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 <I>. 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 expan- sion 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 </>.