TEXTS AND READINGS IN PHYSICAL SCIENCES Field Theories in Condensed Matter Physics Texts and Readings in Physical Sciences Managing Editors H. S. Mani, Harish-Chandra Research Institute, Allahabad. Ram Ramaswamy, lawaharlal Nehru University, New Delhi. Editors Deepak Dhar, Tata Institute of Fundamental Research, Mumbai. Rohini Godbole, Indian Institute of Science, Bangalore. Ashok Kapoor, University of Hyderabad, Hyderabad. Arup Raychaudhuri, Indian Institute of Science, Bangalore. Ajay Sood, Indian Institute of Science, Bangalore. Field Theories in Condensed Matter Physics Sumathi Rao Harish-Chandra Research Institute Allahabad ~o0 HINDUSTAN U l1U UBOOKAGENCY Published by Hindustan Book Agency (India) P 19 Green Park Extension, New Delhi 1 \0 016 Copyright © 2001 by Hindustan Book Agency ( India) No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, induding photocopying, recording or by any informa­ tion storage and retrieval system, without written perm iss ion from the copyright owner, who has also the sole right to grant Iicences for translation into other languages and publication thereof. All export rights for this edition vest excIusively with Hindustan Book Agency (India). Unauthorized export is a violation ofCopy­ right Law and is subject to legal action. ISBN 978-81-85931-31-9 ISBN 978-93-86279-07-1 (eBook) DOI 10.1007/978-93-86279-07-1 Texts and Readings in the Physical Sciences As subjects evolve, and as the teaching and study of a subject evolves, new texts are needed to provide material and to define areas of research. The TRiPS series of books is an effort to doc­ ument these frontiers in the Physical Sciences. One of the principal aims of the series is to make expositions of current topics accessible to the graduate student and researcher through Textbooks and Monographs. In addition, we also feel that publication of lecture notes emanating from a thematic School or Workshop, and topical volumes of contributed articles can go a long way in providing an insight into a rapidly developing field, or an introduction to a new area. The pedagogical value of all these forms of exposition is ines­ timable. 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H S Mani R Ramaswamy Allahabad New Delhi Contents Preface Xlll Introduction 1 1 Quantum Many Particle Physics 7 Pinaki M ajumdar 1.1 Preamble .. 8 1.2 Introduction. 8 1.3 Introduction to many particle physics 10 1.3.1 Phases of many particle systems 10 1.3.2 Quantities of physical interest .. 12 1.3.3 Fermi and Bose liquids. 14 1.4 Phase transitions and broken symmetry 22 1.4.1 Phase transitions and symmetry breaking 22 1.4.2 Symmetry breaking and interactions in BEC 25 1.5 Normal Fermi systems: model problems . 31 1.5.1 Neutral fermions: dilute hardcore Fermi gas 34 1.5.2 Charged fermions: the electron gas ..... 39 1.6 Electrons and phonons: Migdal-Eliashberg theory . 46 1.6.1 Weak coupling theory: BCS ..... 48 1.6.2 The normal state: Migdal theory . 52 1.6.3 BCS theory: Greens function approach. 56 1.6.4 Superconductivity: Eliashberg theory 58 1.7 Conclusion: 'field theory' and many particle physics 63 viii CONTENTS 2 Critical Phenomena 69 Somendra M. Bhattacharjee 2.1 Preamble . 70 2.1.1 Large system: Thermodynamic limit 72 2.2 Where is the problem? . 72 2.3 Recapitulation - A few formal stuff 74 2.3.1 Extensivity 74 2.3.2 Convexity: Stability 76 2.4 Consequences of divergence 78 2.5 Generalized scaling 81 2.5.1 One variable: Temperature 81 2.5.2 Solidarity with thermodynamics 87 2.5.3 More variables: Temperature and field 88 2.5.4 On exponent relations 92 2.6 Relevance, irrelevance and universality 93 2.7 Digression. 95 2.7.1 A first-order transition: a=1 95 2.7.2 Example: Polymers: no "ordering" 97 2.8 Exponents and correlations 99 2.8.1 Correlation function 99 2.8.2 Relations among the exponents . 101 2.8.3 Length-scale dependent parameters . 103 2.9 Models as examples: Gaussian and <p4 105 2.9.1 Specific heat for the Gaussian model 106 2.9.2 Cut-off and anomalous dimensions 107 2.9.3 Through correlations . 110 2.10 Epilogue . 112 3 Phase Transitions and Critical Phenomena 119 Deepak K umar 3.1 Introduction. ..... ., ... 120 3.2 Thermodynamic stability 121 3.3 Lattice gas : mean field approximation. 126 3.4 Landau theory 134 CONTENTS ix 3.5 Spatial correlations ........... 138 3.6 Breakdown of mean field theory .... 141 3.7 Ginzburg-Landau free energy functional 143 3.8 Renormalisation group (RG) ..... 144 3.9 RG for a one dimensional Ising chain . 146 3.10 RG for a two-dimensional Ising model 150 3.11 General features of RG ...... 158 3.11.1 Irrelevant variables . 163 3.12 RG scaling for correlation functions . 164 3.13 RG for Ginzburg-Landau model .. 167 3.13.1 Tree-Ievel approxi~ation .. 170 3.13.2 Critical exponents for d > 4 172 3.13.3 Anomalous dimensions .. 175 3.14 Perturbation series for d < 4 ... 176 3.15 Generalisation to a n-component model 183 4 Topological Defects 189 Ajit M. Srivastava 4.1 The subject of topological defect 191 4.2 What is a topological defect? . 193 4.2.1 Meaning of order parameter. 194 4.2.2 Spontaneous symmetry breakdown(SSB) . 195 4.2.3 SSB in particle physics . 197 4.2.4 Order parameter space . 197 4.3 The domain wall . 198 4.3.1 Why defect? 200 4.3.2 Why topological? . 201 4.3.3 Energy considerations 202 4.4 Examples of topological defects 203 4.5 Condensed matter versus particle physics 209 4.6 Detailed understanding of a topological defect . 213 4.6.1 Free homotopy of maps . 216 4.6.2 Based homotopy and the fundamental group 217 4.7 Classificatiun of defects using homotopy groups 219 4.8 Defect structure in liquid crystals . 227 x CONTENTS 4.8.1 Defects in nematics ........ 228 4.8.2 Non abelian 11"1 - biaxial nematics . 230 4.9 Formation of topological defects ..... 231 5 Introduction to Bosonization 239 Sumathi Rao and Diptiman Sen 5.1 Fermi and Luttinger liquids 240 5.2 Bosonization . .................... 247 5.2.1 Bosonization of a fermion with one chirality 248 5.2.2 Bosonisation with two chiralities . 257 5.2.3 Field theory near the Fermi momenta . 265 5.3 Correlation functions and dimensions of operators 268 5.4 RG analysis of perturbed models . 272 5.5 Applications of bosonization. 231 5.6 Quantum antiferromagnetic spin 1/2 chain . 282 5.7 Hubbard model . 300 5.8 Thansport in a Luttinger liquid - clean wire 309 5.9 Thansport in the presence of isolated impurities 319 5.10 Concluding remarks . ... .. .. .. .. .. 328 6 Quantum Hall Effect 335 R. Rajaraman 6.1 Classical Hall effect . 336 6.2 Quantized Hall effect 337 6.3 Landau problem .. 338 6.4 Degeneracy counting 340 6.5 Laughlin wavefunction 341 6.6 Plasma analogy . 342 6.7 Quasi-holes and their Laughlin wavefunction 344 6.8 Localization physics and the QH plateaux 345 6.9 Chern-Simons theory . 348 6.10 Vortices in the CS field and quasiholes 354 6.11 Jain's theory of composite fermions .. 355 CONTENTS xi 7 Low-dimensional Quantum Spin Systems 359 Indrani Base 7.1 Introduction ..................... 360 7.2 Ground and excited states . 365 7.3 Theorems and rigorous results for antiferromagnets 369 7.3.1 Lieb-Mattis theorem . 369 7.3.2 Marshall's sign rule . 370 7.3.3 Lieb, Schultz and Mattis theorem 372 7.3.4 Mermin-Wagner theorem ..... 376 7.4 Possible ground states and excitation spectra . 376 7.5 The Bethe Ansatz . 387 Preface The idea of holding an SERC school on 'Field theories in con­ densed matter physics' came through discussions with Diptiman Sen, from the Centre for Theoretical Studies in Bangalore, Somen­ dra Bhattacharjee from the Institute of Physics in Bhubaneswar and R. ShankaF- from the Institute of Mathematical Sciences in Chennai. All of us feIt that field theoretic methods are widely used in studying a variety of phenomena in condensed matter physicsj so it would be a good idea to expose graduate students from different parts of the country to these methods. After a few rounds of discussions, we converged on a set of topics and lectur­ ers, and then I wrote up a proposal to the Department of Science and Technology based on this, which got approved. The school was held at the Harish-Chandra Research Institute (formerly Mehta Research Institute of Mathematics and Mathe­ matical Sciences) from February 13 to March 4, 2000. All the lecturers we contacted proved enthusiastic about the idea and came for the school fully prepared for their courses. With over forty participants and twelve lecturers from all over the country, it turned out to be a great experience, where the lecturers as well as the students benefited from the interaction. The success of such a school depends mainly on the studentsj I must thank them for their whole-hearted participation during the school, which was very encouraging for the lecturers and for their warm and genuine feedback. Most of the lecturers agreed to write up their lecture not es for publication and this book is the result of their cooper­ ation.