo-critical phenomena observed here. As we will only be concerned with

1D systems in this thesis, this will not be relevant here, and will not be discussed

(see Barber (1983) for a review).

Ferdinand and Fisher (1969) argued that thermodynamic quantities depart from their bulk values when the correlation length ~ becomes the order of the linear dimension of the system. This gives us a natural estimate for the position of the rounding temperature

~(T*(N)) I"'-' L, (4.3) where L = Na, and a is the microscopic lattice spacing. In the critical region we recall that (4.4) hence we must have

0 = 1/v. (4.5)

In a similar fashion we may estimate the shift exponent. Ferdinand and Fisher

(1969) use the same criterion as (4.3), to obtain

.X = 0 = 1/v. (4.6) FINITE SIZE SCALING 61

In fact this same result may be derived using renormalization group argument

(Barber, 1983), although we do not discuss this here.

4.2 The Finite Size Scaling Hypothesis

We may turn to formulating the finite size scaling hypothesis. Let us consider some arbitrary thermodynamic quantity (such as the specific heat), which around the critical point behaves as (4.7) where t = (T - Tc) /Tc. On a finite lattice however, as we have seen, the critical behaviour is modified according to the lattice size. Our task now is to attempt to write an expression for PN(T). At any given temperature, there are three length scales involved, namely the correlation length ~, the lattice spacing a, and the size. of the system L = Na. If we measure in units of the lattice spacing, naively one would expect that PN(T) is function of the remaining two length scales:

PN(T) = P(L/a, ~/a, T). (4.8)

The finite size scaling hypothesis (Fisher and Barber, 1972) however states that

PN(T) is a function of only one variable

y = L/~(T), ( 4.9) where ~(T) is the correlation length of the bulk system. We can distinguish three cases: y « 1: The correlation length is much larger than the system size. Strong finite- FINITE SIZE SCALING 62

size effects are observed.

y C::'. 1: The cross-over region where finite-size effects become important. This de

fines the rounding temperature T*(N). y » 1: The correlation length is much smaller than the system size. Bulk be haviour is observed.

The finite size scaling ansatz (Fisher and Barber, 1972) may now be written, which states that any thermodynamic quantity may be written in a general form

(4.10)

where w is some exponent to be determined, and Q(y) is some scaling function, also unknown at this stage. Using (4.4), we may rewrite this as

(4.11) where i is defined using the pseudocritical temperature

i = (T - Tc(L))/Tc. (4.12)

The exponent w can be found by requiring that the in the limit N --+ oo ( 4.11) reduces to (4.7). Matching the exponents fort, we require that

X --t 00. (4.13)

In order to have a finite result for N --+ oo, the N dependence must drop out, hence we obtain

w = p/v. (4.14) FINITE SIZE SCALING 63

Quantity Bulk Behaviour Finite-size Behaviour Specific Heat C t-a: Magnetic Susceptibility x t-'Y Correlation Length ~ t-11 Order Parameter r tf3

Table 4.1: Bulk and finite-size dependences of various thermodynamic quantities.

If we now consider finite N, we have seen at the pseudocritical point i -+ 0 there is no singularity. Therefore we require

Q(x) rv Qo X -t 0. (4.15)

We may now combine the results of (4.11), (4.14), and (4.15) to obtain the be haviour of PN(T) at its pseudocritical point:

(4.16)

This equation therefore gives the finite-size dependence of any thermodynamic quantity near its pseudocritical point. We show specific examples in Table 4.1. In particular, note the correlation length does not depend on any exponent, a result that will turn out to be important in the following section.

4.3 Phenomenological Renormalization

Using the scaling ansatz (4.11) and applying it to the correlation length we have

N-+oo. (4.17) FINITE SIZE SCALING 64

Let us now demand that for two lattice sizes N and N', we have temperatures T and T' respectively such that

~N(T)/N = ~N1 (T')/N'. (4.18)

In the limit N ---+ oo, this is satisfied by

T' - Tc= (T-Tc)(N/N') 11v, (4.19) by equating the arguments of ( 4.17). It was first realized by Nightingale (1976) that ( 4.18) can be interpreted as a form of a renormalization group mapping

T---+ T' = RL,b(T), (4.20) where the rescaling factor is b = N / N'. We note that this mapping is actually not a true renormalization group mapping, as ( 4.18) considers two different systems, with different lattice sizes respectively. For this to be a true renormalization group mapping, the same system must be considered

~(T) = b~(T'), ( 4.21)

which is only possible in the infinite lattice N ---+ oo. The "fixed point" of the mapping ( 4.20) satisfies

(4.22) FINITE SIZE SCALING 65 where the fixed point should approach the critical point as

N, N' ----t oo. ( 4.23)

Therefore we obtain a sequence of estimates for pseudocritical points which should converge to the critical point Tc for large lattice sizes. Previous calculations (Hamer and Barber, 1981) suggest that taking N' as close as possible to N (i.e. N' = N + 1) gives the fastest convergence to the critical point.

To obtain an estimate for the critical exponent v, let us differentiate ( 4.17) to give

(4.24)

Applying the same techniques here as for the correlation function, we have

(4.25) from which we obtain a sequence of estimates for v(Ti,m,), which approaches the true value for N, N' ----t oo.

4.4 Quantum Hamiltonian Limit

To use the results of the previous sections, we must transcribe these results from the language of statistical mechanics to the quantum Hamiltonian framework that we work in. A "dictionary" of correspondences between the two formalisms are shown in Table 4.2 (Barber, 1983). The results of the previous section may be transcribed straightforwardly using this table. If we consider a theory where we FINITE SIZE SCALING 66

Statistical Mechanics Field theory Hamiltonian -{J'}-{, = I:r Er Action iS/n = (i/n) f dtL Local energy density Er Lagrangian density (Infinitesimal) transfer matrix Quantum Hamiltonian H Equilibrium state Ground state Partition function Z N = Tre-.B?-t Path integral Z = J DPropagators Free energy Ground state energy of H ( Correlation length )-1 Mass gap~

Table 4.2: Equivalences between statistical mechanics and field theory.

have a coupling g, the pseudocritical points are found by demanding

~N(g*)N = (N + l)~N+1(g*), (4.26)

where ~N is the gap with lattice size N, and g* is the pseudocritical coupling.

To estimate the exponent v, it is in practice more convenient than the method described in the previous section to calculate the Callan-Symanzik {J-function, de fined by

( 4.27) we should therefore have as N ---+ oo

( 4.28) which gives us an estimate for v. Alternatively we can use the limiting behaviour

ln[fJN(9c)/ fJN-1 (gc)] 1 r-v-- (4.29) ln[N/(N - 1)] l/ as N ---+ oo. We call the first of these ratios ( 4.28) the 'linear' estimate, while the FINITE SIZE SCALING 67 second is the 'logarithmic' estimate. We will use these expressions to obtain critical indices in the Schwinger model in Chapter 6.

4.5 An Example - the Transverse Ising Model

In this section we examine a simple case of a model with a phase transition, namely the transverse Ising model. In particular we examine the spectrum of states, order parameters, finite size scaling behaviour, and the property of self-duality.

We discuss this in somewhat more detail than is necessary for the purposes of this chapter, the reason being that the Schwinger model at 0 = 1r will turn out to be closely related to this model. Many of the phenomena that will be discussed here will have strong analogies with the Schwinger model, hence it is beneficial to present all these results in one place. For the disinterested reader, he/she may proceed directly to Chapter 5 and return to this section if necessary.

4.5.1 Energy Spectrum and Order Parameters

The Hamiltonian is

(4.30)

where

Pfeuty (1970) was able to solve the model analytically using the methods of

Lieb, Schultz, and Mattis (1961), for h = 0. We shall not go into details of the solution here, but merely quote results. Pfeuty solves the model by transforming FINITE SIZE SCALING 68 the Hamiltonian into a theory of free fermions,

(4.31) k k where 77k, 'T]k are the transformed fermion creation and destruction operators in momentum space, Ak = V(l - ,\) 2 + 4,\sin2 k, (4.32) and ,\ = J /r. The ground state is then

(4.33)

Looking at the form of the Hamiltonian (4.31) we see that the elementary exci tations must occur around the minimum of Ak, which is at k = 1r. The physical momenta of the excitations can then be more appropriately interpreted as being k' = 1r - k. The first excited state at momentum k' is therefore

E1 = 2ri 1 - ,\ I - r L Ak, ( 4.34) k so that the gap is E1 -r Eo = 211 - ,\I . ( 4.35)

We see that the gap vanishes at the point Ac = 1, which is the critical point. We also see the linear dependence of the gap, giving an exponent v = 1.

To interpret the low-lying eigenstates of this model, let us look at the model in the strong and weak coupling limits (r -+ oo and r -+ 0 respectively). The Ising limit is reached in the limit r = 0, where we have degenerate ground states, FINITE SIZE SCALING 69 namely all spins up I0; r = o) = { I+½) = iiiiiiii (4.36) I - ½) = 11111111 all spins down where the spins are eigenstates of the az matrix. The ground state here has a spontaneously broken symmetry for N --+ oo, and will occupy one of the above states at a time. Choosing I½) to be the ground state, the next highest state is then

11-kink; r = 0) = nn1111 ... (4.37) translations which introduces a "kink" into the state, thereby raising the energy of the state by

2J. The summation is over translations in the position of the kink. The energy gap of this state agrees with Pfeuty's result for r = 0, according to (4.35). This state is however not in the same sector as the ground state, as there is no way to match the boundary conditions at either end, assuming periodic boundary conditions. A

2-kink state 12-kink; r = 0) = iii liiii ... (4.38) translations is possible from the ground state, as the boundary conditions match at either end.

This will be the lowest excitation from the ground state in this sector of the theory.

This has an energy gap of 4J.

Let us look at the limit in the other direction, namely the strong coupling limit r--+ oo. The ground state here is

I0; r = oo) = (i)x(i)x(i)x(i)x(i)x(i)x ... ( 4.39) where (i)x denotes an eigenstate of ax. This ground state is unique. The next FINITE SIZE SCALING 70

e.------.-----.----~----,-----.------,,

4 2-kink

3 2-flip

2 -..... 1-kink

o~--~---~---~------~--~ 0 0.5 1.5 2 2.5 3

Figure 4.2: Spectrum of the transverse Ising model.

state above this is

11-flip; r = oo) = ( 4.40) translations where one x-spin is flipped, which may be called a "1-flip" state. This has an excitation energy of 2f, in agreement with Pfeuty's result (4.35). As was the case for the 1-kink state, this is not in the same sector as the ground state, and cannot be reached by multiple perturbations of -JCJf CJf+1. The lowest excitation in the same sector as the ground state is the 2-flip state, defined by

12-flip; r = oo) = ( 4.41) translations

Figure 4.2 summarises the behaviour of the gaps.

Pfeuty also obtains results for the magnetisation order parameter Mz = (CJz). FINITE SIZE SCALING 71

-h

00 0 AC =1

Figure 4.3: Phase diagram for inverse coupling 1/ A and magnetic field h. The jagged line represents a first-order phase transition. The open circle represents a second order phase transition.

He finds

(l-;2)1/s >.>1 Mz={ ( 4.42) 0 >. ~ 1

We observe the critical exponent /3 = 1/8. We also explicitly see the presence of long-range order for values of coupling >. > 1. It is also helpful for later comparison to draw a phase diagram of the model, for general h. Figure 4.3 shows that at non-zero h we have an induced magnetisation in the direction of the applied field. In the region >. > >-c the magnetisation undergoes a discontinuity, and therefore a first-order phase transition. The first-order line ends in a second order critical point, where the magnetisation becomes continuous. If we associate 1/>. with a temperature-like quantity, then this is precisely the same phase diagram as one would obtain with a 2D or 3D classical Ising model. FINITE SIZE SCALING 72

4.5.2 Finite Size Behaviour

In this section we will show the leading behaviour of the finite size corrections to the gap (4.34) for two cases i) A =J Ac ii) A= Ac= 1. The allowed values of the momenta k depends upon the boundary conditions imposed upon the system. For periodic boundary conditions we have for the ground state (Hamer and Barber,

1981) 2N-1 ( k) Eo/f = - L A ;N , ( 4.43) k=O {k odd} and for the first excited state

2N-1 ( ) Ei/f = 2(1 - A) - L A ;i . ( 4.44) k=O {k even}

The gap can therefore be written

F(A, N) = w1 - wo = 2(1 - A)+ 2N[T2N(A) - TN(A)], (4.45) where 1 N-1 (7rk) TN(A) = N ~ A N . ( 4.46)

We have used the following results

'f:1 A(;!) ( 4.4 7) k=O {k even} 'f:1 A(;!) ( 4.48) k=O {k odd} FINITE SIZE SCALING 73

It is shown in Appendix C that for large N, the finite size corrections are of form (Hamer and Barber, 1981)

e-2Nwc F(A-/= Ac, N) ~ 2(1 - A)+ const. -/N ( 4.49)

We see that the gap has exponential corrections to the limiting value, which is typical of behaviour away from the critical point.

At the critical point, the behaviour is quite different. Substituting A = Ac = 1 into (4.45) gives

2N-1 ( ) F(A = 1,N) = 2 ~ (-ll+lsin ;i . ( 4.50)

This sum may be evaluated using a power series I:~=O xn = 1 / ( 1 - x) to give

2 sin ( 2~) FA=lN( ) =-----""-'- (4.51) ' 1 + cos (2~) for large N--+ oo this may be approximated (Hamer and Barber, 1981),

(4.52)

The,..__, 1/N behaviour agrees with the finite size scaling arguments presented earlier

(Table 4.1).

4.5.3 The Self-duality of the Transverse Ising Model

We now discuss an important property of the transverse Ising model, called self-duality. The first instance of self-duality was discovered by Kramers and Wan nier (1941), who located the position of the critical point in the 2D square Ising FINITE SIZE SCALING 74

(a) (b)

' ' ' --- --<;>---' --<;>---' --<;>---' --<;>------

' ' ' --<;>--- --<;>---' --<;>---' --<;>---' ---

' --- --<;>---' --<;>--- --<;>--- --<;>------

Figure 4.4: Two dimensional Ising lattices and their duals. Solid circles denote original lattice spins. Open circles denote dual lattice points.

model by noticing that there was a symmetry linking the high and low temperature behaviour. Onsager (Wannier, 1945) was able to generalise these principles to a

2D Ising model of arbitrary geometry, a property that in general is called duality.

The key idea of duality is that a symmetry exists between certain pairs of 2D Ising lattices geometries, some of which are shown in Figure 4.4. A dual lattice may be constructed by placing a dual lattice point within each elementary polygon of the original lattice, and constructing bonds on the dual lattice by bisecting each bond on the original lattice. Therefore a triangular lattice has a dual hexagonal lattice, and a square lattice has another square lattice as its dual, in the latter case we say that they are self-dual. Using this construction a relationship between the partition function of the original and dual lattice may be derived, which was found to be (Wannier, 1945),

Z(T) Z*(T*) (4.53) 2N/2 ( cosh J / kT)(N+N*-2)/2 2N*/2(cosh J/kT*)(N+N•-2)/2 · where Z(T) denotes the partition function, kT is a Boltzmann factor, N is the number of spins, and a * denotes the analogous quantity on the dual lattice. The FINITE SIZE SCALING 75 key relationship is that

sinh(J/kT)sinh(J/kT*) = 1, ( 4.54) which relates the high temperature behaviour to low temperature. Therefore from

(4.53) we see that the high temperature behaviour of the original lattice is equiv alent to the low temperature behaviour of the dual lattice, and vice versa. In the case of the square self-dual lattice, the high temperature behaviour of the lattice is related to the low temperature behaviour of the same lattice. Kadanoff and Ceva

(1971) used this symmetry principle to show that a disorder parameter could be defined for a self-dual lattice, in terms of magnetic dislocations in the lattice. The key property of a disorder parameter is that whenever an order parameter (a) is zero, a disorder parameter (µ) is non-zero, and vice versa.

Fradkin and Susskind (1978) were able to find explicit operator equivalences for the self-duality of the transverse Ising model (4.30), with h = 0. Dual lattice operators were found to be

z z µf aiai+l (4.55)

µ[ II aJ. (4.56) j

It is easily shown that the µ variables obey exactly the same spin commutation relations as the a's. The Hamiltonian can be transformed

( 4.57) which is exactly the same Hamiltonian as (4.30) up to a multiplicative factor, and the replacement ,\ +-+ 1/,\. Interpreting ,\ as a temperature parameter, we see that FINITE SIZE SCALING 76 the high temperature regimes are related to low temperatures, analogous to the 2D classical Ising model. This property of the Hamiltonian may be summarised

H(a; .X) = .XH(µ; 1/ .X). (4.58)

A direct consequence of this is that if E(.X) is some energy eigenvalue of a state, then .XE(l/ .X) is also an energy eigenvalue. We may verify this statement by using the exact result for the energy gap (4.35),

(4.59) which satisfies (4.58).

We discussed previously the order parameter

(4.60)

for the transverse Ising model. From the definition of the Hamiltonian ( 4.5 7) in terms of dual variables it is clear that (µz) must obey

(4.61)

where replacement A ---+ 1 / A was made according to the coupling appearing in the

Hamiltonian. We see that (µz) is non-zero whenever (az) is zero and vice versa, which suggests that (µz) is an appropriate disorder parameter for the model. If we look at the structure of µf (4.56), we see that its action is to flip all spins FINITE SIZE SCALING 77

(a) (b) (c)

A>> 1 A=l llllllllll ll!lllll!l ll!!!ll!ll!ll!! d a d a

Figure 4.5: ( a) The ground state is a perfectly ordered eigenstate of az. (b) The ground state is dominated by the ordered state, with isolated kink-antikink pairs. The size of the kink-antikink pairs a is much smaller that the distance between the kink-antikink pairs d » a. (c) A large number of flipped spins result in a kink condensate. The size of the kink-antikink pairs become comparable with the separation d ~ a.

to the left of site i. This is precisely the "kink" state discussed previously ( 4.37):

µf [rnnrnJ =1111 nn . (4.62)

We may then say that the dual order parameter, or the disorder parameter is the expectation value of a kink creation/ destruction operator. Furthermore, a single spin flip nn1 rnr (4.63) can be associated with a kink-antikink pair. Under periodic boundary conditions, we must only have such kink-antikink pairs, since any single kink will violate the boundary conditions. We can then think of such kinks as a kind of particle in some sense, as their number is conserved.

Fradkin and Susskind (1978) interpreted the ground state for ,\ < 1 as a "kink condensate". The reason for this interpretation can be understood in Figure 4.5.

In the limit ,\ - oo, the ground state is a perfectly ordered state and there are no kink-antikink pairs. At large but finite A, the ground state is still dominated by the ordered state, but due to the perturbative effect of the ax, there are some flipped FINITE SIZE SCALING 78 spins, which in the kink language are kink-antikink pairs. The crucial point to note here is that the separation d between the kink-antikink pairs is much larger than the size of the kink-antikink pair itself a. We can imagine in this regime widely separated kink-antikink pairs, somewhat akin to molecules in a gaseous state. As we approach the critical point, there are many more flipped spins, which means that the separation between the pairs becomes comparable to the size of the pairs d ~ a. The kinks and the antikinks therefore disassociate, and the gas we had for >. > 1 condenses. In this analogy, for >. < 1, we then have a kink-antikink

"liquid" , or a "kink condensate" . In this interpretation a kink condensate is a highly disordered state, a fact that is well reflected by the disorder parameter (µ 2 ) being non-zero in this region. Chapter 5

The Density Matrix

Renormalization Group Method

In this chapter we introduce the DMRG method, which is the primary technique used throughout this thesis. It will be instructive to first examine the historical roots of DMRG, namely the numerical renormalization group (NRG) procedure, as implemented by Wilson (1975). Wilson originally used this procedure as a way of examining the Kondo problem, which turned out to be rather successful for this particular model, but problematic for other models. Attempts were made on the

Hubbard and Heisenberg models, but in each case the results were unreliable, and the method was only used occasionally. The reasons for this failure were examined by White and Noack (1992), which are discussed in Section 5.2. A solution to these problems gave rise to the DMRG method that is finally described in Section 5.3.

79 THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 80

5.1 The Numerical Renormalization Group Pro

cedure

In this section we introduce the NRG procedure, following the explanation given by White (1993). For concreteness, let us say we want to examine an N-site

Heisenberg chain,

N-1 l H = Ls:s:+1 + 2(st si+l + si- st1), (5.1) i=l with open boundary conditions. The general idea of the NRG is to tackle the N site problem by starting with a much smaller number of sites, say 2 sites, then build up the lattice incrementally. The basic scheme is shown in Figure 5.1. At any time in the NRG procedure, the lattice is split into two regions of equal size, called blocks, which we label by B1 and B2. In the very first step, each block contains one site each. A single NRG procedure then consists of combining the two blocks together

B' = B1B2 , in process called augmentation. Two copies of the augmented block are made, B~, B;, and the NRG procedure then may be repeated. At the end of the first NRG iteration, each block contains two sites each, therefore four sites in total. Each NRG iteration thus doubles the total system size. This process can be repeated indefinitely until the desired lattice size is reached.

Let us now examine what happens to the states in a single augmentation. Since the lattice is broken into two parts, we must also split the Hamiltonian of the system in the same way. We clearly get three parts

H = HBl + HBl-B2 + HB2, (5.2)

where the H BI and H 82 represent the intra-block terms of the Hamiltonian, while THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 81

BI B2 ------

! ~I-·-==-· I ------I• ------· I !

I'------·--=-· ---=--·,______,-· I ---+-*-I_·----=--· _ --_·-=::_,· I

Figure 5.1: A schematic picture of a Numerical RG iteration. Blocks B 1 and B2 are augmented to form the block B', which are copied to form blocks B~ and B~.

the Hm-s2 is an inter-block term. For example, in terms of the Heisenberg Hamil tonian (5.1), this inter-block term is

(5.3) where the subscripts 1, R refer to the right-most site of block 1, and 2, L the left most site of block 2. We must also write the basis states of the system in terms of the arrangement of blocks B1 and B2 . We assume that the basis states for each block are obtained from the previous NRG iteration, or if it is the very first step, they are put in by hand. The basis states in the combined block B' are then

ns1, ns2 = 1, ... mRc (5.4)

where nm, nB2 labels the mRc states in each block. The states in the combined system then run from ns, = 1, ... , m~c- Now that the Hamiltonian is specified and a basis set is given, a standard diag onalisation can be performed to obtain the lowest lying eigenstates of the system

IEn), n = 1, ... , m~c- These eigenstates are described by some linear combination THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 82 of the basis states Ins,). The idea now is to abandon the basis states Ins,) in preference of the newly found states !En)

n = 1, .. . mnc, (5.5) where Ins,) refers to the new set of states which will be used. Furthermore, note that we truncate the Hilbert space down from size mh0 ,to the mnc lowest eigen states of !En). The assumption here is that since we are keeping the mnc lowest eigenstates, the shed states give negligible effect on describing the system, since we are generally interested in the ground state and low lying excitations. Later we will see that this assumption is incorrect, and that these are not the optimum set of states that can be used.

Such a change in basis must affect all information held in the algorithm, specif ically the inter- and intra-block matrix elements. To translate the matrix elements a rotation matrix O is needed, where the rows of O are the !En) states. To take into account the truncation, we only keep the lowest mnc states of these, hence 0 is a mnc x mhc matrix. For example the intra-block Hamiltonian matrix elements are rotated according to (5.6)

Similarly the inter-block terms (for the Heisenberg Hamiltonian) are transformed as

Si oszI,L ot (5.7) (5.8)

The s± terms are transformed in a similar way. Note that the left-hand operator THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 83 for B' comes from the left operator for block 1, while the right-hand operator for B' comes from the right operator for block 2. These matrix elements are then passed onto the next NRG iteration, and the process repeats itself. This can be repeated indefinitely until the desired number of sites is reached.

5.2 Problems with the Numerical RG Procedure

As mentioned before the NRG method was found to be generally unsuccessful with the exception of the Kondo problem. The reason for this failure was suspected to lie in a problem with the basis states passed down through the NRG procedure.

To examine this problem in more detail, White and Noack (1992) applied the algorithm to the 1D tight-binding lattice, which in the continuum limit reduces to a particle-in-a-box problem, with plane wave eigensolutions. The idea was to try and identify any problems with the basis states with a model where the eigenstates are already known.

Figure 5.2 shows the main result of their work. The figure considers the situation where two 8-site blocks B1 , B2 are combined to form one 16-site block B'. The basis states of the blocks B 1, B2 originate from the previous iteration, and are plane wave solutions to the 8-site problem. The lowest of these basis states are shown as the black circles in the figure. The basis states of the 16-site system are now formed from the outer product of the two sets of 8-site basis states, according to (5.4).

Note that every basis state for the 8-site system will have a node at each end of the block, according to the open boundary conditions. Therefore any product of basis state of the 8-site system will necessarily have a node in the middle of the lattice.

This is in fact highly undesirable, as the ground state of the 16-site lattice, has an anti-node in the middle of the lattice (shown by the white squares). Therefore THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 84

• • • • • • • D D • D D D D • D • • D D D •

D D • D • • D • D D

Figure 5.2: The ground state eigenvalues for two 8-site blocks (black circles) and a single 16-site block (white squares), reproduced from White and Noack (1992).

nearly a complete set of states must be retained in order to preserve any accuracy.

In short, our basis set is a highly inefficient one, with undesirable features at the boundary conditions.

Two solutions to the problem were proposed. The first was called the "combi nation boundary conditions" method, which involves diagonalising the combined block B' for a variety of different boundary conditions (e.g. periodic, anti-periodic), then using a selection of states from each case as the new basis set Ins,). These states are then orthogonalised, then passed to the next iteration. Imposing several types of boundary conditions thus allows for the fact that different states in the spectrum are suited to different boundary conditions. For example, the ground state is suited to imposing periodic boundary conditions on the block B', as this prevents the problem of the node in the centre of the lattice in the next iteration, as seen in Figure 5.2. However for the first excited state, open boundary conditions tends to be more suitable, as this state has node in the centre. Therefore loosely speaking, one obtains a broader set of states, which are more likely to describe the THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 85

ii • • 11 • • Ill~·~· I '---;------' B'

Figure 5.3: An example of the superblock method. Blocks B 1 and B2 are augmented to form B', and passed onto the next iteration. Block B3 serves as an environment block.

eigenstates of the larger system, by choosing a variety of boundary conditions. This method was found to produce good results, for various combinations of boundary conditions (e.g. perio?ic/ anti-periodic, fixed/free).

Another solution was proposed, which was called the "superblock" method.

Instead of diagonalising just two blocks BB, one puts other blocks around the two blocks in question, and diagonalises the whole system. The combination of all the blocks in question is called the superblock. An example of how this is done is shown in Figure 5.3. By adding another block B3 to the chain, B2 is no longer at the open end of the chain, and hence the wave function will no longer have a node at one end. Block B3 acts as a buffer in a sense, that it keeps B 2 from being at the boundary, and is called the environment block. Blocks B1 and B 2 are the blocks which will eventually be passed to the next iteration, and are called the system blocks. The combined environment and system blocks are then called the universe.

This method was also found to work quite effectively, with the advantage that the lattice did not need to be diagonalised multiple times, as in the combination boundary conditions approach. The reason for the success was attributed to the way the blocks act once they are put together. The interface of the blocks was found to simulate the effect of providing a variety of boundary conditions, and hence in turn gives a good "spread" of basis states.

While this fixes up the problem associated with the boundaries of the eigen- THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 86 states, it turns out the set !En) is not the optimum set of states that can be passed on (White, 1993). The optimum set of states can be shown to be the eigenvectors of the density matrix of the system. To see how this is so, let us say that we are trying to describe some state of an arrangement of blocks shown in Figure 5.3,

(5.9) i,j

Ji), i = 1, ... , imax refer to the states of the system blocks, i.e. the blocks B1 and

B2 • Jj), j = l, ... , ]max refer to the states of the environment block. We are looking for another basis set in favour of Ji), namely Ju°'), a = 1, ... , amax, which more efficiently describes the system, in the sense that we can take less of these states

(amax < imax), and still have

m 11/J) ~ li]j) = Laalua)lv°') (5.10) a=l where Iv°') is the corresponding set of optimum Jj) vectors. Another way of saying this is that we want to minimise the error

(5.11)

Using the notation uf = (iju°') and vf = (jJva), and taking (u°'lua') = c5aa' and (vajva') = c5aa' we get

(5.12) THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 87

Minimising this expression over uf and vf, we obtain

L 'lpijaavj (5.13) j L'lpijUf (5.14)

These may be combined to give

(5.15)

This is an eigenvalue equation with respect to the matrix

Pii' = L 'lpij'lpi'j (5.16) j which is precisely the definition of the density matrix (Feynman, 1972). The eigen vectors of the density matrix are the uf, with corresponding eigenvalues Wa = a;. The density matrix has the property that all its eigenvalues Wa 2". 0 and La Wa = 1.

The eigenvalues Wa have the interpretation of the probability that the system is in the state jua). In short, to minimise the error S, we must have the basis states jua) such that they are eigenvectors of the density matrix Pii'. Furthermore, the most likely states which j1ji) is in are the eigenstates with the largest eigenvalues

Wa, In short, to get the best possible representation of J1ji), we must use the density matrix eigenvectors jua) with the highest eigenvalues Wa,

To summarise, we saw that there were two main problems with the NRG method. Firstly the boundary conditions were not properly considered, and hence the basis states passed onto the next iteration had undesirable features at the ends of the blocks. Secondly the states passed on were not the optimum set of states that could be passed on to the next iteration. In the next section, these improvements THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 88

B, B' :------..... / : . . . ·.

.., , • • •

Figure 5.4: An augmentation process in the standard DMRG method

will be incorporated into the NRG method, which gives rise to the DMRG method.

5.3 The Infinite Lattice DMRG Method

There are several variants of DMRG being used today, here we describe the original "infinite lattice" method (White, 1992), which will be the technique used throughout this thesis. This method may be formulated with both open and peri odic boundary conditions (White, 1993), here we describe the periodic case.

We first give a schematic description of the method. As with the NRG method one splits the chain into blocks, but this time in such a way that there are two blocks B1 and B2 , and two single sites S1 and S2 (see Figure 5.4). At the end of a single DMRG iteration, the block B1 and the site S 1 combine to form the augmented block B' = B 1S1. Note that only one site is augmented to the block, rather than an entire block. This augmented block B' then becomes the blocks

B1 and B2 in the next iteration, therefore in each DMRG iteration the block size increases by one site. Normally one starts with each block containing one site each, so that in the very first iteration the superblock contains four sites. This procedure can be repeated indefinitely until the desired number of sites is reached.

The details of the procedure are very similar to that of the NRG method as THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 89 described in Section 5.1. As described there, the Hamiltonian is split into the inter and intra-block terms

H = HBl + Hs2 + Hs1 + Hs2 + HBI-s1 + HB2-s1 + HB1-s2 + HB2-s2- (5.17)

The basis states are also written using the arrangement of the blocks

nBl, nB2 = 1, ... , mDMRG (5.18)

where moMRG is the number of states in a block, and n 81 and n82 label the single sites basis states. For the S = 1/2 Heisenberg chain a site contains a single spin, which can be in one of two states, ns1 , ns2 = 1, 2. These basis states are inherited from the previous iteration, unless it is the first iteration, where they are put in by hand.

Using this one constructs the Hamiltonian matrix, which is diagonalised using a suitable algorithm, in our case we use a conjugate gradient method, although any may be used. So far the procedure is identical to NRG, apart from the different configuration of the blocks. The main differing feature in DMRG is that a density matrix is formed using the eigenstates of the Hamiltonian at this point. One can form a density matrix from any of the eigenfunctions of the Hamiltonian, although it is common to use the ground state. The state, which is used to form the density matrix, is called the target state, as this usually produces a more specialised set of basis states. The target state may be written in terms of the basis states in (5.18)

(5.19) THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 90

The density matrix is then defined as

(5.20) Pna1ns1n'.e1ns1 = L 1Pna1ns1nB2ns21Pn'.e1ns1nB2ns2· nB2ns2 following from (5.16). Note the summation runs over the degrees of freedom in the environment block, n82 , n82 . The density matrix is then diagonalised to form the set of eigenvectors lu°'). These eigenvectors are in the same Hilbert subspace as

1nm)@ lns1). Note that there is some truncation involved here as we only take the moMRG best eigenstates (i.e. the ones with the largest eigenvalues).

All that remains is to transform system blocks B1S1 in terms of the new basis vectors lu°'). A transformation matrix O is formed, taking us from the old basis to the new basis. The intra-block part of the Hamiltonian for B' originates from the augmented blocks B 1S 1 hence we evaluate

(5.21)

The inter-block terms are also transformed in a similar way. The basis vectors lu°') then replace the old set of basis states Jnm)@ lns1 ). These can now be relabelled In 8,), and passed onto the next iteration. As long as the Hamiltonian preserves translational symmetry ( as is the case for equation 5.1), blocks B 1 and B2 are identical, and hence one may replace them by B', and the process is repeated from the start. The sites S1 and S2 remain unchanged throughout the iteration.

To summarise, the DMRG method is nothing but a modification of the original

NRG method, with two vital differences: first an implementation of the super block method, and second an incorporation of a density matrix to optimise the states for the next iteration. In practice this turns the method into an extremely robust, reliable, and accurate algorithm that has been tested on a large number of very THE DENSITY MATRIX RENORMALIZATION GROUP METHOD 91 different models, generally giving excellent results. We have only discussed the "infinite lattice method" here, but there are several other variants: the "finite lat tice method" (White, 1993), the "four-block method" (Bursill, 1999), momentum space DMRG (Xiang, 1996), local Hilbert space reduction schemes (Zhang et al., 1998), to name a few. Here we only use the infinite lattice method, as it is the most appropriate for this thesis. Many of these alternate methods are reviewed in Hallberg (1999). We will discuss the specific applications to the Schwinger and weak link Heisenberg models in Chapters 6 and 9. Chapter 6

Results for the Massive Schwinger

Model at 0 == 1r

In this chapter we show our results for the massive Schwinger model at the point

0 = 1r. Many of the results contained in this and the next chapter may be found in Byrnes et al. (2002a). Section 6.1 begins by discussing the modifications, which are necessary to the method described in Section 5.3 to implement this model.

Section 6.2 demonstrates the accuracy of the DMRG compared to existing methods, including exact diagonalisation and strong coupling series expansions. We present the first of our results in Section 6.3, where we obtain the critical point and critical exponents to good accuracy. Order parameters and various mass gaps are found in

Section 6.4. We give concluding remarks in Section 6.5.

6.1 Implementation of the DMRG

We now discuss how to implement the DMRG method for the lattice Schwinger

Hamiltonian given in (3.42). Examination of this Hamiltonian reveals two crucial differences with conventional spin Hamiltonians such as that of the Heisenberg

92 RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 93

n=l n=2 n=3 n=4

n=l n=2 n=3 n=4 n=5 n=6 n=7 n=8 .•.•....•..•...... ' .... ------L(O) I • L(l) • L(2) • I L(3) i• L(4) • L(5) I • L(6) • L(7) • I

Augmentation 1 Augmentation 2

Figure 6.1: Augmentation process for the Schwinger model with OBC imposed. Heavy dots indicate the location of the lattice spins. n labels the sites on the lattice, while L(n) labels the electric fields on the links.

model:

• The Schwinger model Hamiltonian has the presence of electric fields L(n) in

addition to spin degrees of freedom.

• lnequivalence of odd and even sites.

Let us first deal with the first point above. For simplicity we shall examine the case with open boundary conditions (OBC) first, then consider periodic boundary conditions (PBC). Foll?wing the method discussed in Section 5.3, we may put forward a candidate arrangement of blocks and sites as shown in Figure 6.1. We see that each block contains spin degrees of freedom as well as electric field degrees of freedom. However the electric field degrees of freedom are completely specified for a particular spin configuration according to Gauss' law (3.57). Therefore once the initial electric field L(O) is specified there are no electric field degrees of freedom.

The implication is that the basis set may be constructed in the usual way by considering spin degrees of freedom only, and the electric fields everywhere may be worked out implicitly. L(O) is simply set to the value of the background field which RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = n 94

n=l n=l n=2 n=3 ,,------., • L(l) • L(2) • L/ii______,1;{0) L/si

n=4·:.. _ _) n=2 -----c-- n=l< _) n=4 ',,,,

L(3) L(7) L(4) • L(6) • L(S) • n=3 n=7 n=6 n=S

Figure 6.2: As for Figure 6.1 but with PBC.

is being calculated at the time, i.e. L(O) = a, where a was given in (3.39). For this study we only consider the two cases, a = 0 and a = 1/2. The situation with PBC is only a little more complicated (see Figure 6.2). First we note that since now we demand L(O) = L(N), the fermions must be in the charge zero sector, according to Gauss' law. In this case we cannot set L(O) to a constant, as it is possible to have loops of electric flux winding around the lattice.

To see how this is possible let us consider the lattice initially with zero electric field everywhere (see Figure 6.3). If a fermion pair now emerges from the vacuum, and traverses the lattice to annihilate, we are left with a unit electric flux throughout the lattice. In this way, fermion configurations with different loop values must be incorporated into the basis for PBC. Each fermion configuration will therefore have several copies of itself in the basis set, with different loop values through it (see

Figure 6.4). Due to the ,....., L2 (n) dependence of the energy, it is only necessary to consider relatively small loop values. The magnitude of the maximum loop flux added to a state is denoted Lmax, hence for a given fermion configuration there will be 2Lmax + 1 states, in the range [-Lmax, Lmax]- We discuss convergence with Lmax in Section 6.2.

Let us now consider how to deal with second point raised previously. Odd and even sites are not equivalent in this model, as they refer to quarks and antiquarks RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 95

a) b) c) - ~ L=l

--t> + ~

L=O L=O L=l

Figure 6.3: Formation of a loop of electric flux. a) Initially there is no electric flux throughout the ring. b) A qij pair forms with electric flux L = 1 between them c) The pair annihilate to leaving an electric flux L = 1.

a) L=O

L=O L=l L=2 L=l b) + t----1 + t----1

c) L=2

Figure 6.4: Three basis states with the same fermion configuration, but with different loops of electric field. The left and right ends of the line are connected by PBC. Taking the state labelled as (b) as the base configuration, we may obtain states (a) and ( c) by adding or subtracting a unit flux respectively. In this example, we therefore have Lmax = 1. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 96 respectively (see Section 3.4.1). Let us consider the OBC case shown in Figure 6.1 first. Recall that in the standard algorithm of Section 5.3, we augment a single site to a block at a time. In our context this would mean adding a single quark or antiquark at a time. We prefer here to adopt a more symmetrical approach, by augmenting two sites to a block in a single iteration, corresponding to a quark an tiquark pair. This method offers several advantages in addition to simply aesthetic reasons. For example, for the OBC case, at any stage in the augmentation the left hand block will always have an odd site at its right boundary. In a similar way, the right hand block will always have an even site at its left boundary. In a standard algorithm if one augments one site at a time, the sites alternate in parity, which makes the implementation somewhat more complicated. We note that there is no disadvantage to our augmentation scheme in terms of accuracy, as the degrees of freedom being added to the block are still relatively small (4 degrees of freedom, compared to 2 in the standard algorithm). One thing to note here is that we can not simply make a copy of block 1 onto block 2, as they are inherently different.

Block 1 has an odd site at the open boundary, while block 2 has an even site at the boundary. We must then construct a density matrix for each block separately, and diagonalise each on its own.

For PBC the augmentation process is shown in Figure 6.2. Again we augment two sites at a time in the way shown in the figure, this time on either side of a block. This allows one to have the same type of site (i.e. both even or both odd) on either end of the block, which again simplifies the implementation of the code. The two blocks in this case are exactly equivalent, and hence only one density matrix is necessary. Apart from these modifications the implementation of the code is identical to the method described in Section 5.3. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 97

N Eo/x Eifx Hilbert Space E~xact/x oEgMRG /x E~xact /x oEgMRG /x Exact DMRG 4 -2. 76419540 round off -2.64600481 round off 61 61 8 -5.15743875 round off -5.04514549 round off 677 677 12 -7. 65089022 round off -7.54661694 round off 8578 8578 16 -10.16576620 round off -10.0719183 round off 115237 115237 20 -12.68838261 6.8 X 10-ll -12.60393996 3.9 X 10-ll 1600406 1255905 24 -15.21455674 2.5 X 10-lO -15.13772319 1.6 X 10-lO 22709186 491149

Table 6.1: Comparison of exact finite-size ground state energies with DMRG estimates using PBC, with 0 = 1r, m/g = 0.2, x = 100 and Lmax = 5. DMRG results use moMRG = 930 states in a single block. The difference 8E[?MRG = E[?MRG - E~xact between the DMRG and the exact results are quoted as "round off" if they agree to within machine precision, which is rv 1.0 x 10-11 .

6.2 Convergence Tests

As an initial step, we would like to test that the algorithm presented in the previous section a) works correctly, and b) produces results with a reasonable error. To test our DMRG algorithm, we obtain independent results using exact diagonalisation, which can produce results up to 24 lattice sites. Table 6.1 compares our DMRG data with the exact diagonalisation for the ground state energy and the first excited state, using PBC. Table 6.2 includes similar results for the ground state energy and the order parameter fll< = (L+a)0 , using OBC. Perfect agreement is seen for all sites (to round-off error) up to 16 sites, where there is no truncation of Hilbert space involved. For larger sites, there is a small discrepancy between the two results as there is a truncation in the Hilbert space, but the agreement is still excellent. In particular, for 24 sites energies are accurate to within 1 part in

1010. The efficiency of the DMRG may be demonstrated by noting that this level of accuracy may be obtained by using l"v 2% of the total Hilbert space, in the case of the PBC calculation. Order parameters typically have a larger error due to the larger round-off error associated with calculating the overlap of two wavefunctions RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 98

N Eo/x f£l< = (L + n)o Hilbert Space E~xact oE8MRG p:t /x /x Exact or~MRG Exact DMRG 4 -2.23146282 round off 0.275528 round off 6 6 8 -4.74542614 round off 0.292918 round off 70 70 12 -7.27328772 round off 0.306353 round off 924 924 16 -9.80478296 round off 0.316405 round off 12870 12870 20 -12.33762004 1.7 X 10-lO 0.323621 round off 184756 139194 24 -14.87103922 4.8 X 10-10 0.328555 7.2 X 10-6 2704156 125971

Table 6.2: Comparison of ground state energies and two order parameters between exact diagonalisation and DMRG with OBC, with 0 = 1r, m/g = 0.3, x = 100. DMRG results use moMRG = 400 states in a single block. The difference oE"f]MRG is defined in the caption for Table 6.1, and similarly oroMRG = roMRG - rExact· The difference between the DMRG and the exact results are quoted as "round off" if they agree to within machine precision, which is oE"{]MRG rv 1.0 x 10-11 • For order parameters, the round-off level is rv 1.0 x 10-6 .

N E0 /Nx rcl< = (L + n)o f 5 = (i'1/ry5'!/J)o 4 -.5541925 0.4570936 0.14909561 8 -.5373622 0.4543641 0.15195378 12 -.5362547 0.4542326 0.15198401 16 -.5361298 0.4542260 0.15198394 20 -.5361126 0.4541429 0.15195627 24 -.5361099 0.4542252 0.15198384 Series -.5361094 0.4542256 0.15198394

Table 6.3: Comparison of DMRG data to strong coupling series estimates (Hamer et al., 1997), for x = l, m/g = 0.5, 0 = 1r, using PBC and Lmax = 5. Order parameter estimates for the strong coupling series are supplied courtesy of Z. Weihong.

(see also caption for Table 6.2).

As a further check we may compare our DMRG results to lattice strong coupling series estimates (Hamer et al., 1997). For small x values it is possible to obtain accurate results to compare to our DMRG data. Table 6.3 displays convergence of the DMRG data to the series estimates for the ground state. energy and two order parameters for x = 1, m / g = 0. 5, 0 = 1r. For these parameter values we see very fast convergence to the bulk limit, such that virtually all quantities have converged at 24 sites. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 99

Lmax Eo/x E1/x 3 -12. 70028300228 -12.62773305713 4 -12. 70028312386 -12.62773379147 5 -12. 70028312385 -12.62773379146 6 -12. 70028312385 -12.62773379146

Table 6.4: Convergence with Lmax for a sample calculation with x = 100, m/g = 0.3, 0 = 1r, N = 20 using PBC and calculated using exact diagonalisation.

In Section 6.1 we discussed the cutoff Lmax which is imposed on the maximum value of the electric loop present for a given basis set. In any given calculation we must first determine a sufficiently large value of Lmax such that the truncation of the Hilbert space has the smallest possible impact on accuracies. In practice it is possible to achieve full machine precision accuracy by choosing an appropriate Lmax,

2 due to the large cost in energy ( l"v £ ( n)) for basis states with large loop values.

The exact value of Lmax necessary for this depends on the point in parameter space

(x; m/g; N; 0) chosen. As a general guide both large x and small N requires a large

Lmax; near the critical region (m/g ~ 0.3) for 0 = 1r also requires large Lmax· An example of the convergence with varying cutoffs in Lmax is shown in Table 6.4. In this example, full machine precision may be obtained by choosing Lmax = 5, which is a fairly typical value used in many of our calculations.

An estimate for the error on DMRG data may be obtained by analysing the convergence with moMRG, the number of basis states retained per block. An example of this is shown in table 6.5 for a calculation using PBC. We see the ground state energy can be resolved to 1 part in 106 , while the two-particle gap is resolved to 1 part in 103 . Order parameters have a reduced accuracy due to reasons explained previously: here we have 3 figures. It is known that the energy generally converges to its asymptotic value linearly (White, 1993) with the density matrix truncation eigenvalue, which corresponds to the sum of the eigenvalues thrown away from the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 100

moMRG Eo/2Nx !:l2/g f°' r5 244 -.31676292 .26833 .28068 .29104 324 -.31677001 .25687 .27598 .28147 416 -.31677263 .25279 .27417 .27735 555 -.31677385 .25054 .27319 .27528 742 -.31677428 .24963 .27273 .27444 932 -.31677440 .24933 .27266 .27423

Table 6.5: DMRG estimates of the ground state energy density Eo/2N x, the "2-particle" gap 6.2 / g, and two order parameters, the mean field r 0 = ((L + a))o and axial density r 5 = (i{ry51/J/g)0 as functions of moMRG, the number of basis states retained per block. These results are for PBC, at x = 100, m/g = 0.3, 0 = 1T, and N = 256 sites. density matrix. In Figure 6.5 we plot ground state energy versus the truncation eigenvalue, which in this case confirms this behaviour. By performing a simple linear extrapolation and taking the intercept, we may obtain a further improved result. In our case, we perform two-point extrapolations with moMRG = 650 and 820 to obtain our final results for PBC, and moMRG = 330 and 400 for OBC. The error estimate is obtained by taking the difference in the extrapolant and the result with the largest moMRG· The accuracy of the DMRG calculation is strongly dependent on the parameter values x and m/g; in particular near critical regions the accuracy is reduced. We find that OBC provide more accurate results than PBC, which is consistent with previous DMRG studies (White, 1993). The results shown here are for parameters quite near the critical region, and hence these may be regarded as "worst-case scenario" values of the errors, and are not typical of the accuracies achievable with DMRG.

Table 6.6 shows similar convergence data, this time for OBC. We again see excellent results for the ground state energy, which is accurate to nearly machine precision, and the 2-particle mass gap to 1 part in 107 • Again due to round-off errors, we are limited in accuracy for the order parameters, so a similar accuracy to the PBC result is achieved here. A point against using OBC is that the finite-size RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 101

p:t moMRG E 0 /2Nx !::l.2/ g rs 107 -.31611009314180 .19536745 .256555 .257439 162 -.31611009382342 .19535871 .256594 .257479 199 -.31611009385738 .19535765 .256606 .257489 244 -.31611009386852 .19535733 .256782 .257658 327 -.31611009387277 .19535723 .256751 .257627 396 -.31611009387248 .19535720 .256734 .257604

Table 6.6: As for Table 6.5, with OBC.

corrections are much larger for this case. We use OBC to calculate the 1-particle gap and the order parameter estimates in the continuum limit, and PBC otherwise.

6.3 Analysis of Critical Behaviour

6.3.1 Position of the Critical Point

Let us first examine the physics of the model near to the critical region. Figure

6.6 shows the "loop gap" !::l.0/ g at a coupling x = 25 for various m/g in the critical region. The plot shows that the loop gap approaches a finite value form/ g :S 0.30, but collapses to zero for larger m/ g, signalling the presence of a critical point. The work of Hamer et al. (1982) suggests that the critical point for this particular x value is approximately (m/g)c(x) ~ 0.3, which agrees with the results shown here.

Furthermore we see that the m/ g = 0.3 case approaches the bulk limit nearly perfectly in ,...., 1/ N, agreeing with the finite-size scaling prediction !::l. ,...., 1/ ~ ,...., 1/ N

(see Table 4.1).

To obtain the location of the critical point, we use the finite-size scaling methods outlined in Section 4.4. Using (4.26), we demand that the ratio

R ( / ) = (N + l)!::l.N+1(m/g) (6.1) Nm g N!::l.N(m/g) ' RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 102

-0.3167730 ,U

-0.3167732

-0.3167734

-0.3167736

X z -0.3167736 ~ 0 8 -0.3167740

-0.3167742 -0

-0.3167744

-0.3167746 0 -0.3167746 o·

-0.3167750 0 0,5 1,0 1.5 2.0 2,5 3.0 D.M. truncation error (x10 -a)

Figure 6.5: Dependence of the estimated ground state energy density E0 /2N x on the density matrix truncation eigenvalues, for x = 100, m/g = 0.3, 0 = 11', using PBC.

0.16 mfg = 0.26 ---0--- 0.14 mfg = 0.28 ---0-- m/g=0.30 o----6---< 0 mfg = 0.32 ____, 0 0.12 0 mfg= 0.34...... , 0 D D 0.1 D D 8. llofg 8. 8. 0.06 8. 8. • • 0.06 • • • • • 0.04 •

0.02

0 0 0.005 0.01 0.015 0.02 0.025 1/N

Figure 6.6: The "loop gap" 6. 0 / g for various m/ g near to the critical region with x = 25. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 103

0.335 X=100 ,-.0.... X=44.4>-0-< 0.33 0 X=25 >--8--< 0 0 0 0.325 0 0 oO O • z t 0.32 D D D D ODD D D 0.315

0.31 l!,.68 6 I!,. I!,. I!,.

0.305 0 2.0 4.0 6.0 8.0 10.0 1/N 3 (x10-e)

Figure 6. 7: Pseudocritical points for three couplings x = 100, 44.4, and 25.

is equal to unity at the point m / g = (m / g) N· (m / g) N is then the pseudo critical point at some lattice size N. To obtain the pseudocritical point we follow the method given by Hamer and Barber (1981). We calculate RN(m/g) using the loop gap ,6.0 / g for a cluster of five points straddling the pseudocritical point, then use a polynomial interpolation to find (m/g)iv- In this way we obtain a pseudocritical point for every lattice size in the range N = 4 to 256, in steps of 4 sites. The five m/ g values are chosen in such a way that they straddle the critical point for a given x value, in steps of 6:..(m/g) = 0.02. A rough guide of the location of the critical point for each x may be obtained by examining Figure 9 of Hamer et al.

(1982). Finally to obtain the continuum values we take our critical point values for each (m/g)c(x) and extrapolate to x -t oo. We choose x values lying in the range 1/ Jx = [0.1, 0.5] with steps of 0.05 between points. We use lattice sizes of N = [4, 256] in steps of 4, according to the augmentation procedure shown in

Section 6.1. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1T 104

Figure 6. 7 shows the pseudocritical points obtained for three x values in this range. Empirically we find that these approach the bulk critical values as r-v 1/N 3.

This r-v 1/N3 dependence was also found by Hamer and Barber (1981) in a finite lattice study of the transverse Ising model. A simple quadratic extrapolation may be performed to extract the bulk values, with an associated error of 1 part in 104 for all x. Some numerical instability creeps in for the larger lattice sizes, where the change in the gap energy with lattice size N becomes so slow that round off errors become appreciable. The continuum limit is then estimated from these bulk critical points, as shown in Figure 6.8. Numerical values are also given in Table 6.7.

On the same plot we show previous estimates obtained by Hamer et al. (1982). A quadratic fit in 1/ ,Jx extracts the continuum limit a - 0 or x - oo, which we estimate to be

(;) C = 0.3335(2). (6.2)

This is consistent with the previous estimate by Hamer et al. (1982) of (m/g)c = 0.325(20), or Schiller and Ranft (1983), (m/ g)c = 0.31(1), but with two orders of magnitude improvement in accuracy. Our DMRG method allows us to obtain estimates for larger lattice sizes, which in turn allows larger x values, which results in the much improved result.

6.3.2 Critical Indices

The considerations of Section 4.4 tell us how to obtain the critical indices for the model. The 'linear' and 'logarithmic' ratios given in ( 4.28) and (4.29) are calculated for the same lattice sizes N and couplings x as discussed in the previous RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = n 105

0.36 .---~---.---...--~---.---r-----r----.---.------,

This work ---0-- 0.34 Hamer et al. --0--

0.32

0.3 -e> E 0.20

0.26 D- ·o 0.24 .. 0.

0.22 ·-o

0.2 .___ _.__ ___._ __.__ _ _.__ ___._ __...._ _ _.__ _.___ ._____, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ga

Figure 6.8: Critical line in the m/ g versus 1/ ,/x = ga plane. Open circles are our present estimates, and squares are the previous results of Hamer et al. (Hamer et al., 1982), which are in good agreement. The dashed line is a quadratic fit to the data in ga.

section. In order to evaluate the ,B-function, we use the expression

,B(N, m/g) 1::1(N,m/g) (6.3) m/g (1::1(N,m/g) - 2(m/g)i::1'(N,m/g)) as given by Hamer and Barber (1981), where i::l(N,m/g) is the gap, in our case we choose here the loop gap 1::10 / g. Analysis of the data shows that the 'logarithmic' version (4.29) is more stable numerically, and converges more quickly towards the bulk limit. We show an example of this convergence in Figure 6.9 for a particular lattice spacing 1/ y'x = ga = 0.45. We see monotonic convergence towards -1 / v -+

-1, until round off errors become appreciable for large lattice sites.

A similar method may be used to estimate the critical index ,B. As shown in

Table 4.1, order parameters should scale like ,...., N-!31 11 • 'Linear' and 'logarithmic' ratios are again constructed from these quantities, and analysed in the same way RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 106

-0.1

-0.2 -1/v >--0--< -0.3 -Plv >-0--< VI 0 -0.4

~ -0.5 ~2 E .c -0.6 ~ Cl --0.7 0 :-.I __g------ -0.8 -0-----0------

--0.9 0-0--0--0--0-- ~~j.:7.::7 -1

-1.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 1/N

Figure 6.9: 'Logarithmic' ratio estimates of critical indices -1/v and -(3/v for lattice spacing 1/../x = ga = 0.45. Quadratic fits in 1/N provide the bulk extrapolations. We estimate here 1/v = 1.00(2) and (3/v = 0.125(5).

as for the ,8-functions. Again the 'logarithmic' ratios

ln[f N( (m/g)N )/fN-1 ( (m/g)N )] .8 rv -- (6.4) ln[N/(N - 1)] V seem to do better than the 'linear' ratios, in terms of numerical stability and convergence. Figure 6.9 shows the estimate for .8 / v, using the 'logarithmic' ratios calculated using rN = ((L + a))o- We see here -.8/v - -0.125. Table 6.7 displays our results for the critical exponents, for each coupling x.

We see essentially no variation in the exponents with x, to within the accuracy of our calculations. Our best estimates for the critical exponents are thus

V - 1.01(1) (6.5)

,8/v 0.125(5). (6.6) RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 107

1/y'x (m/g)c(x) 1/v {3/v 0.5 0.2724(2) 0.99(1) 0.126(5) 0.45 0.2784(1) 1.00(2) 0.125(5) 0.4 0.2845(1) 0.99(2) 0.125(5) 0.35 0.2905(1) 0.99(2) 0.125(6) 0.3 0.2965(1) 1.00(4) 0.126(6) 0.25 0.3027(1) 0.99(3) 0.126(6) 0.2 0.3088(1) 0.99(3) 0.127(6) 0.15 0.3149(1) 0.97(4) 0.123(6) 0.1 0.3211(3) 1.0(1) 0.12(1)

Table 6.7: Estimates for critical points (m/g)c(x) and critical exponents 1/v and (3/v as functions of the lattice spacing parameter 1/ Jx = ga.

These results provide reasonably conclusive evidence that the Schwinger model transition at 0 = 1r lies in the same universality class as the one-dimensional trans verse Ising model, or equivalently the 2D Ising model, with v = 1, {3 = 1/8.

6.4 Mass Gaps and Order Parameters

Our aim in this section is to obtain the low-lying mass gaps, namely the loop gap

6.o/9, 1-particle state 6.if g and the 2-particle state 6. 2/g, and order parameters f°' and f 5 which were introduced in Section 3.4.2. We start with the two particle gap, 6.2/g.

6.4.1 Two Particle Gap /::,,.2/ g

To obtain the two particle gap we generate data for points in the set 1/ y'x =

[0.1, 0.5] (in steps of 0.05), N = [4, 256] (in steps of 4), and m/ g = [O, 1] (in steps of O.1). Let us first examine the extrapolation to the bulk limit N ----+ oo. Figure 6.10 shows our results for m/g = 0.0. We see that even for 256 sites there are still fairly strong finite size effects, and some extrapolation procedure is necessary to RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 108

0.85 .------.....----~------,-----.------, X= 100 o-0--< X=44 .4 --0--, 0 0.8 X=25>-fr--< ,0- 0 0 0.75 0 00 c:P0 QI 0.7 ,0

0.65

0.6

0.55

0.5 L--_____._ ____ ...... ______.__ ___ ...... ______,

0 0.002 0.004 0.006 0.008 0.01 1/N

Figure 6.10: Bulk extrapolations for the 2-particle gap 6.2/ g for m/ g = 0,0, Dashed lines are the fits to the data, according to (6,7)

obtain the bulk limit, At large couplings in x, the figure shows stronger finite size effects, a trend generally true for all m/ g, To guide us in performing this extrapolation, we use the results obtained in the previous section, which revealed that the critical behaviour of this model is closely related to that of the transverse Ising model, The finite-size behaviour of the transverse Ising model was discussed in Section 4.5.2, where it was found that corrections converge exponentially to the bulk limit, modulo half-integral powers of N. We therefore apply the fitting form

(6.7)

to the data, where a0 , a 1, a2 , a3 , a4 are fit parameters. The dashed lines in Figure

6, 10 show that this form in fact fits the data very well, for all values of m/ g. As a double-check of the results of the fit, we also apply a VBS sequence ex- RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 7r 109 trapolation routine (Vanden Broeck and Schwartz, 1979; Barber and Hamer, 1980) to the finite-lattice sequences, which provides an independent estimate of the bulk limit. The VBS procedure is able to obtain the asymptotic value of a sequence by starting with an input sequence and generating another sequence, which should also converge to the same asymptotic limit. Repeating the procedure one obtains further sets of sequences, which may be used to create a table of results, such as that shown in Table 6.8. In our case we choose 8 finite lattice values evenly spaced in N, with the last point in the sequence always fixed at N = 256. The spacing of the points requires some trial and error to give a table of results, which converge to a consistent value. In the results shown in Table 6.8, we use results for the 2-particle gap in the range N = 32 to 256, with 32 lattice spacings between each value.

Errors for the VBS estimates were obtained by examining the columns of the

VBS extrapolants, and comparing the results with different VBS parameters a.

Typically the VBS and the fit results agree to better than 1% accuracy, as long as we are not near the critical region where it becomes difficult to extract a reliable estimate. In the case of the results shown in Table 6.8, our VBS extrapolation gives us a result of ~ 2 / g = 0.511(1), compared to the fitting method which produces a result of ~ 2 / g = 0.5105(3), which is in good agreement.

Figure 6.11 shows the extrapolation to the continuum limit x ----+ oo, or a ----+ 0.

We see that generally there is good agreement between the VBS extrapolation and the fit (6.7), except towards small lattice spacings where a discrepancy opens up.

An extrapolation to the continuum limit is now performed by a simple polynomial fit in powers of 1/ vx = ga. The double extrapolation therefore introduces quite large errors into our final results, compared to the original DMRG eigenvalues where the errors are at worst of order 0.1 %. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = n 110

(a) 0.930873871 0.640914917 0.548877059 0.571052551 0.528087812 0.517787012 0.544448853 0.519884820 0.513920286 0.511078677 0.531677246 0.515866407 0.511662299 0.511385935 0.524612427 0.513504961 0.511400753 0.520294189 0.512268375 0.517486572 (b) 0.930873871 0.640914917 0.548877059 0.571052551 0.528087812 0.514538583 0.544448853 0.519884820 0.512007564 0.504875934 0.531677246 0.515866407 0.510139516 0.510684490 0.524612427 0.513504961 0.510908965 0.520294189 0.512268375 0.517486572

Table 6.8: VBS extrapolations for the 2-particle gap with m/ g = 0.0, x = 16. The two tables are for the VBS parameters a) o:v 8 s = 1 and b) av8 s = 0. Here our extrapolation is Di2/g = 0.511(1).

0.58 ...---..--~-~-~--.---.--~-~-~-~

using fit >-0-< 0.56 using VBS >-0-<

0.54

0.52 .:gi

0.48

0.46

0.44 .___ _.___....___ _.___ _.___ _.__ _,___ __,__ __,__ ___.,_ ___, 0 0.05 o. 1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ga

Figure 6.11: Continuum extrapolations for the 2-particle gap for m/g = 0.0 and 0 = 1r. Data sets obtained through separate VBS and fit extrapolations. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = n 111

Repeating this procedure for a set of m/g values in the range [0,1], we obtain the result shown in Figure 6.12. We see that the 2-particle gap is zero at the critical point, and non-zero on either side, with a nearly linear dependence with m/ g. We see that the 2-particle gap at m/ g = 0 agrees with the analytic result of the Schwinger boson Ll2 1 - = ~ ~ 0.56419, (6.8) g y?T compared to the DMRG result of fl2/ g = 0.57(1). For m/ g # 0, we may compare our results with a semi-classical expansion around m/ g = 0. The appropriate form of the theory in this limit is the bosonized Hamiltonian at 0 = n, given in (2.67). For small m, the potential term U() has a single minimum, and hence may be expanded around = 0. Expanding the cosine we get 2 1 2 1 2 1 ( g '-) 2 mcg] 1t = N [ -II + -(81) + - - - 4mcgy?T + - . (6.9) µ2 2 2 ?T fo

The term in the round brackets is then equal to the mass squared (fl2 ) 2 , hence to leading order we have

0 (6.10) ~ [1-v,ie (;) +o (;)']

0 564 - 1. 78 (;) + 0 (;)' (6.11)

Now we may compare this to our numerical results. A linear fit through the points in the range m/g = [0, (m/g)c] gives

gfl2 ~ 0.569 - 1.72 (m)g , (6.12) which agrees very well with the prediction made in (6.11). Note, however, that the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 112

2------.------r-----~------,,------,

Loop energy, l:10/g >-0--<

1-particle gap, l:11 /g >-{}-<

1.5 ,' 2-particle gap, 1:1 2 /g >-fr--<

m/g

Figure 6.12: Final estimates for gaps in the 0-particle, 1-particle and 2-particle sectors at 0 = 1r, Dashed lines are merely to guide the eye.

behaviour is not exactly linear.

6.4.2 Loop energies !:l.o/ g

We now repeat the procedure performed for the 2-particle state and apply it to the 0-particle, or the loop state D.0 / g. Figure 6.13 shows the convergence of the data to the bulk limit N -t oo for various couplings x. The plot reveals that even for the largest coupling x, we see that finite size effects have all but completely vanished, and we may simply "read off" the bulk values with no extrapolation. The points converge in this fashion for all m/ g values, except for near to the critical region 0.25 ~ m/ g ~ 0.4. Near to the critical region we must use the fitting function (6.7) and the VBS extrapolation of the previous section to obtain bulk limits, particularly for large x.

Performing similar continuum extrapolations x -t oo as that described for the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 7r 113

0.4.------.------,------~----~ X = 4.0 ,-.0-. X=6.25 '-D X = 11. 1 •---8---< • 0.35 X = 25.0 ,--e---, • • • X=100.0 ...... •• ••••• 0.36

0.34 !:io/9 ------······ ......

0.32

0.3 I IIIOIOIOIOIOOOllllllllllllllll 111 ILJOOOO O O O O O 0

0.28 , Ml»»>iimmnn»rni»» l OXDOOOO O O O O O O O 0

0 0.005 0.01 0.015 0.02

1/N

Figure 6.13: Convergence of lattice data for the loop gap l:10/ g for various coupling x at m/g = 0.1. At N = 256, all curves converge to within errors of the points.

2-particle gap, we obtain the spectrum for the loop gaps shown in Figure 6.12. We see that this gap vanishes at the same point as calculated in (6.2), and is zero for all m/g > (m/g)c, as discussed in Section 3.4.2 and predicted by Coleman (1976). For m/g < (m/g)c the loop gap appears to be degenerate with the 2-particle gap.

In particular, at m/ g = 0, the loop gap is found to be l:i0 / g = 0.5643(2), which agrees with the result of the Schwinger boson mass (6.8). The degeneracy of the loop gap and the 2-particle gap is a somewhat unexpected result as these have very different interpretations in the lattice strong coupling limit. One would naively expect the 2-particle gap to have a larger gap energy, due to the 2m/ g energy required to excite a pair in the strong coupling limit. This picture certainly seems to hold true in the large m/g region, where the 2-particle spectrum asymptotically approaches l:i 2/ g '"'-' 2m/g. Below the critical region this picture would seem to be wrong even qualitatively. A further interesting feature is that for m/g = 0, it RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 114

------·g/2 ---10--·---g/2

Figure 6.14: The electric fields at either end of the chain in the presence of a single particle are g/2, -g/2 respectively.

would appear that there are two degenerate Schwinger boson states. At m/g = 0 we have seen from Section 2.3 that the spectrum should contain an integer number of Schwinger bosons, independent of 0. Here we see a degeneracy of states equal to the Schwinger boson mass g / ,Jir. Such a degeneracy is not seen at 0 = 0, and would seem to be particular to 0 = 1r.

6.4.3 I-particle gap ~i/ g

The I-particle gap must be calculated using open boundary conditions, due to the mismatch in electric fields at either end of the chain. Figure 6.14 shows the situation. The presence of a single charged particle will shift the electric field by ±g, and hence the electric fields are necessarily different at either end. It is impossible therefore to apply PBC in the presence of a single particle state, as one cannot match the electric fields at the two ends. In the lattice spin formulation, a single particle corresponds to one flipped spin, hence is found in the S:C,t = 1 sector. We thus find the I-particle gap by finding the difference between the ground states with s:ot = 1 and s:c,t = 0. There is a further complication in the case with OBC, in that applying a back ground field of either o: = 1/2 or -1 /2 gives a different result for the ground state energy for a finite lattice. To see this, consider a simple two-site system as that shown in Figure 6.15. The figure lists all the possible fermion configurations that can be present on a two-site system, which form a complete set of basis states. We RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 115

a) b)

L=l/2 L=l/2 L=l/2 L=-1/2 L=-1/2 L=-1/2

L=3/2 L=3/2 L=-1/2 (±) L=l/2 L=l/2 L=l/0 • • L=l/2 L=l/2 8L=-l/2 L=-1/2 • L=-1/2 8 L=-3/2 L=l/2~ L=l/2 L=-1/2-+ ~ -- L=-1/2

Figure 6.15: Basis states for the two-site Schwinger model in the staggered lattice formulation. We see that setting a background field of a) a = 1/2 and b) a = -1/2 do not produce equivalent basis states.

see that depending on whether the electric field is set to a = ±1/2, we have an inequivalent set of basis states. In particular, the state with both an electron and a positron has an electric field of L(l) = 3/2 or 1/2, depending on the field imposed on the first site. Due to the L L2 (n) term present in the Hamiltonian, we then obtain a different energy eigenvalue according to the basis set used.

The origin of this discrepancy is due to the "staggered" lattice formulation we adopt in our Hamiltonian. In the figure we have chosen that a positive charge occupies the very first site of the lattice n = l. Therefore depending on the applied fields a = ±1/2 we immediately have an asymmetry in the outgoing field L(l) = 1/2 or 3/2. The only consistent definition of the one-particle gap for a finite lattice is therefore

(6.13)

where E 1 is the 1-particle energy, Et is the ground state energy with a= 1/2, and E0 that for a= -1/2. Apart from this complication the procedure in extrapolating to the continuum RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 116 limit is the same as for the 2-particle state. We use data from the set 1/ Jx = [0.1, 0.5] in steps of 0.05, lattice sizes N = [4,256] in steps of 4, and m/g = [O, 1] in steps of 0.1. Our results are shown in Figure 6.12. We see that the gap vanishes for m/g < (m/g)c, while for m/g > (m/g)c the 1-particle gap is very close to half the

2-particle gap. For large m/g we see the 1-particle gap asymptotically approaches

~i/g,...., m/g. Once again, overall the behaviour is very nearly linear in m/g. The pattern of eigenvalues exhibited in Figure 6.12 bears an extraordinary resemblance to that of the transverse Ising model (Fradkin and Susskind, 1978;

Hamer and Barber, 1981), even down to the (almost) linear behaviour with m/g.

In particular, we see that the energy of the 1-particle state vanishes at the critical point, and then remains degenerate with the ground state for (m/g) < (m/g)c This is exactly the behaviour of the "1-kink" state of the transverse Ising model, as shown in Figure 4.2. For the loop gap ~ 0 / g, the spectrum is reversed, the gap being zero for m/g > (m/g)c, and non-zero for m/g < (m/g)c- This is exactly the behaviour of the "1-flip" state of the transverse Ising model. The 2-particle state gapped on both sides of the critical point, in a similar fashion to the "2-flip" and "2-kink" states of the transverse Ising model. The analogy goes further, if we interpret the 1-particle state in the Schwinger model as a "kink" in the Bose field, as in Section 2.5.2. In this language the 1-particle state is explicitly a single kink state. We shall discuss further aspects of this in Section 6.5.

As a further check on our results of Section 6.3.1, we may calculate the position of the pseudocritical points (and hence the critical point), using the 1-particle gap ~i/g instead of the loop gap ~ 0 / g. The method of calculation is identical to that presented in Section 6.3.1. Our results are shown in Table 6.9, which may be compared to those listed in Table 6. 7. We see that the these results are in good agreement to those obtained previously, although the accuracy is considerably RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 117

1/y'x (m/g)c 1/v 0.5 0.273(1) 0.9(1) 0.45 0.2786(3) 0.9(2) 0.4 0.2841(3) 1.0(2) 0.35 0.2903(3) 1.0(2) 0.3 0.2968(3) 1.0(3) 0.25 0.3024( 4) 1.0(5) 0.2 0.3095(10) 1(1) 0.15 0.3145(10) 1(1) 0.1 0.320(1) 1(1.5)

Table 6.9: Estimates for critical points and the critical exponent 1/v, using the I-particle gap bi.if g, as a function of 1/ .Ji= ga. reduced. The loop gap is therefore found to be more suitable for this calculation.

6.4.4 Order Parameters r5, px

We may also obtain estimates for the order parameters f 5 and p:ie as functions of m/ g. It is possible to obtain estimates using either PBC and OBC (see the discussion in Section 3.5.2), although we chose to use OBC in our calculations as these were found to be more numerically stable than those with PBC. The reason is that the overlap matrix element (3.64) involves using the symmetric and antisymmetric ground states, which for large N become very nearly degenerate. These small differences in energy cause the conjugate gradient method to develop numerical instabilities, which degrades the accuracy of the results. The OBC case does not have this problem as only one state is involved in calculating the matrix element. The extrapolations to the bulk limit N - oo are shown in Figure 6.16.

We see that the results for a particular coupling x converge quickly to the bulk limit, for any x. We have used the same data set as for Section 6.4.3 to generate these results. The ambiguity which was present for the I-particle state (see Figure

6.15) is again relevant here, hence we must calculate the order parameters for the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 118

0.438 ••••••••••• 0.436

0.434 •••••••••••• 0.432

0.43 ra 0.428 I IIIIOOOllilllllllllllllllllllliii ii I nJDDDD D D D D D D

0.426

0 0 0.418 .______...______. ______.______0 __,

0 0.005 0.01 0.015 0.02 1/N

Figure 6.16: Extrapolation to the bulk limit for the order parameter r 0 = (L + a) 0 , form/ g = 0.5 and various couplings x, and an applied field of a= 1/2.

two cases a = ±1/2. In the bulk limit it is found that both cases extrapolate to the same value, to within errors ..

Performing the extrapolation to the continuum limit in the same way described previously, we obtain our estimates for the continuum values as shown in Figure

6.17. Both order parameters are zero, within errors, for m/g < (m/g)c, Near the critical region, particularly for m/ g < (m/ g )c it becomes quite difficult to obtain accurate estimates, but it appears that both order parameters turn over and drop abruptly to zero as the critical point is approached from above, consistent with the small exponent f3 = 1/8 found in Section 6.3.2. The axial density r 5 decreases steadily towards zero at large m/g. We may see the reason for this by using simple semi-classical methods on the bosonized potential (2.68). For large mass m --t oo RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 119 the potential is dominated by the cosine term,

U( cp) ~ cmµ cos(2'1ir), (6.14)

hence the two vacua approach cp ---+ ±.Jir/2 in this limit. The order parameter behaves with cp as

r 5 rv (sin(2v-ir) )o, (6.15) which will be zero for both vacua.

For large m/g all quantum fluctuations are suppressed due to the large energy required to excite fermions out of the vacuum. We therefore expect that the ground state is a completely "static" state, with zero fermion density everywhere in the line.

The expectation value of the electric field then simply is equal to the background field, and hence r 0 is expected to approach the value of 1/2 in this limit, as seen in the figure.

We sh_ow our complete results for the loop energy ~ 0 /g, 1-particle gap ~i/g,

2-particle gap ~ 2 /g, and order parameters r 0 = ((L + a))o and r 5 = (ii/;157/J/g)o in Table 6.10 for future reference.

6. 5 Concl us ion

We have performed a DMRG calculation for the massive Schwinger model at

0 = 1r, using the Hamiltonian lattice formulation up to 256 sites. We find that the DMRG method works extremely well in obtaining the quantities of interest, namely the critical parameters, gaps, and order parameters. In particular the loop gap ~ 0 /g was used to locate the position of the critical point at (m/g)c = 0.3335(2) using finite size scaling theory. This is two orders of magnitude better than the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 120

0.5 ...------,.------.------.------r-----, . -0· .. -0· ... -0 ...... 0.45

0.4

~ 2 0.35 Q) E I!! 0.3 ~ Q> 0.25 1:! 0 0.2

0.15

0.1

0.05

0.4 0.6 0.8 mfg

Figure 6.17: Order parameters r'" = ((L + a:))o, r 5 = (ii/ry5 '1/;/g) 0 near the critical region. Dashed lines are merely to guide the eye.

mfg ~o/9 ~i/g ~2/9 f°' rll 0.0 0.5643(2) 0.0 ± 10-5 0.57(1) 0.0 ± 10-4 0.0 ± 10-4 0.05 0.4756(2) 0.48(1) 0.1 0.3883(2) 0.0 ± 10-6 0.40(1) 0.0 ± 10-3 0.0 ± 10-2 0.15 0.3020(5) 0.30(2) 0.2 0.2173(5) 0.0 ± 10-5 0.23(4) 0.00(2) 0.000(5) 0.25 0.134(2) 0.16(4) 0.3 0.05(2) 0.0 ± 10-2 0.03(7) 0.0(3) 0.0(2) 0.4 0.0 ± 10-3 0.105(3) 0.22(1) 0.376(1) 0.302(5) 0.5 0.0 ± 10-4 0.246(3) 0.49(1) 0.421(1) 0.270(5) 0.6 0.0 ± 10-4 0.3764(6) 0.758(8) 0.4430(5) 0.238(5) 0.7 0.0 ± 10-4 0.5020(2) 1.006(4) 0.4566(5) 0.211(5) 0.8 0.0 ± 10-5 0.6224(1) 1.249( 4) 0.4657(5) 0.189(5) 1.0 0.0 ± 10-5 0.8530( 4) 1.711(4) 0.4769(5) 0.155(5)

Table 6.10: Our results for the loop energy Di.of g, 1-particle gap Di.if g, 2-particle gap D.2/ g at background field 0 = 11'. We also quote our results for the order parameters r'' = ((L + a:))o and r 5 = (ii/vys'l/;/g)o. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 121 previous estimates of Hamer et al. (1982) and Schiller and Ranft (1983), which demonstrates the power of the DMRG method. This may be attributed to the ability of DMRG to obtain very accurate results for large lattice sizes - an exact calculation is limited to a lattice size of approximately N = 24 sites, whereas we are able with relative ease to extend this to N = 256. This in turn allows much larger x values, corresponding to small lattice spacings a. The larger the x values, the more accurate the results one obtains, since this reduces the amount of extrapolation required, the largest source of errors in the procedure.

We have calculated the 0-, 1- and 2-particle gaps of this model around the critical region. All gaps reduce to zero at the critical point, as expected. In particular, by locating the 1-particle gap, we have identified the existence of the

"half-asymptotic" particles predicted by Coleman, which has not explicitly been calculated before, as far as we are aware. The "half-asymptotic" particles exist in the region m/g > (m/g)c, and disappear for m/g < (m/g)c, in exact agreement with the semiclassical predictions of Coleman. We have also calculated the critical indices of the model and find that this model for 0 = 1r lies in the same universality class as the transverse Ising model (v = 1, {3 = 1/8). We calculated the spectrum of two order parameters, f°' and f 5 , and found their behaviour is consistent with the exponent {3 = 1/8. Order parameters also behave as expected for large m/ g, which tend towards f°' .- 1/2 and f 5 .- 0.

The spectrum of states, shown in Figure 6.12 is found to have an amazing resemblance to the spectrum of the transverse Ising model (Figure 4.2). The 1- particle state was seen to be analogous to the 1-kink state of the transverse Ising model, the loop state was analogous to the 1-flip state, and the 2-particle state was analogous to the 2-kink and 2-flip states. The analogy is even stronger if one notices that in the Bosonic form of the theory, the 1-particle state is a kink state, RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 122 i.e. a transition between the double well of Figure 2.10.

This leads us to a new interpretation of the ground state for m/g < (m/g)c. In the transverse Ising model the ground state for >. < Ac was described as a "kink condensate", where kinks could be excited out of the vacuum with no cost in energy.

The kink picture of the 1-particle state therefore points to a similar interpretation of the ground state for m/g < (m/g)c, as a "kink condensate". In this picture we have a condensate, or a liquid of kink-antikink pairs, or alternatively, quark antiquark pairs. This state is a state of high disorder, characterised by the non-zero expectation value of the disorder parameter (see Section 4.5.3). This points to the existence of a "dual symmetry" and "dual order parameters" in the Schwinger model, which have not been recognised as far as we are aware. In our calculations we have calculated two order parameters, both related to the expectation value of the bosonic field (cp). If we consider this to be the "order" parameter, then the "disorder" parameter should have a spontaneously broken phase in the low mass region, and collapse to zero at the critical point. By analogy with the Ising case, one would expect the disorder parameter to be the expectation value of the kink creation/ destruction operator. However, we have not been able to verify this by explicit computation so far. The close relation between the Schwinger model at

0 = 1r suggests that it may be possible to write an "effective" Hamiltonian of the

Schwinger model relating the two models. Again, we have not been successful in doing this as of yet, although this may give additional insight into this model.

There are some puzzling features that still are in need of explanation. The degeneracy of the loop gap and the 2-particle gap for m / g < (m / g) c, is of some concern as no simple interpretation can be given in any limit of the theory, and not present in the Ising model spectrum. Therefore any correspondence to the Ising I model is most likely to be only qualitative. The other puzzling feature was the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 1r 123 degenerate Schwinger boson mass at m/ g = 0. This is also unexpected in terms of the spectrum predicted by analytical solutions, which does not predict such a degeneracy for all 0. As of yet, we have not found a satisfactory explanation for the origins of these features, although this may be pursued in future studies. Chapter 7

Results for the Massive Schwinger

Model at 0 == 0

7.1 Non-relativistic Series

In Section 2.2 we derived a diagrammatic expansion of the Hamiltonian valid for weak coupling. This may be used to obtain a non-relativistic expansion of the vector and the scalar states, which were obtained to first order by Hamer (1977).

This was extended to third order by Sriganesh et al., (2000), but on re-examination of their result an error was found in their calculation, hence in this section we present a re-derivation of the calculation.

The first step in this calculation is to expand the Hamiltonian (2.26) for large mass to obtain a Schrodinger equation. We keep all terms to O(m-3 ). The free

Hamiltonian can be written

H[/- 2Ep = 2(p2 + m2) 112 182 1 84 ~ 2m------(7.1) m 8x2 4m3 8x4 ·

124 RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 125

The remaining terms may be expanded using the following results

1 1 1 a2 ~ -+-- (7.2) EP m 2m3 8x2 1 1 1 a2 ~ -+--- (7.3) E2 2 4 2 p m m 8x ·

The Coulomb term can be expanded

(7.4)

This may be simplified using [fx, lxl] = Sgn(x),

(7.5)

The self-mass term may be expanded

(7.6) while the annihilation term is

(7.7) RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 126

7.1.1 First Order Schrodinger Equation

To reproduce the results of Hamer (1977) let us only keep leading order terms in the kinetic and potential parts of the Hamiltonian:

1 a2 g2 HR = 2m - --+ -lxl + O(m-1 ). (7.8) m8x2 2

Dropping the constant 2m gives us a Schrodinger equation

1 82 92 ] [ ---+ -lxl '!/J(x) = E'ljJ(x). (7.9) m8x2 2

Rescaling the variables according to

94 ) 1/3 X = (-2 )1/3 z E=- ( - ,\ (7.10) mg2 4m ' gives us two standard differential equations

z>0 (7.11)

z < 0, (7.12) which have Airy function solutions in each region

'!/J(z) = ±c1Ai(z + -\) z>0 (7.13)

'!/J(z) = ±c2Ai(-z + -\) z < 0, (7.14)

where c1, c2 are positive constants to be fixed by normalisation. Depending on the sign chosen in each region, we have either symmetric or antisymmetric solutions. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 127

Matching the wavefunctions at z = 0 gives the conditions

Ai' ( -X!) = 0 (symmetric) (7.15)

Ai (.x;;) = 0 (antisymmetric) . (7.16)

These can be solved numerically

.xt = -1.01879, -3.24820, .. . (symmetric) (7.17)

.x;; = -2.33811, -4.08795, .. . (antisymmetric) . (7.18)

The two lowest excited states are known to be the vector and the scalar states, which correspond to the lowest symmetric and antisymmetric solutions respectively. Rescaling according to (7.10) gives the binding energies

1 113 ~ = 0.642 (!) (vector) (7.19)

~2 = 1.473 (!) 1/3 (scalar), (7.20) which is the result obtained by Hamer (1977).

7.1.2 Second Order Schrodinger Equation

We now repeat the calculation, but keep all terms to O(m-3). The Schrodinger equation is

1 fJ2 92 92 1 a4 92 92 a2 ] [ ---+ -lxl --- --+ -b(x) - --- 1/J(x) = E'lj;(x) m 8x2 2 1rm 4m3 8x4 4m2 21rm3 8x2 (7.21) RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 128

Integral Symmetric Antisymmetric (%z4) -0.577655 1.093349 (b(z)) 0.490777 0.0

Table 7.1: Results of numerical integration for various operators.

If we use the same change of variables (7.10), we have

(Ho+ V) 1/;(z) = A1/;(z), (7.22) where a2 Ho --lzl 8z2 4 2 ) 1;3 ( 4 ) 1;3 84 ( 4 ) 1;3 ( 6 ) 1;3 82 V ( 1r3~ 2 + 25~m4 8z4 - 3fm4 b(z) + sJm6 8z2 .

At this point we see that the last term in Vis of higher order than the other terms, and therefore can be discarded. The effect of the new terms in the Hamiltonian may be calculated by perturbation theory,

(7.23) where 1/;± refer to the symmetric and antisymmetric solutions. Contributions from each term may be evaluated by numerical integration, the results of which are shown in Table 7.1

Using these results, and rescaling back to the original variables using (7.10) gives us our final results: for the vector state

E1 ( g ) 1/3 1 ( g ) ( g ) 5/3 [ ( g ) 1 /3] 9 = 0.642 m - ;: m + 0.155 m + 0 m , (7.24) RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 129 and for the scalar state

E2 ( g ) 1/3 1 ( g ) ( g ) 5/3 [ ( g ) 1 /3] 9 = 1.4 73 m - ; m - 0.109 m + 0 m . (7.25)

We will compare these results to existing data in later sections.

7.2 DMRG Results

We may straightforwardly use the DMRG methods developed in Chapter 6 and apply them to the zero background field case 0 = 0. As outlined in the introduction, many authors have studied the case of zero background field already, hence our purpose here is to demonstrate the power of the DMRG method applied to this case. The most accurate results to date are those of Sriganesh et al. (2000), who used exact diagonalisation methods applied to the same lattice Hamiltonian (3.42). The largest lattice size calculated by these authors was N = 22, hence we expect to do much better using our DMRG algorithm, which can go to much larger lattice sizes.

7.2.1 Vector Gap

For the 0 = 0 case, we find that convergence with lattice size is much more rapid than 0 = 1r. Usually there is essentially no extrapolation necessary to obtain the bulk limit. Figure 7.1 shows some sample data for m/g = 0. For a particular lattice spacing x, we have very good convergence with lattice size, resulting in 6 figure convergence for x = 4 through to x = 100. Even for the largest coupling x = 400, the gaps can be resolved to 4 figures. Our final continuum estimate is obtained by the method of linear, quadratic and cubic extrapolants used by Sriganesh et al. (2000) in their estimates of the RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 130

0.61 PlOO"AAAA A 8 X= 100 ....-0-, A 0.605 A X=44.4 --0-, A X=25 >-&--< 0.6 I MIOm0111 ! I IJ:tt:Jo D 0.595 D D D 0.59 E1l9 D 0.585 ~o D 00 0.58 0 0 D 0.575 0

0.57 0 0

0.565 0 0.02 0.04 0.06 0.08 0.1 1/N

Figure 7.1: "Vector" gap binding energies Eif g at m/g = 0, for various lattice spacings.

vector and scalar states. We now give a brief summary of this method. Our aim is to extract the continuum limit ga ---+ 0 of a sequence of points, such as that shown in Figure 7.2. Linear extrapolants are found by taking sets of three consecutive points (e.g. ga = 0.2, 0.25, 0.3), and fitting a straight line to the points.

Three points are used in the linear extrapolation ( as opposed to two), so as to obtain an error bar for each point. The extrapolation of this line provides us with an estimate of the continuum limit. Similarly, taking sets of four consecutive points, and extrapolating in the same way find the quadratic extrapolants. Cubic extrapolants are found using sets of five consecutive points. This gives multiple estimates of the continuum limit, which are shown in Figure 7.3. We see that the extrapolants themselves converge towards the continuum limit, as ga---+ 0. In this example, our final estimate then lies somewhere between the two dashed lines in the figure. Our final estimate of Ei/g = 0.56419( 4) for this case agrees extremely well with the analytic result (6.8). RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 131

0.68

0.66 .0 0

0.64 0 .0· E1/g 0.62 .0' ,0·

0.6 0 .o 0.58 .0·

0.56 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ga

Figure 7.2: Continuum extrapolation of "vector" gap binding energies Ei/ g for m/ g = 0.

0.5656 ----~---~------~---~---~ ) 0.5654 linear o--Q-o quadratic --0-- 0 0.5652 cubic >--A--< 0

0.565 0

0 0.5646

0.5638 .______,______,_ ____..______.______.. ______, 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 7.3: An example of our extrapolation procedure for the "vector" gaps Ei/ g for m/ g = 0. Circles, squares and triangles show linear, quadratic and cubic extrapolants respectively. Dashed lines show the upper and lower bounds for our final estimate. Here we estimate Eif g = 0.56419(4). RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 132

m/g This work Sriganesh Crewther & Kroger & 3rd Order (DMRG) et al. Hamer Scheu Non-relativistic 0 0.56419(4) 0.563(1) 0.56(1) 0.125 0.53950(7) 0.543(2) 0.54(1) 0.528 0.25 0.51918(5) 0.519(4) 0.52(1) 0.511 0.5 0.48747(2) 0.485(3) 0.50(1) 0.489 1 0.4444(1) 0.448(4) 0.46(1) 0.445 0.478 2 0.398(1) 0.394(5) 0.413(5) 0.394 0.399 4 0.340(1) 0.345(5) 0.358(5) 0.339 0.340 8 0.287(8) 0.295(3) 0.299(5) 0.285 0.286 16 0.238(5) 0.243(2) 0.245(5) 0.235 0.236 32 0.194(5) 0.198(2) 0.197(5) 0.191 0.193

Table 7 .2: Comparison of bound-state energies for the "vector" state Ei/g with previous works. The results of Sriganesh et al. (2000) and Crewther and Hamer (1980) were obtained through finite-lattice studies, and Kroger and Scheu (1998) used light- cone methods to obtain their results. We also show our 3rd order non-relativistic calculation of Section 7.1.2 for comparison.

Table 7.2 summarises our results, which show between one and two orders of magnitude improvement in accuracy over previous best estimates for small values of m/g. We must note, however, that for values of m/g > 1 there is little or no improvement in accuracy over previous results. To explain this, first note that the structure of the eigenvalue function in x shifts towards large x for large m/g. But at large x and large m/ g there are many "intruder" states below the vector state for finite N: artefacts of the finite lattice corresponding to states with no fermion excitations but with loops of electric flux winding around the entire lattice. This restricts the range of x that can be used, and hence the accuracy of the calculation.

In a future calculation, a way of eliminating these intruder states must be found. Comparison of our results with previous works is shown in Figure 7.4. In the low mass region, we see excellent agreement between our results and the series expansion around m/ g = 0. Our results are also fairly consistent with the results of Sriganesh et al., obtained by exact diagonalisation. The "fast-moving frame" results of Kroger and Scheu seem to be consistently a little low in this region. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 133

I 1st order non-relativistic expansion - 0.55 I 3rd order non-relativistic expansion - · - 2nd order expansion around mfg = 0 - - - 0.5 Sriganesh et al. >-0-< Kroger & Scheu 6. This work -0-- 0.45

0.4 E1/g

0.35

0.3

0.25

0.2

.3 ·2 ·1 0 2 3 4 5

Figure 7.4: Comparison of our results for the "vector" state binding energies Ei/ g with other works. Squares mark the results of Sriganesh et al. (2000) and triangles the results of Kroger and Scheu (1998). The results of Vary, Fields and Pirner (1996) and Adam (1996) were used for the expansion around m/ g = 0. The 1st and 3rd order non-relativistic expansions of Section 7.1 are also shown for comparison.

We now compare our third order non-relativistic calculation of Section 7.1 to our numerical results. We see from Figure 7.4 that compared to the leading term, the third order result dramatically improves the agreement with the numerical data. In particular we have better than,...., 0.5% agreement to the numerical results all the way up to m/g ~ 2, in comparison to the first order expansion which seems to diverge already at fairly large masses. We expect the non-relativistic expansion to give the most precise results for very large masses; some points are evaluated in Table 7.2 for comparison. The best agreement is seen between our own DMRG results, and the quasi-light-cone approach of Kroger and Scheu. Our results are perhaps a little high in this region, and also have large error bars, which may be attributed to the problem with the intruder states discussed above. RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 134

7.2.2 Scalar Gap

Using the same method we also calculate the scalar state binding energies Ed g using DMRG. The problem with the intruder states that was present for the vector masses is compounded in this case, as we are now targeting a state of higher energy.

We therefore concentrate our attention for small masses, in the range m/ g :S 4, as we expect little improvement over previous studies in the large m/ g regime. Even for small m/ g, the procedure is not as straightforward as for the vector state. A typical excitation spectrum is shown in Figure 7.5. We see a large number of higher momentum states of the vector state cut in below the scalar state, hence a large number of excitations must be calculated to obtain results for large lattice sites. To identify the state we must track the state "by eye" to find the bulk values. The vector state does not have this problem as by definition it is the lowest bound state. We choose x values such that the finite size behaviour is small enough that values converge to their bulk values without any extrapolation. This corresponds to a range 1/ y'x = [0.2, 0.55] form/ g :S 0.125 gradually tapering down to 1/y'x = [0.1,0.45] for mid-range 0.25 :S m/g :S 4. All 1/y'x values step in units of 0.05.

By choosing appropriate x values we obtain bulk values for each x to an accuracy of typically,...._, 0.01%. For larger x values the accuracy diminishes to,....., 0.1%. The largest source of error results from the second extrapolation x -. oo, where for small m/ g, quite a long extrapolation is needed. We again use the method of linear, quadratic, and cubic extrapolants to obtain our final estimate and error values. Our final results are given in Table 7.3. We see that for mid-range m/ g, we obtain our best results, improving estimates by more than an order a magnitude compared to the previous best estimates of Sriganesh et al .. For small m/g we only marginally improve our values, due to the long extrapolation that is needed in x for RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = O 135

2 • i=7i=6:1: • i=3 i=B •• ~:1i=4 i=9'"*-' o* * i=5 i=10 >---v'---< 1.5 0 * * * * +++ E/g + + + + +

XX X X X X X 0.5 X

0 .______._ ____ ....______. ____ _,______,

0 0.02 0.04 0.06 0.08 0.1 1/N

Figure 7.5: A typical excitation spectrum for binding energies E / g for the scalar state. Here we choose x = 25 and m/g = 0.125. Each excitation is labelled with an index i. Here the vector state approaches Ei/ g ~ 0.584, while the scalar state approaches E2/ g ~ 1.269.

these values. In Figure 7.6 we compare our DMRG results with other works, and also plot the 3rd order non-relativistic result (7.25). Our results agree quite well with previous studies, particularly with those of Sriganesh et al. for m/g ~ 0.125.

For m/g < 0.125 we are only able to compare to the mass perturbation series result of Adam (1996). According to (2.52), we would expect that we see better agreement between the results, as the series is known to O((m/g)2). However the coefficient of the 0( ( m/ g )2) is quite large in this series, and it is possible that results only converge very slowly. Otherwise we have not found reasons for this discrepancy.

The 3rd order non-relativistic expansion of Section 7.1 on the other hand agrees very well with the numerical data. Good agreement is seen up to m/g ~ 2, where agreement is seen with the DMRG results to rv 3%. The series diverges away for RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 136

1.4 1st order non-relativistic expansion - ' 6 3rd order non-relativistic expansion - · - 1.3 ' ' \ t:,. Expansion around m/g = 0 - - - Mo & Perry t; \~ 1.2 0 Kroger & Scheu *6 ~ Sriganesh et al. • ·D- ·• 1.1 * I ~ This work ... o-, \ I ,...... ---, I I / -,~ I I . E2/g o.s I' " I " 1:l_ 0.8 i , I " 0.7 ; I " 'i ~" 0.6 I, " " n ~ 0.5 i : '-: I 0.4 -5 -4 -3 -2 -1 0 2 3 4 5

log2(m/g)

Figure 7.6: Comparison of our results for the "scalar" state binding energies E 2 / g with other works. Squares mark the results of Sriganesh et al. (2000) and triangles the results of Kroger and Scheu (1998). The results of Adam (1996) were used for the expansion around m/ g = 0. The 1st and 3rd order non-relativistic expansions of Section 7.1 are also shown for comparison.

smaller values of m/ g.

7.3 Conclusion

We have in this chapter studied the vector and scalar gaps using two methods:

DMRG and a non-relativistic expansion. We find our DMRG results significantly improve estimates for the vector mass gap compared to previous works, particularly in the small mass region. In this region we typically see two orders of magnitude improvement over previous estimates. We note that this improvement is obtained with a fairly moderate computational effort - each data point in m/ g is obtained typically by calculating a spread of ten x values, each run taking approximately 12 hours on the APAC supercomputer. These can typically be run in parallel, hence RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 137

m/g This work Sriganesh Crewther & Eller Mo& Kroger & et al. Hamer et al. Perry Scheu 0.03125 1.168(8) 0.0625 1.193(9) 0.125 1.215(7) 1.22(2) 1.11(5) 1.35 1.16 1.314 0.25 1.228( 4) 1.24(3) 1.12(5) 1.25 1.19 1.279 0.5 1.201(1) 1.20(3) 1.15(5) 1.19 1.17 1.227 1 1.118(1) 1.12(3) 1.19(5) 1.13 1.12 1.128 2 0.989(3) 1.00(2) 1.10(5) 0.98 0.99 0.991 4 0.84(1) 0.85(2) 0.93(5) 0.84 0.84 0.837

Table 7.3: Comparison of bound-state energies for the scalar state E2/ g with previous works. We display here the results of Sriganesh et al. (2000), Crewther and Hamer (1980), Eller et al. (1987), Mo and Perry (1993), and Kroger and Scheu (1998).

our complete set of results were obtained in approximately 2 weeks of wall time.

By simply increasing the number of states kept in a DMRG block mnMRc, we may systematically improve our results, which is easily possible if one is prepared to spend more computer time.

In the large mass region however, our DMRG results do not improve on previous estimates. This is due to the presence of intruder states, artefacts of the lattice corresponding to states with loops of electric flux through the system. We did not find a simple way of eliminating these states, which would be necessary if one required similar levels of accuracy in the large mass region, compared to the low mass region. Calculation of the scalar mass states were inflicted with the same problem, hence we directed our efforts in the range of small to mid-range m/g. We find here good results particularly for mid-range m/g values, where an order of magnitude improvement was obtained. For smaller m/ g, the structure of the spectrum moves to smaller x, and hence a longer continuum extrapolation is needed. This reduces the accuracy of the results, although even here we improve on past studies by at least a factor of two.

We also performed a non-relativistic expansion to third order, correcting an RESULTS FOR THE MASSIVE SCHWINGER MODEL AT 0 = 0 138 error m a previous work. This provides a valuable check of various numerical approaches, including our own DMRG methods. We find that including second and third order terms dramatically improves the range of the expansion, such that

0.5% accuracy is maintained tom/g ~ 2. The quasi-light-cone approach of Kroger and Scheu and our DMRG results seem to agree best with the expansion. The scalar state expansion also compares favourably with the numerical data, up to m/g ~ 2. In an upcoming paper (Byrnes et al., 2003), we use the results of this calculation with the method of Feynman-Kleinert approximants (Janke and

Kleinert, 1995). This method can perform extrapolations of series results, in a similar way to Pade approximants, or alternatively interpolate between strong and weak coupling series. We do not present results of these as it is beyond the scope of this thesis. Chapter 8

The Persistent Current

We now turn to the second part of this thesis, where we study the phenomena of persistent currents. Our starting point will be the well-known Aharonov-Bohm effect, which will serve to introduce several important concepts for understanding the origins of the persistent current. Section 8.2 examines the persistent current properties of a system of non-interacting electrons, where many of the qualitative features can be already seen. Section 8.3 examines the effect of electron-electron interactions.

8.1 The Aharonov-Bohm Effect

Let us first briefly review the origins of the Aharonov-Bohm effect ( Aharonov and Bohm, 1959). An example of an experimental set up is shown in Figure 8.1.

An electron passes into the ring from the left, where it has equal probability of entering the upper or lower channel. The wavefunctions of the electrons combine as they leave the ring. A magnetic flux pierces the ring, but no magnetic force is felt by the electrons, as the magnetic field is only present in the region labelled by . There is however, a magnetic potential A experienced by the electrons, as

139 THE PERSISTENT CURRENT 140

Figure 8.1: An Aharonov-Bohm ring with a central magnetic flux (and magnetic potential A). An electron enters from the left and has equal probability of entering the upper and lower channel. An interference effect is seen when the two paths recombine.

shown in the diagram.

The direction of the A field is parallel to the electron propagation in the top half of the apparatus, and antiparallel below. This induces a phase difference in the electron wavefunction depending on the path the electron takes. Therefore, as the two wavefunctions recombine, there is an interference effect in the conductance of the ring. The period of the oscillations is in units of the elementary flux quantum

0 = hc/e. This is the nature of the Aharonov-Bohm effect. In fact, one can view the Aharonov-Bohm effect as a special case of a phe nomenon that occurs for rings pierced by a magnetic flux. Byers and Yang showed

(1961) by quite general considerations that all energy levels, and hence all mea surable quantities must oscillate with period 0 for such rings. Let us consider a simple conducting ring of some material, with a magnetic flux pierced through it (see Figure 8.2). The magnetic potential then modifies the Schrodinger equation through L-1 [ -in-+a ~A(xj) ] 2 'lp + V'i/J = E'i/J. (8.1) . 2m oxi c J THE PERSISTENT CURRENT 141

I ((

Figure 8.2: A conducting ring with a magnetic flux threading the axis.

We can transform this to a gauge with A(xj) = 0, so long as we transform

(8.2)

This eliminates the potential A(xi) from the Schrodinger equation. The penalty is that the condition for periodic boundaries

(8.3) where L is the length around the perimeter of the ring, is modified to so-called

"twisted boundary conditions"

(8.4)

where 0 = he/ e, and (8.5)

The conclusion is that any flux penetrating the ring can be viewed as a change in boundary conditions of the ring. It is explicit from the form of this condition that all energy levels must be periodic in units of 0 , and hence all measurable quantities, including the persistent current, must also have the same periodicity. THE PERSISTENT CURRENT 142

In calculations that we will perform later, it is more convenient to impose twisted boundary conditions than deal with a magnetic potential.

An important conceptual development in connection to these boundary condi tions was made by Buttiker, Imry and Landauer (1983). They noticed that the boundary condition (8.4) was similar to the Bloch function of a periodic lattice. In fact the two problems are identical, if one makes the replacement kL - 21r/0 .

We can therefore immediately draw a schematic spectrum of the model, using this knowledge (see Figure 8.3). For a ring with no disorder (i.e. V = 0), we have a set of parabolas, periodically arranged in 0 . This is shown by the dotted lines in the figure. In the presence of disorder, we have bands of occupied energy ranges, with larger gaps opening in the presence of stronger impurities. In fact, one can imagine that effectively an electron experiences a periodic potential on a ring, since it 'sees' the same potential with each revolution. The circumference can then be regarded as the unit cell of the 'periodic' structure.

We must also obtain an expression that defines the persistent current. A ther modynamic derivation is presented by Bloch (1970). If we consider an external voltage V through the ring, this will induce a current J, hence the rate of change of the free energy is dF/ dt = IV. If we consider the origin of the voltage to arise from a changing magnetic, then V = -(l/c)d/dt. The current must therefore be

dF l=-c (T > 0). d

For our purposes, we limit ourselves to zero temperature, where the free energy coincides with the ground state energy. Our definition will therefore be

I= -cdEo (T = 0). (8.7) d THE PERSISTENT CURRENT 143

Energy ,

~: __ ---

0 o/2

Figure 8.3: 'Band' structure of a ring with varying magnetic flux . For an impurity free ring ( dashed lines), the energy levels are determined by a sequence of parabolic potentials. For a ring with impurities (solid lines) gaps open at level crossings of the energy.

8.2 Persistent Current Properties for Non-interactill/

Electrons

Now we turn to evaluating the persistent current for a simple model. It turns out that the simplest possible case that of non-interacting electrons confined to

a ring is enough to show most of the relevant physics of the persistent current

(Cheung et al., 1988). We will look at the system in two limits, one with and

without the presence of impurities. THE PERSISTENT CURRENT 144

8.2.1 No Disorder

As there are no interactions, the Hamiltonian is trivially,

(8.8) where we have imposed twisted boundary conditions

· '1/J(x) = '1/J(x + L)ei21rif?/if?o_ (8.9)

The electrons are non-interacting, hence we need only to find the single electron wavefunction, and sum the energies. We therefore have

n,2k2 n (8.10) 2m -27r ( n+- ) (8.11) L o

with n = 0, ±1, ±2, ... , with energies periodic in 0 • Using the definition (8. 7), the current of a single state is

21ren ( ) In= - mL2 n + o . (8.12)

Since electrons are fermions, adding electrons one by one fill the energy levels from lowest to highest. The total current is then the sum of the contributions from each level. However many of these contributions will cancel, due to the positive and negative values of n. The behaviour of the current will then depend upon the last filled state, i.e. the electron at the Fermi level. This cancellation is a characteristic of the persistent current, and is present for other types of system, not only the case studied here, and is called the "parity effect". In fact, this could already have been THE PERSISTENT CURRENT 145 deduced from Figure 8.3, as for a given , the gradients of the energy levels (which gives the current) alternate, giving rise to a cancellation effect. It follows then the current must depend on whether the total number of electrons

Ne is odd or even. One easily obtains

-l 2if! 0 for Ne odd, -0.5 ::; : 0 < 0.5 l( ) = { if!o (8.13)

-lo ( ~~ - 1) for Ne even, 0.0 ::; : 0 < 1.0.

where 10 = evF / L, and VF denotes the Fermi velocity, which in our case is

nkF n1rNe VF=-=--. (8.14) m mL

A plot of the current (8.13) is shown in Figure 8.4. The current reaches a maximum exactly 0 /2 out of phase for the odd case with respect to the even case.

It is possible to write the current (8.13) as a Fourier sum (Cheung et al., 1988)

(8.15) which will be convenient for later.

8.2.2 Single Scatterer

To study the effect of disorder, we again consider the simplest possible case, that of a single o-function potential placed in the ring. The potential term to be added to the Hamiltonian (8.8) is

V(x) = 1:o(x). (8.16) THE PERSISTENT CURRENT 146

N0 = odd N0 = even lo [' [' ' I' ' ' ' ' ' ' ' ' ' ' ' ' ' ' I 0 ' ' ' ' ' ' ' ' ' ' ' ' I ' ' ' I ,1 ' ' ---! ' '\J -I o

-o -o/2 0 o/2 o

Figure 8.4: The persistent current for a ring of non-interacting electrons with no impurities. Solid lines show systems with an odd number of electrons, dashed lines for systems with an even number of electrons.

Cheung et al. (1988) studied such a system using transfer matrix methods (Erdos and Herndon, 1982; Anderson and Lee, 1980). It is possible to obtain an expression for the current in the limits of strong (E / E* » 1) and weak ( E/ E* « 1) impurity strengths, where the typical strength of an impurity is defined by E* = Neh2 /mL. For weak impurities they obtain

I - -lo-24> [ 1 - --1 ( E ) 2 [ 1 ( ) - o 21r2 E* sin2 (2;;) while for strong impurities they obtain

7r E* 27r

I() = I -~ sin ( o ) + ( ~ ) 2 sin ( o ) +... , (8.18) 02 [ l THE PERSISTENT CURRENT 147

,· 0 ,· £1£ =0.4 ,· ,· : ,f·-·, __ ,· ' I \ ' ,· ,' ,' £1£' = 0.8 , ' , · ' I ,· 'I , · 'I .,,,--,-·--- , · : : ,/ ...... :: I o~.·~-r,'__ ~-~----_-_-_-_-_-_-_-_-_--_-_-_-_~_-_-.::-- ___-~~-""""'-=-----£~!~£·_=_1~0'----~:: --? ::I I ------.,,,/,.,, I 'I ...... /" I' 1 I '----7--" : : : : E/E·= 5 ,' : 'I I ' ,, I ' 'I ,I .,,I ,I £1£'=0 -I o " --- 0

Figure 8.5: The effect of 8-function impurity on the persistent currents present in a ring of non-interacting electrons. In the limit of strong impurity, the current reduces to a sine function.

where 10 = evp / L. Both expressions are valid for Ne odd, and terms of order 1/Ne have been dropped.

The behaviour of these expressions can be seen from the plot shown in Figure

8.5. We see that in the presence of an impurity, the current gradually departs from the sawtooth behaviour to that of a sine function. The effect of disorder is then to

"round off" the discontinuities of the sawtooth function. Comparison with (8.15) shows that for strong impurities, the effect is to suppress the higher harmonics of the Fourier expansion, the dominant effect arising from the fundamental frequency

"'sin(2n /o). THE PERSISTENT CURRENT 148

8.3 The Effect of Electron-Electron Interactions

The study of persistent currents in systems with electron-electron interactions

was motivated at least in part by the discrepancy found between the magnitude

of the current between theory and experiment. The first experiment performed by

Levy et al. (1990) on an ensemble of rings found a current of order,...., 10-2evp/ L,

which was in good agreement with theory. However, another experiment on a

single ring (Chandrasekhar et al., 1991) found a much enhanced current, of order,....,

evp/ L. The discrepancy between the two experiments was attributed (Imry, 1997)

to the presence of electron-electron interactions (i.e. repulsive Coulomb forces)

present in the ring. As an example of such a system we shall examine in this section

the XXZ Heisenberg chain, which will be the subject of our numerical analysis later.

Again we study two cases, with and without the presence of impurities in the ring.

-8.3.1 No Impurity

The Hamiltonian that we examine in this section is

(8.19)

where we impose periodic boundary conditions, and the number of lattice sites is

N. This may be used as a model of electrons in a ring, as the Jordan-Wigner

transformation

j-1 s+ [ l J exp irr L clck c} (8.20) k=l s-.- J (8.21)

(8.22) THE PERSISTENT CURRENT 149 transforms the Hamiltonian into a theory of spinless fermions with a short range Coulomb interactions:

(8.23)

where p( i) = cJ ci - 1/2. An explicit expression for the current in this model was derived by Loss (1992), using bosonization methods. The result for the current in the zero temperature limit was found to be

(8.24) where 1; = ev"p/ L, with v"p = v* /2K*. v* and K* are parameters characterizing the Luttinger liquid. In the non-interacting case we have v* = Vp and K* = 1/2. Comparison of (8.24) with (8.15) reveals that the form of the current for the interacting and non-interacting cases is exactly the same, up to the magnitude of the current. Furthermore, the period of oscillation of the current is also 0 , as was the case for non-interacting electrons. Hence all the qualitative features of the persistent current that were presented in Section 8.2 still hold and only the magnitude of the current is modified.

Similar studies of various other models have shown similar conclusions. The ideal Wigner crystal with no pinning potential (i.e. no impurities) was found to have precisely the same expression as (8.24) for the current (Zvyagin and Krive,

1995). Such a result can be intuitively understood by considering the degrees of freedom for electrons in a chain (Miiller-Groeling et al., 1993). If we consider the systems of electrons in a ring as a whole, we may consider the movement of a particular electron as having "internal" and "external" degrees of freedom. The THE PERSISTENT CURRENT 150 interactions present between the electrons must only affect the "internal" degrees of freedom, as there is no link to the outside world. The magnetic flux on the other hand affects the "external" degrees of freedom of the electrons, which gives rise to the persistent current. Therefore, the presence of interactions does not change the persistent current, up to a renormalization of the Fermi velocity.

We may see the effect of the strength of the interactions on the current by calculating the charge stiffness, defined by (Kohn, 1964)

2 D _ Nd Eol . (8.25) 2 d 4>=0

The stiffness is therefore proportional to the slope of the current dl / d at = 0. The ground state energy under twisted boundary conditions was obtained using Bethe Ansatz methods by Hamer, Quispel and Batchelor (1987). They obtain for the ground state energy density

2 2 [ N() _ oo(O)] /J ~ _ 7f Sinµ 7rSin(µ)(/o) eo eo 6µN2 + 4µ(1r - µ)N2 ' (8.26) where

.6. = - cosµ (8.27) and 1 e~ (0) / J = 4 - ln 2. (8.28)

Using the definition of the stiffness (8.25), we obtain (Sutherland and Shastry, 1990) D = 1rvpsinµ (8.29) 8µ(1r-µ)' where Vp = 2 in these units. For the non-interacting case (.6. = 0), the charge THE PERSISTENT CURRENT 151 stiffness is a maximum, while for all other µ values it is suppressed. At the isotropic point (~ = -1), the stiffness is suppressed by a ratio rr/4 ~ 0.785 compared to the non-interacting case. The presence of interactions therefore has the effect of diminishing the size of the current.

8.3.2 Single Impurity

We now consider the effect of a single impurity placed in the ring. Kane and

Fisher (1992a; 1992b) examined a model of spinless interacting fermions in the continuum limit, which may be written after a bosonization procedure

H = v (!(v7)2 + 2~ (v70)2). (8.30)

where v is the sound velocity, 0 and are bosonic fields satisfying

[(x), O(x')] = -i8(x - x'), (8.31) hence v70 (v7) is the momentum conjugate of (0). The Hamiltonian (8.30) is a theory of interacting bosons, parameterised by g, an effective interaction strength. g = l describes a non-interacting Fermi gas, g < l describes repulsive interactions, g > l describes attractive interactions. Using this Hamiltonian the conductance was found to be e2 G = gh. (8.32)

We see that in the non-interacting case, the well-known quantisation of conductance in units of e2 / h is reproduced, with the factor of g modifying the conductance according to the interaction strength.

Now we may introduce a defect into the ring. This may be modelled by a po- THE PERSISTENT CURRENT 152 tential barrier V(x) placed at a point in the ring. The barrier height may be char acterised by a parameter t, which may be regarded as the transmission amplitude through the barrier. A very large barrier corresponds to a very small transmission amplitude, and hence a very weak link, and vice versa. The perturbation for a small barrier ( t rv 1) is written

oH = JdxV(x)'lj;t(x)'lj;(x), (8.33) where V(x) is non-zero only for x near zero, and has a maximum amplitude much smaller than the Fermi energy. The main results are summarised in Figure 8.6.

Arrows on the figure denote renormalization group flows for the perturbation. In the limit of large t we see that for attractive interactions g > 1, the barrier is an irrelevant perturbation and we have perfect transmission. For repulsive interactions g < 1, the perturbation is a relevant perturbation, and for g = 1 it is marginal.

Kane and Fisher obtain to order ,\2 = (1 - t)2,

(8.34)

In the DC current limit w - O,.for g > 1 the conductivity reduces to (8.32), while for g < 1 the perturbation theory breaks down. The authors also examine the limit of a large barrier potential (t small). This may be analysed by considering two semi-infinite lines, connected with the matrix element oH ~ -t ['lj;t(x = O)'lj;_(x = 0) + H.c.] , (8.35) where 1P± denote the operators in each of the two semi-infinite regions. Again the results are shown in Figure 8.6. The perturbation in this case is irrelevant for THE PERSISTENT CURRENT 153

e2 t % G=O G=~

0 0 1 g

Figure 8.6: Renormalization group flows as a function of the electron coupling g and the transmission t. For repulsive interactions g < l, the conductance renormalizes to G = 0, while for attractive interactions g > l, the conductance renormalises to that of zero barrier height.

g < 1, hence renormalises to zero conductance. For g = 1 again we have a marginal perturbation.

For the point g = 1, it is possible to solve for the conductance for all barrier heights, they obtain e2 4t2 G=--- (8.36) h (1 + t2 ) 2 .

In the renormalization group language, this corresponds to a fixed line. Kane and Fisher thus suggest that it is quite plausible that one can join together the behaviour in the two perturbative regimes, and conclude the following:

• For g < 1, there is zero transmission for barriers of arbitrary height.

• For g > 1, there is perfect transmission for barriers of arbitrary height.

No rigorous proof is available for the above statement, however for g = 1/2 Kane and Fisher are able to exactly solve the model to support the above statements.

Several studies have been undertaken to confirm these results. In particular, Eggert and Affleck (1992; 1995) have used renormalization group and numerical finite size THE PERSISTENT CURRENT 154

scaling methods to examine the effect of impurity terms on the 11-h model at the critical point 12/ 11 = .2411. They consider single and double impurities located in the lattice and find indeed that such impurity terms renormalise to an open chain.

As discussed in the introduction, we propose to examine the Heisenberg chain

(8.19) at its isotropic point b.= 1, with a single weak link. In particular we will also attempt to confirm the work of Kane and Fisher by examining various strengths of the weak link, and compare to the results of Eggert and Affleck. These results are contained in the following chapter. Chapter 9

Results for Heisenberg Chain with a Weak Link

In this chapter we show our results for the persistent current properties of the

Heisenberg chain with a weak link. Many of the results in this chapter may be found in Byrnes et al. (2002 b). Section 9 .1 discusses the implementation of the D MRG, while Section 9.2 discusses our methods for calculating the quantities of interest.

Section 9.3 shows our results including convergence data, spin stiffness estimates, and correlations across the weak link. Section 9.4 ends with some concluding remarks.

9.1 Implementation of the DMRG

The Hamiltonian that we intend to study, which we repeat here for convenience, is (9.1)

155 RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 156

.------Augmentation 1 • J'

Block l / Site 2

Augmentation 2

Figure 9.1: DMRG augmentation process for the Heisenberg chain with a single weak link J'.

In terms of the implementation of the DMRG, the main problem associated with the weak link is that it breaks the translational symmetry of the lattice. To see why this is significant, let us consider the arrangement of blocks and sites shown in Figure

9.1. Here we show a DMRG augmentation similar to that discussed in Section 5.3, but with a single weak link placed between Site 2 and Block 1. In the standard method, one augments Block 1 and Site 1 and constructs a density matrix, which in turns yields a new set of basis states. In a system with translational symmetry this set of basis states may be copied and inserted in both Blocks 1 and 2, since each block is identical. Here, translational symmetry is broken, as Block 1 is always directly adjacent to the weak link, while Block 2 is one site away from the weak link. The reader may verify that this is true even for the weak link placed in other positions relative to the block. The solution to this problem is to perform two augmentations per DMRG iteration, as shown in the figure. Each augmentation therefore produces its own density matrix, which in turn is diagonalised to yield its own set of basis states. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 157

9.2 Quantities of Interest

Our main objective is to study the effect of the weak link with respect to the persistent current properties of the ring. If one were to explicitly calculate the current, according to the definition

J() = _ dEo d one must calculate the ground state energy at set intervals in , then numerically differentiate this function. Our situation here may be simplified considerably using the result that for the defect free Heisenberg chain the dependence of the ground state energy is purely quadratic (see Equation 8.26):

(9.3)

Assuming that this relation holds for all values of the weak link, then the current must be linear in , a result already seen in Section 8.3. The slope of the current may be found by picking any two points on the parabola. For our purposes it is convenient to choose the points = 0 and = 7r, which correspond to open and anti-periodic boundary conditions respectively. Anti-periodic boundary conditions are defined by

S1v+1 = St (9.4)

The expression for the current is then

J() = -D (9.5) L RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 158 where D is the charge stiffness

8L[ _ +] D =

Ps = :~[E0(N) - Et(N)], (9.7) where the flux periodicity

9.3 Numerical Results

9.3.1 Convergence Tests

As a first step let us examine the accuracy of the D MRG algorithm by comparing to independently obtained exact diagonalisation results, up to 24 lattice sites. The comparison between the DMRG and the exact results are shown in Table 9.1.

For lattice sizes less than 16 sites we expect to see perfect agreement as there is no truncation of the Hilbert space. Lattice sizes larger than 18 sites have some truncation of the Hilbert space, and we may expect to see some discrepancy between the exact and the DMRG estimates. We see that for 24 sites, where the Hilbert space is reduced to 2% of the size of the exact calculation, the agreement is better than 1 part in 108 .

For larger lattice sizes ( N > 24), we can no longer rely on exact diagonalisation, as the Hilbert space grows beyond the computational resources currently available. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 159

N J' I J = l (PBC) J'/J = 0 (OBC) Hilbert Space Egxact / J <5EDMRG Egxactj J <5EDMRG 0 0 Exact DMRG 4 -2.00000000 round off -1. 61602540 round off 6 6 8 -3.65109341 round off -3.37 493260 round off 70 70 16 -7.14229636 round off -6.91173715 round off 12870 12870 20 -8.90438653 round off -8.68247333 9.8 X 10-9 184756 76022 22 -9.78688065 round off -9.56807588 2.0 X 10-8 705432 58090 24 -10.67001452 1.3 X 10-lO -10.45378576 3.1 X 10-8 2704156 77680

Table 9.1: Comparison of exact finite-size ground state energies with DMRG estimates, where the difference 8E[?MRG = (E[?MRG -E~xact)/ J. The difference between the DMRG and the exact results are quoted as "round off" if they agree to within machine precision, which is "' 1.0 x 10-11 . The Hilbert space size is also quoted for the two calculations.

N Hilbert Space Exact DMRG 32 -14.20652744 2.0 X lQ-8 6.0 X 108 74868 64 -28.37 430 3.0 X 10-6 1.8 X 1018 76256

Table 9.2: Comparison of exact finite-size ground state energies obtained by Bethe ansatz with DMRG estimates for J' / J = l, where 8E[?MRG = (E[?MRG - E~xact)/ J.

For the point J' / J = l, we may compare to Bethe ansatz results (Avdeev, 1990), shown in Table 9.2. We again see excellent agreement, with a relative error of 10-7 for 64 sites. The Hilbert space size used by each calculation is also shown, which again demonstrates the efficiency of the D MRG.

In a particular D MRG calculation we may estimate the error by varying the truncation parameter moMRa, which determines the number of basis states retained in a block. This in turn modifies the total Hilbert space of the calculation, and hence the accuracy of the calculation. Table 9.3 lists the convergence of the ground state energy and correlation (SfvSi) for J' / J = 0.5 using periodic and anti-periodic boundary conditions. We see the energy converges to better than 1 part in 106 for both periodic and anti-periodic cases. The correlations converge to better than

1 part in 104 . The larger error on the correlations is expected as the calculation RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 160

moMRG Periodic Anti-periodic Eo/1 (SivSi) Eo/J (SivSi) 96 -28.217652 -0.060888 -28.212779 -0.0560186 164 -28.217938 -0.061194 -28.213082 -0.0562742 234 -28.217964 -0.061219 -28.213123 -0.0563028 342 -28.217970 -0.061222 -28.213132 -0.0563091

Table 9.3: DMRG estimates of the ground state energy Eo/ J and correlation (SivSi.) for N = 64 sites at J' / J = 0.5 as a function of m, the number of states retained per block for periodic and anti-periodic boundary conditions imposed.

Lattice sites, N [4, 64] steps of 2 Weak link parameter, J' / J [0, 1] steps of 0.1 Boundary conditions periodic, anti-periodic Basis states kept per block, m 350 Total Hilbert state size "-' 100000

Table 9.4: A summary of the technical parameters used for the calculation.

involves evaluating the overlap of two wavefunctions, which introduces larger round off errors. In this example our final estimate for the ground state energy with PBC is E0 / J = -28.21797(1). In a collaboration with A. W. Sandvik we were able to obtain results using the stochastic series expansion (SSE) method, which is a variant of quantum Monte Carlo (Sandvik, 1999). For particular values of J', we are able to compare our DMRG results with SSE results. For this case the SSE method yields E 0 / J = -28.2178(3), agreeing perfectly to within errors.

9.3.2 Surface Energy

We now show the results of our calculations. Our calculational parameters are summarised in Table 9.4. To see the effect of the weak link, we examine the dependence of the ground state energy on varying strengths of the weak link J'. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 161

N=4 --0- N=8 -0- N=16 ----A---- N=32 -e-

N=64 ----- Extrapolation - - -

0.5

0.3

0.2

0.1

0 ,______....______.______.______. _____

0 0.2 0.4 0.6 0.8 J'/J

Figure 9.2: f:1EN = (Eo(N; J') - Eo(N; J' = J))/ J as a function of J' / J for the ring with periodic boundary conditions. The data are extrapolated using a polynomial fit to obtain the bulk limit.

0.4 ,------"T'""----"""T""------.------,.------, ~-~-. N=4 --0- N=8 -0- 0.35 N=16 ----A---- N=32 -e- N=64 ----- Extrapolation - - -

0.25

0.15

0.1

0.4 0.6 0.8 J'/J

Figure 9.3: As for Fig. 9.2, but with anti-periodic boundary conditions. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 162

Figure 9.2 displays a plot of the energy shift in the ground state energy

6EN = (Eo(N; J') - Eo(N; J' / J = 1))/ J (9.8) as a function of the strength of the weak link. Analogous results are shown for anti-periodic boundary conditions in Figure 9.3. We see that the presence of the weak link has a positive energy shift 6EN for all values of J' and N, and that the curves for the finite lattices converge to some bulk limit 6E00 ( J'). This curve may be estimated by performing a simple polynomial fit to extrapolate the bulk value for each J' value, shown as the dotted lines in the figures.

The point J' / J = 0 on this curve may be compared to existing analytical results. The finite size behaviour of the ground state energy density with open boundary conditions was obtained by Hamer, Quispel and Batchelor (1987) to be

2 . N oo f oo 7r smµ eo = eo + N - 24µN2 (9.9)

whereµ and e0 are defined in (8.27) and (8.28) respectively, and f 00 is

1r sin µ cos µ sin µ 100 f 00 = --- -·--- dk[l - coth(nk/4) tanh(µk/2)]. (9.10) 2µ 2 4 -00

This is a "surface" energy term, as it originates from the free ends of the chain. Combining this with the result for the J' / J = 1 case (8.26) and substituting into (9.8) we find that in the limit N - oo the energy difference at J' / J = 0 must be

6E00 (J'/J = 0) = Joo, (9.11) hence the energy difference arises entirely from the surface energy. This result RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 163

0.35 J'IJ=0.0 --0-- J'IJ=0.2 --0--, J'/J:0.4 0.3 ~ ...... J'/J=0.6 ...... • 111uu•••"' J'/J=0.8 ...... J'/J=1.0 ...... _, 0.25 • • 0.2 • • P, ,,,-·· 0.15 ,,,,,,,.... • • • • 0.1 8 8 /'88 8 0.05 0 0 ~ooo 0 0

0 0 0.05 0.1 0.15 0.2 0.25 1/N

Figure 9.4: The stiffness factor Ps as a function of 1/N, for lattice sizes N = 4 to 64.

should hold for both periodic and anti-periodic boundary conditions. We calculate our numerical results at the isotropic pointµ= 0, hence we obtain

7r - 1 .6.Eoo(J' / J = 0) = lim J00 = -- - ln 2 ~ 0.378. (9.12) µ->O 2

We see that the numerical results shown in Figures 9.2 and 9.3 agree with this result at J' / J = 0.0.

9.3.3 Spin Stiffness

The spin stiffness is calculated according to (9. 7), and our results are shown in Figure 9.4. There is a marked difference between the behaviour of the curves with J' / J = 1 and the others. At couplings other than J' / J = 1 the values trend steadily down towards zero as the lattice size N increases. The curve J' / J = 1 on the other hand approaches a non-zero limit, and the strong curvature towards zero RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 164 is not apparent. An exact result is available for the J' / J = 1 case, as obtained by Hamer, Quispel and Batchelor (Hamer et al., 1987)

Ps(J'/J = 1) = 1· (9.13)

The apparent discrepancy between the data and this result can be attributed to the presence of logarithmic corrections to the ground state energy, and hence the stiffness, as shown by Woynarovich and Eckle (1987). Such logarithmic effects were observed numerically by Laflorencie, Capponi and S0rensen (2001), who calculated the stiffness for the isotropic model using the Bethe ansatz. Their plot of the stiffness is reproduced here in Figure 9.5. Even for a lattice size of 10000 sites, the stiffness was still above 0.26. A similar effect was noted by Alvarez and Gros (2002), who studied the zero momentum limit of the Drude weight ( which approaches the spin stiffness), and found it well above 0.25 even for 512 sites. Our DMRG results are therefore consistent with these previous findings.

For J' / J < 1, following the discussion in Section 8.3.2, we expect that the presence of a weak link (or a barrier, in the language of conductance) of arbitrary magnitude acts as a completely insulating link in the bulk limit. The persistent current must therefore also reduce to zero for any strength of weak link, J' / J < 1.

Our data shown in Figure 9.4 confirms this prediction, as for all J' / J < 1 the stiffness, and hence the current approaches zero.

To examine the scaling behaviour of the stiffness, we plot the same data as shown in Figure 9.4 using a logarithmic plot, shown in Figure 9.6. The results are in accordance with a scaling behaviour

PN( J' / J) "'a( J' / J)N-a (9.14) RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 165

o Bethe Ansatz 0.275 - AG

0.27 0 0 0 p/J 0.265

0.26

0.255

0.25 -~~_._-~_ __.____..____,_~--'---~---' 0 0.001 0.002 0.003 0.004 0.005 1/L

Figure 9.5: The spin stiffness obtained by Laflorencie, Capponi and S0rensen (2001) by Bethe ansatz and renormalization group methods.

where <7 ~ 2/3 according to the plot. In a collaboration with A. W. Sandvik, inde pendent results using SSE Quantum Monte Carlo methods (Byrnes et al., 2002b) were able to reproduce this 2/3 exponent result. Figure 9.7 shows that a simple scaling form

(J') rv 2.6J' / J N-2/3 N --+ 00, (9.15) PN (l-J'/J) , fits the data quite well. It is likely, however, that the true asymptotic correction to-scaling behaviour is again being disguised by logarithmic corrections, and that the v_alue <7 ~ 2/3 is only an "effective exponent" valid over the present range of

N values, rather than the true asymptotic exponent.

9.3.4 Spin-Spin Correlation across the weak link

To further examine the effect of the weak link, we calculate the spin-spin cor relation (SfvSi). Our results are shown in Figure 9.8. We see for all values of the RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 166

-1 ...... A A A &AA&A•••••••••111HHH -1.5 • • • • • • • • ••••• -2 • • • 6 • ••• 6 6 ••••••• 6 6 -2.5 6 6e,. - ln(p.) D -3 D """""~ D D D J'/J=0.2 --0-- Doo -3.5 J'/J=0.4 • 6.- ' 0 J'/J=0.6 .... , Do ~ J'/J=0.8 ,__., -4 J'/J=1.0 .....,,.._.

-4.5 1 1.5 2 2.5 3 3.5 4 4.5 ln(N)

Figure 9.6: The same data as Figure 9.4 but plotted on a log-log scale.

0.40

0.30

0.20 0----0 J'/J= 1/32 -t::. 0.15 ._. J'/J= 1/1 6 0.. [3-----£) J'/J= 1/8 __. J'/J= 1/4 l'r--6 J'/J= 1/2 0.10 lt------1. J'/J=3/4 0--0 J'/J= 7/8 +--+ J'IJ= 15/16 +---* J'/J=31 /32 --- p/X=2.6[(1-x)312Nrz13

Figure 9. 7: Stiffness Ps obtained through SSE Quantum Monte Carlo methods, courtesy of A. W. Sandvik. x = J' / J in the figure. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 167 weak link the correlation approaches a finite value in the bulk limit, except for the special case J' / J = 0 where the link is open. At J' / J = 1 the correlations approach the expected value of

(SNS{)=~(~ - ln2) ~ -0.148, (9.16) where the term in the bracket is the ground state energy density (8.28), and the

1/3 follows since we are at the isotropic point ~ = 1. The finite size corrections for this point must also be rv 1/N 2 , according to (8.26). Theoretical expectations

(I. Affleck, private communication) are that the correlation function for all J' / J should approach its bulk limit like 1/N 2 , up to logarithmic corrections. Figure 9.8 shows our results are consistent with this prediction.

Our results in this section may appear to contradict our previous statement that a weak link should renormalize to an open chain in the bulk limit. One may expect according to these principles that the correlation across the weak link will approach zero for all J' / J < 1 in the bulk limit N ----. oo, which was the case for the stiffness. This expectation is in fact incorrect, due to a critical difference: the stiffness is a bulk quantity, while the correlation calculated here is a local quantity.

It is integrated out in the early stages of renormalization and then remains fixed, while longer-range correlations will keep renormalizing. It is only bulk properties such as the spin stiffness that scale to the value of the open chain. Nevertheless, the data of Figure 9.8 do show a weak renormalization effect. While the data for

J' / J = 1 remain essentially flat and independent of N, the data for lower values of J' / J exhibit a weak upturn at large N. RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 168

0 "-00 0 0 0 0 0 0 0

-0.02 r.1111111111111111 Doo o o D D D D D D J'/J = 0.0 ,-0-. -0.04 J'tJ = 0.2 • D · ...... _. J'/J = 0.4 ~ 8- ' MAe.~6 6 6 8 8 8 8 8 8 ·0.06 J'/J=0.6 ~---· J'/J =0.8 ...... J'/J = 1.0 ··•··' -0.08 - ...... • • • • • • -0.1 - -0.12 •••• • • • • • •

-0.14 IIUUU,&&& & ......

·0.16 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 1/N 2

Figure 9.8: The correlation (SNSi.) across the weak link for various J' / J for the periodic chain.

9.4 Conclusion

We have performed a DMRG study of the Heisenberg ring with a single weak link J' up to 64 lattice sites. The spin stiffness was calculated by evaluating the difference between ground state energies of periodic and anti-periodic rings, for var ious weak link strengths J' / J. This gives the correct value of the stiffness assuming that the dependence of the energy is purely quadratic in the twist parameter (mag netic flux) . We calculate the stiffness as an indirect measure of the strength of the persistent current in the equivalent Jordan-Winger transformed chain of short range interacting spinless fermions. The stiffness was found to scale to zero in the bulk limit N - oo, for any value of the weak link J' / J < 1. This agrees with the renormalization group predictions of Kane and Fisher (1992a; 1992b ), and Eggert and Affi.eck (1992; 1995). Theory predicts that the stable fixed point of the system corresponds to an open chain, which is in agreement with our data. The stiffness RESULTS FOR HEISENBERG CHAIN WITH A WEAK LINK 169 approached a constant value where there was no weak link present (J' / J = 1), as expected. In terms of the persistent current, we therefore find that a non-zero cur rent will only be present for the case with no weak link, and for any other strength of weak link ( J' / J < 1), the current will drop to zero. The spin-spin correlation was also measured across the weak link. A finite anti ferromagnetic correlation remains in the bulk limit, depending on the coupling

J'. Unlike the stiffness, here the correlations only scale to zero for a true open chain J' / J = 0. The renormalization group arguments in this case do not apply, as the correlation is a "local" quantity, while the stiffness and current are "bulk" quantities.

Both these quantities suffer from the presence of logarithmic finite-size scal ing corrections, and therefore the true asymptotic behaviour is disguised to some extent. To circumvent this problem, Eggert and Affleck (1992; 1995) include next nearest neighbour terms at the point 12/ J1 = .2411. At this special point logarith mic corrections vanish, and hence the "true" asymptotic behaviour is more easily observed. The next-nearest neighbour term acts as a marginal operator, and hence is included to eliminate the effects of the logarithmic corrections. For our purposes however, we found this unnecessary. Chapter 10

Conclusion

In this thesis we applied the DMRG method to two lattice models, the massive

Schwinger model and the Heisenberg chain with a weak link. We find in both models the DMRG to be an extremely effective tool in studying quantities such as the ground state energy, gaps and various expectation values. In particular for the Schwinger model we are able to improve by up to two orders of magnitude the accuracy of several quantities, for example the position of the critical point in the case of 0 = 1r, and the "vector" state mass gaps for 0 = 0. The power of the method lies in its ability to obtain highly accurate results for very large systems sizes N ~ 256, with relative computational ease. This is in comparison to competing methods such as exact diagonalisation, which is restricted to much smaller lattice sizes, and Monte Carlo methods, which are plagued by the minus sign problem.

Let us now summarise the main results of this thesis. The Schwinger model was studied in two regimes, 0 = 1r and 0 = 0. The most interesting results were obtained for 0 = 1r, where we studied the region close to the critical point. We found:

170 CONCLUSION 171

• Location of the critical point to excellent accuracy (m/ g )c = 0.3335(2)

• Identification of critical exponents v = 1.01(1) and /3/v = 0.125(5), putting the Schwinger model in the same universality class as the transverse Ising model

• Location of "half-asymptotic" particles, as predicted by Coleman (1976)

• Calculation of the spectrum of 0-, 1- and 2-particle gaps, and order parame

ters ro: and r 5

• Interpretation of the ground state as a "kink condensate" for m/g < (m/g)c

• Identification of a possible "dual symmetry" or "dual order parameter"

Particularly interesting was the analogy of the spectrum of states with the spectrum of the transverse Ising model. The 1-particle state was found to have a close analogy with the 1-kink state of the Ising model. In terms of the bosonized fields, the 1- particle state is a soliton solution of the field equations, and hence is itself a "kink" state between the degenerate ground states. This led to the interpretation of the ground state in the low mass region as a "kink condensate", and the interpretation of a possible "dual symmetry" and "dual order parameter" in the model. We have unfortunately not found the explicit form of the symmetry, or the disorder parameter, but these are possible avenues of investigation in the future. Our results for the Schwinger model at 0 = 0 were

• Extension of the non-relativistic expansion to third order

• DMRG calculation of the "vector" and "scalar" states, improving the accu

racy by two orders of magnitude in the low mass region CONCLUSION 172

Our main motivation here is to improve upon previous results, and demonstrate the applicability of the DMRG method. To our knowledge this is the first appli cation of DMRG to a non-trivial model in high-energy physics, and our success is encouraging for future studies of related models. Of particular interest is the two-flavor Schwinger model, which has a global SU(2) symmetry that can be iden tified with isospin. The particles of the theory therefore reside in isomultiplets, with pion-like excitations (Kenway and Hamer, 1978), hence has some additional interesting properties that may be examined. QCD (l+l)D is another possibility for an application of D MRG.

Unfortunately, the comparative advantage DMRG has in (l+l)D does not exist in higher dimensions. There is no reason a priori why DMRG should not work in higher dimensions, hence a large effort is currently in progress to develop new 2D methods (Xiang et al., 2001; McCulloch et al., 2001). In light of our success with the Schwinger model, an efficient higher dimensional DMRG algorithm would pro vide an excellent alternative to current Monte Carlo methods, with the additional advantage of no minus-sign problem. We are currently in investigation of various

2D DMRG algorithms.

Finally we studied the Heisenberg ring with a single weak link. Our main results were

• Verification of the renormalization group predictions of Kane and Fisher via

observation of the spin stiffness renormalizing to zero for all J' / J < l

• Calculation of the spin-spin correlation across the weak link, which does not

renormalize, due to the "local" nature of this quantity

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21, 533. Appendix A

Weak Coupling Expansion of the

Massive Schwinger Model

Our task is to expand the Hamiltonian (2.24) in the two particle subspace. For reference the Hamiltonian is repeated here:

(A.1) where

Ho Jdxi/;(-h181 + m)'l/J (A.2) H1nt - - g: Jdxdyjo(x)jo(Y)lx - YI (A.3) H[nt - -Fg Jdxxjo(x). (A.4)

Let us start by writing down a few things about operators in (l+l)D.

183 WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL184

A.1 Dirac Fields in (l+l)D

The Dirac field has an expansion

'ljJ(x) = ~ Jdp [a(p)u(p)eipx + bt(p)v(p)e-ipx] , (A.5) where a(p) and bt (p) are destruction and creation operators satisfying

{a(p), at (p')} = {b(p), bt (p')} - <5(p - p') (A.6)

{ a(p), a(p')} = {b(p), b(p')} 0. (A.7)

We also choose normalisation such that ut(p)u(p) = vt(p)v(p) = 1. The u(p) and v(p) satisfy the Dirac equation, if we choose a representation

(A.8)

we can write

u(p) (A.9)

v(p) (A.10)

where Ep = (p2 +m2 ) 112 . These are normalised such that ut(p)u(p) = vt(p)v(p) = 1.

We will consider matrix elements in the two-particle subspace (Coleman, 1976), containing one quark of momentum p and an antiquark of momentum q. Such a WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL185 state is normalised such that

(p', q'Jp, q) = 8(p' - p)8(q' - q). (A.11)

In a center of mass reference frame, q = -p, so a Hamiltonian matrix element is

(p', q'IHJp, -p) = 8(p' + q')(p'IHRIP), (A.12) where HR is the reduced centre of mass Hamiltonian.

A.2 Evaluation of Matrix Elements

A.2.1 The Free Hamiltonian Hf

We may now derive the reduced free Hamiltonian H{; shown in (2.26). The calculations shown in this section will be quite explicit, as later sections are simple extensions of the method used here. Explicitly substituting (A.5) into the matrix element gives

(p', q'IHolP, -p) 2~ Jdxdp1dp2(0la(p')b(q')

x [at (P1)ut (p1)e-ipix,'o (-h181 + m) a(p2)u(p2)eip 2 x

+at (P1)ut (P1)e-ipix')'o ( -i')'181 + m) bt (P2)v(p2)e-ip2x

+b(p1)vt(P1)eipix (-h181 + m) a(p2)u(p2)eip 2 x

+b(p1)v t (p1)eipix ( -hi 81 + m) bt (p2)v(p2)e-ip2x] at (p )bt ( -p) JO). WEAK COUPLING EXPANSION OF THE MASSIVE SCHWINGER MODEL186

Only the first and last term in the square brackets give nonzero matrix elements, with restrictions on the momenta such that

(p', q'IHolP, -p) j dp1dP2

x [6(p' - P1)6(p2 - p)6(p1 - P2)6(q' + p)ut(P1ho (-i2'Y1P2 + m) u(p2)

-6(p - p')6(p1 + p)6(p2 - q')6(p1 - P2)vt(P1ho (i2')'1P2 + m) v(p2)]

- 6(p - p')6(q' + p) [ut(p) b1P + m) u(p) - vt(-p) b1P + m) v(-p)]

We can use the relations

p u t (p ho1'1 u (p) - Ep p vt(-p),0')'1V(-p) EP m ut(p),ou(p) EP m vt(-p),ov(-p) E'p which finally gives

(p', q'IHolP, -p) = 26(p - p')6(p + q')(p2 + m2) 112 . (A.13)

In operator form, this is

(A.14) as given in (2.27). WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL187

A.2.2 The Interaction Hamiltonian H{nt

Now let us examine the next term in the Hamiltonian H!nt, which may be written as

(A.15) where colons denote normal ordering. Again we substitute explicit expressions of the field operators (A.5). If we evaluate this in the two particle subspace as in the previous section, many of the terms immediately give zero, as particle numbers are not conserved. Only six of the 16 terms survive, giving

(p', q'IH1ntlP, -p) = - l61rg2 Jdxdydp1dp2dp3dp4lx - YI (Ola(p')b(q') 2

X [: at(P1)ut(P1)a(p2)u(p2) :: at(p3)ut(p3)a(p4)u(p4): ei[(p2-Pi)x+(P4-P3)Y]

+: at(P1)ut (P1)a(p2)u(p2) :: b(p3)vt(p3)bt(p4)v(p4) : ei[(p2-Pi)x+(P3-P4)Y]

+: b(p1)vt(P1)a(p2)u(p2) :: at(p3)ut(p3)bt(p4)v(p4) : ei[(pi+P2)x-(P3+P4)Y]

+: b(p1)vt(P1)bt(P2)v(p2) :: at(p3)ut(p3)a(p4)u(p4): ei[(pi-p2)x+(P4-P3)Yl

+: b(pi)vt(p1)bt(p2)v(p2) :: b(p3)vt(p3)bt(p4)v(p4): ei[(P1-P2)x+(p3-p4)y]

+: at (P1)ut (p1)bt(P2)v(p2) :: b(p3)vt(p3)a(p4)u(p4) : ei[-(pi+P2)x+(P3+P4)Y]]

xat (p)bt (-p)IO) (A.16)

It is easily verified that the various contractions of the terms above give the di agrams (a) (b) (c) (d) shown in Figure 2.2. The remaining two diagrams in the figure originate from interaction Hamiltonian H[nt. Let us now explicitly evaluate contributions to the four diagrams of Figure 2.2. WEAK COUPLING EXPANSION OF THE MASSIVE SCHWINGER MODEL188

Coulomb Force H f

Contributions to the Coulomb scattering term Figure 2.2(a) originate from the second and fourth terms in (A.16). Once the contractions associated with Coulomb scattering are made, we obtain

(p', q'JH1IP, -p) = - 1::2 / dxdydp1dp2dp3dp4jx - YI

X [ - 8(p1 - p')8(p2 - p)8(p3 + p)8(p4 - q')ut(P1)u(p2)vt (p3)v(p4)ei[(p2 -Pi)x+(P3 -P4 )Y]

-8(p1 + p)8(p2 - q')8(p3 - p1 )8(p4 - p)vt(P1)v(p2)ut(p3)u(p4)ei[(pi-P2 )x+(PrP3 )Y]].

Exchanging dummy variables reveal that the two terms in the brackets are equal.

Changing variables to x+ = ½(x+y) and x_ = x-y, and performing the momentum integrations we obtain

92 (p', q'IH1IP, -p) - 2 Jdx_dx+lx-1 81r x u t (p')u(p )v t ( -p )v( q') exp [-i(p' + q')x+] exp [i(p - ½(p' - q') )x_]

Using the result

1 1 (A.17) (p+iE)2 (p-iE)2 -2P_!__ (A.18) p2 where principal value symbol in the second line symbolically denotes the complex integration which must be performed. This gives us

92 (p',q'JH1IP,-P) = - (, p )2ut(p')u(p)vt(-p)v(q1 )8(p' +q'), (A.19) 27r p -p WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL189 which correctly reproduces the result obtained by Coleman (1976). Using the algebraic result (derived from equations (A.9) and (A.10) ),

(A.20) we obtain

(A.21)

If we note that

(A.22) we can write the operator form of H f:

(A.23) which is the result (2.28).

Self-Mass Hf

Contributions to the self-mass diagrams shown in Figure 2.2(6) and 2.2(c) orig inate from four terms in (A.16), namely the first, third, fifth and sixth terms. Each of these contribute equal amounts and similar calculations to the previous sections gives

x exp [i(p2 - p')x + i(p - P2)Y] b(p + q'). (A.24) WEAK COUPLING EXPANSION OF THE MASSIVE SCHWINGER MODEL190

Evaluating the x and y integrals with a change of variables gives

where the result (A.20) was used in the second line. The integration can be per formed following the steps given in the appendix of the paper by Coleman (1976).

We then obtain 2 (p'IHflP) = - gE o(p - p'). (A.25) 7r p

This can be written in operator form

(A.26)

Annihilation Force H!;

The matrix element for the annihilation diagrams shown in Figure 2.2( d) orig inate from contractions in the third and sixth terms in (A.16). The two terms are equal in magnitude and give

(p', q'IH3lp, -p) = -::2 Jdxdylx - ylvt(-p)u(p)ut(p')v(q')e-i(p'+q')x_ (A.27)

Again, changing variables and integrating gives

g2 2P (p', q'IH3lp, -p) =-;- (p' + q')2 vt(-p)u(p)ut(p')v(q')o(p' + q'). (A.28)

We encounter a problem here as substitution of expressions (A.9) and (A.10) give vt(-p)u(p) = ut(p')v(-p') = 0, with a simultaneous zero in the denominator. The WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL191 correct procedure (Coleman, 1976) is to assume a small nonzero total momentum D. of the quark and antiquark, and take the limit to zero. This involves a recalculation evaluating the matrix element (p' + D., q'IHalP + D., -p). This gives

2 2P (p' + D., q'IHalP + D., -p) = ~ (2D.) 2 vt(-p)u(p + D.)ut(p' + D.)v(-p')6(p' + q'). (A.29)

To leading order we have

(A.30) hence we obtain (A.31)

In operator form (A.32) as.given in (2.30).

A.2.3 The Interaction Hamiltonian H{nt

We now turn to the second term in the interaction Hamiltonian, describing the interactions with the background field F

H[nt = -gF Jdxx: 'if}(x)'ljJ(x) : . (A.33) WEAK COUPLING EXPANSION OF THE MASSIVE SCHWINGER MODEL192

Inserting the explicit expressions for the field operators, and evaluating the Hamil tonian in the two particle subspace gives only two nonzero terms

(p', q'IH{ntlP, q) - -;: Jdxdp1dp2x(Ola(p')b(q') [ : at (p1)ut (P1)a(p2)u(p2) : ei(pz-pi)x

+ : b(p1 )v t (P1 )bt (P2)v(p2) : ei(pi -pz)x] at (p )bt (q) IO), ( A.34) where we have not specialised to the case of total momentum zero, and assigned arbitrary momenta p and q for the initial quarks. Contractions of these two terms directly give the two diagrams shown in figure 2.2(e) and (f), which we will denote by H4 from here onwards. Similar procedures to previous sections give

(p', q'IH41P, q) = -;: Jdxx [ut (p')u(p)ei(p-p')xo(q - q') - vt (q)v(q')i(q-q')xo(p - p')] . (A.35) Using the result 1-: dxxeipx = -i21ro' (p), (A.36) where o'(p) = d~o(p) = -o(p)/p, we obtain

(p', q'IH41P, q) = igF [o'(p - p')o(q - q') - o'(q - q')o(p - p')l. (A.37)

Using the relation o'(a)b(a + b) = b'(a)b(b) - b(a)b'(b), and letting q = -p, the matrix element is

(p', q'IH41P, -p) = igFb'(p - p')b(p' + q'). (A.38)

We may write this in operator form using the result

(p'lxlp) = -ib'(p - p'), (A.39) WEAK COUPLING EXPANSION OF THE MASSIVE SCHWING ER MODEL193 hence we obtain Hf = -gFx, (A.40) which agrees with (2.31). Appendix B

The Transfer Matrix

The action from which we want to construct a transfer matrix from is

(B.1)

where we have simplify slightly by replacing 01 - 0. The general procedure followed here is generalised from the review article by Kogut, (Kogut, 1979). First consider the transition amplitude Z = JV0exp[-S], (B.2) where

(B.3)

Multiplying out the time product,

Z = J( [[ dO(nx, n, = 1)) ([[ d0(nx, n, = 2)). ([[ d0(nx,n, = N,)) e-s (B.4)

194 THE TRANSFER MATRIX 195 let us symbolically denote this as

(B.5) where

(B.6)

If we call

T(0n,+l, 0n,) = exp [ ~/Ja,. ~?0(n,,, nr + 1) - 0(n., nr )) 2] , (B. 7) then we can rewrite the transition amplitude

Z = JII [d0n,.T(0ndl, 0n,.)]. (B.8) n,-

Our task is then to find an operator T such that we can write

Z = JII [d0n,. (0n,.+I ITl0n,.)] , (B.9) n,- where (B.10)

To find such an operator, we must introduce a canonical momentum operator £( nx), obeying commutation relations

[£(nx), 0(nx)] = -i. (B.11) THE TRANSFER MATRIX 196

Following (Kogut, 1979), we guess that a suitable expression for the transfer matrix may be related to the quantity exp[Lnx i,2(nx)]. Let us evaluate its matrix element:

(0'1 exp [ ~?'{nx)] 10) = j dLdL' (0'IL')(L'I exp [ ~?'(nx)] IL)(Ll0)

- JdLdL' ( TI exp [iL'(nx)0'(nx)]) ( TI exp [L2(nx)])

x6(L' - L) ( TI exp [-iL(nx)0(n,,)])

JdL ( TI exp [iL(nx)(0'(nx) - 0(nx))]) ( TI exp [L2 (nx)]) - const. IT exp [0(nx) - 0'(nx)], (B.12) where we used the result

(Ll0) = IT exp [-iL(nx)0(nx)]. (B.13)

We can therefore conclude that the transfer matrix is

(B.14)

The Hamiltonian is related to the transfer matrix via

A 1 A H = --lnT, (B.15) aT

hence the Hamiltonian is

(B.16) Appendix C

Finite Size Behaviour of the

Transverse Ising Model

In section 4.5.2 we obtained an expression

F(>.., N) = 2(1 - >..) + 2N[T2N(>..) - TN(>..)] for the gap of the transverse Ising model on a finite lattice. Following reference (Hamer and Barber, 1981), we change notation to

F(>.., N) = 2(1 - >..) + 2N~[T2N(>..) - TN(>..)], (C.2) where (C.3) with

A.(0) = ../a 2 + 2 - 2cos0 (C.4)

197 FINITE SIZE BEHAVIOUR OF THE TRANSVERSE ISING MODEL 198 and

ci = (1 - -\) 2 /,\. (C.5)

Let us expand A.(0) in a Fourier series

00 (C.6) n=-oo where the coefficients are

Cn = ~ 11r e-inO A(0)d0. (C.7) 27!' -Tr

If we substitute (C.6) into (C.3), we obtain

oo N-1 TN(A) = t L Cn L ei21r(n/N)k. (C.8) n=-oo k=O

We may use the result that for any integer p and q,

f ei21r(p/q)k = { q p / q = integer (C.9) k=O O otherwise, since the sum is over points on a unit circle. The only allowed values of n in (C.8) therefore are n = 0, ±N, ±2N, .... This gives

CX) TN(,\) =co+ L(clN + c_lN)- (C.10) l=l

For large number of lattice sites N ----> oo, the second term above will give a van ishing contribution, as the exponential in ( C. 7) oscillates very quickly. The limit of TN(,\) is therefore c0 for N ----> oo. However we are in fact interested in the finite size behaviour, and since c0 is independent of N, we shall concentrate on obtaining FINITE SIZE BEHAVIOUR OF THE TRANSVERSE ISING MODEL 199

(a) (b) lm(w) Im(w) X 7t X 7t ½ (-w c• rr/'2) Re(w)(-wc, n!ZJ ------J------J Re(w)

___--4_,______: JRe(w)=~

X -7t X

Figure C.1: (a) Contour of integration for (C.14). Crosses indicate non-analytic points given in (C.13). (b) The deformed contour C2.

an expression of ck for large k. To do this, we make a change of variables w = i0 /2 and We= sinh-1 (n:/2), which gives

(C.11) where

A.(0) = 2Jsinh(we + w) sinh(wc - w). (C.12)

We now must identify any non-analytic points in the integrand. The square root function is non-analytic at points where its argument is zero, which gives non analytic points at

Re(w) = ±we Im(w) = 0, ±1r, ±21r, ... (C.13) and is analytic everywhere else.

Let us make another change of variables, w' = w - We, such that the contour runs along C1 , as shown in Figure C.l (a). This gives

2e-2kwc 1 2 112 Ck· . e- kw' [sinh(-w') sinh(w' + 2we)] dw', (C.14) 7r'l C1 FINITE SIZE BEHAVIOUR OF THE TRANSVERSE ISING MODEL 200 where we will for the moment assume that k is positive. The integral may be eval uated by deforming the contour of integration to run along C2, as shown in Figure

C.l (b). A little working shows that the contribution from points (-wc,1r/2)--.

( oo, 1r /2) cancels with the contribution from ( oo, -1r /2) --. ( -we, -1r /2). Likewise, the two paths along Re(w) = oo contribute zero each, as long as k > 1/2. This leaves only the integrations above and below the real axis, which gives

2e-2kWc I [ • 1/2 ck . 100 e-2kw (e2ik€sinh(-w'+ic)sinh(w'+2wc-ic)) 7r1, 0 - ( e-2ik€ sinh(-w' - ic) sinh( w' + 2wc + ic)) 112 ] dw'. (C.15)

We want to obtain an expression for this integral for large k. Due to the factor of e-2kw' in the integral, the dominant contribution comes from w' ::;; 1/k « 1. We can then expand

sinh(-w' ± ic) ~ -w' ± it: ~ -w'e~iE/w', (C.16)

as long as E « w'. Substituting we obtain,

2e-2kwc {oo , [ ( . , ) 1/2 Ck ~ 'Tri Jo e-2kw ei(1r+2kE-E/w )w' sinh(2wc)

112 - ( ei(1r-2h+E/w')w' sinh(2wc)) ] dw'. (C.17)

Since 1/w' 2 k, the exponentials become

112 . / , ] 1/2 [ ei(1r+2kE-E/w')] ~ [ei(1r-€ w) ~ +i

[ei(1r-2h+E/w')] 112 ~ [ei(11-+E/w')] 1/2 ~ -i. FINITE SIZE BEHAVIOUR OF THE TRANSVERSE ISING MODEL 201

We therefore obtain

-2kwc( • h(2 ))1/2100 ,..,_, 4 e Sln We -2kw' ( ') 1/2 d t Ck,.,_,------e W W. (C.18) 7f 0

Using the standard integral

(C.19) we obtain 2e-2kwc(sinh(2wc) )112 (C.20) Ck ~ y'7r(2k )3/2

Now we can see the behaviour of the finite size corrections to F(A, N). For large N, we may approximate in (C.10)

(C.21)

since ck = c_k as A( 0) is an even function. This gives

(C.22)

Substituting into (C.2) gives

e-2Nwc F(\ N) ~ 2(1 - A)+ const . .JN , (C.23) as given in (4.49).