TEXTS AND READINGS IN PHYSICAL SCIENCES

Field Theories in 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

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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. We thus hope, in this series of books, to provide both a forum for the physical scientist to give a personal account and definition of his field, and a source of valuable learning material for the student wishing to gain insight and knowledge.

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

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. My thanks to all of them. I also thank Peali Majumdar, Dileep Jatkar and Radhika Vathsan for their help with the prepa• ration of the lecture notes for publication. It goes without saying that the organisation of such a school is XIV Preface impossible without the efforts and cooperation of several people. Thanks are definitely due to my two condensed matter colleagues, Dattu Gaitonde and , who were with me right from the beginning. I also received a lot of practical organisa• tional help from our students and post-doctoral fellows, Pradeep Mohanty, Saurabh Bannerjee, Sourin Das, Tribikram Gupta and Sanjeev Kumar. Special thanks are also due to our highly effi• cient administrative staff, who ensured the smooth running of the school. I would also like to thank the rest of the academic mem• bers of the institute for participating in the school; in particular, I would like to thank for constant encouragement and for attending all the lectures with great interest. Finally, I must thaIik Prof. H.S. Mani, the director of the institute who was sup• portive of the school, right from its inception to its culmination now in the form of this book. List of Lecturers

1. Prof. G. Baskaran, Institute 01 Mathematical Sciences, Chennai 2. Prof. Somendra M. Bhattacharjee, Institute 01 Physics, Bhubaneswar 3. Prof. Indrani Bose, J. C. Bose Institute, Kolkata 4. Prof. P. Durganandini, Dept. 01 Physics, Pune University, Pune 5. Prof. , Jawaharlal Nehru University, New Delhi 6. Prof. Pinaki Majumdar, Harish-Chandra Research Institute, Allahabad 7. Prof. R. Rajaraman, Jawaharlal Nehru University, New Delhi 8. Prof. T. V. Ramakrishnan, Indian Institute 01 Science, Bangalore 9. Prof. Sumathi Raa, Harish-Chandra Research Institute, Allahabad 10. Prof. R. Shankar, Institute 01 Mathematical Sciences, Chennai 11. Prof. Diptiman Sen, Indian Institute 01 Science, Bangalore 12. Prof. Ajit Srivastava, Institute 01 Physics, Bhubaneswar

List of Participants

1. Mr. Atul Bharadwaj, , New Delhi

2. Ms. Sarmishtha Bose, Indian Statistical Institute, K olkata

3. Mr. Saugata Bhattacharyya, Indian Association for the Cultivation of Science, Kolkata

4. Dr. Sourabh Banerjee, Harish-Chandra Research Institute, Allahabad

5. Mr. Subhashish Banerjee, Jawaharlal Nehru University, New Delhi

6. Mr. Anirban Chakraborti, Saha Institute of Nuclear Physics, Kolkata

7. Mr. Indrajit Chakravarty, Harish-Chandra Research Institute, Allahabad

8. Ms. Emily Chattopadhyay, J. C. Bose Institute, Kolkata

9. Ms. Jayita Chatterjee, Saha Institute of Nuclear Physics, Kolkata

10. Mr. Amit Dutta, Saha Institute of Nuclear Physics, Kolkata

11. Mr. E. Krishna Das, MG University, Kottayam

12. Mr. Sourin Das, Harish-Chandra Research Institute, Allahabad 13. Mr. Sreedhar Dutta, Institute 0/ Mathematical Sciences, Chennai

14. Mr. Sankalpa Ghosh, Jawaharlal Nehru University, New Delhi

15. Mr. Tarun Kanti Ghosh, Institute 0/ Mathematical Sciences, Chennai

16. Dr. Dattu Gaitonde, Harish-Chandra Research Institute, Allahabad

17. Mr.'Iribikram Gupta, Harish-Chandra Research Institute, Allahabad

18. Mr. Sandeep K. Joshi, Institute 0/ Physics, Bhubaneswar

19. Mr. R. S. JohaI, Punjab University, Chandigarh

20. Ms. Kavita Jain, Tata Institute 0/ Fundamental Research, Mumbai

21. Mr. Brijesh Kumar, Indian Institute 0/ Science, Bangalore 22. Mr. Rajesh Karan, Indian Institute 0/ Science, Bangalore

23. Ms. Raishma Krishnan, Hyderabad University, Hyderabad

24. Mr. Sandeep Krishna, Indian Institute 0/ Science, Bangalore

25. Mr. Sanjeev Kumar, Harish-Chandra Research Institute, Allahabad

26. Dr. Anjan Kundu, Saha Institute 0/ Nuclear Physics, Kolkata

27. Mr. Siddhartha LaI, Indian Institute 0/ Science, Bangalore

28. Ms. Tulika Maitra, Indian Institute 0/ Technology, Kharagpur 29. Mr. Joydeep Majumdar, Harish-Chandra Research Institute, Allahabad 30. Ms. Poonam Mehta, Delhi University, New Delhi,

31. Mr. Pramod Kumar Mishra, Banaras Hindu University, Varanasi

32. Mr. Pradeep K. Mohanty, Harish-Chandra Research Institute, Allahabad

33. Mr. Part ha Mukhopadhyay, Harish-Chandra Research Institute, Allahabad

34. Mr. Swagatam Mukhopadyay, Indian Institute of Technology, Kanpur

35. Mr. Nandan Pakhira, Indian Institute of Science, Bangalore

36. Ms. Debjani Pal, Indian Statistical Institute, K olkata

37. Mr. Arun Paramekanti, Tata Institute of Fundamental Research, Mumbai 38. Mr. Rajarshi Ray, Institute of Physics, Bhubaneswar

39. Mr. Suryadeep Ray, Harish-Chandra Research Institute, Allahabad

40. Ms. Sumithra Sankararaman, Institute of Mathematical Sciences, Chennai

41. Mr. Prabudha Sanyal, Indian Institute of Science, Bangalore

42. Ms. Soma Sanyal, Institute of Physics, Bhubaneswar

43. Dr. Asok Kumar Sen, Saha Institute of Nuclear Physics, Kolkata

44. Dr. Parongama Sen, Department of Physics, Calcutta University, Kolkata 45. Mr. S. Shankaranarayan, Inter University Centre /or Astronomy and Astrophysics, Pune 46. Mr. V. Venkatesh Shenai, Institute 0/ Physics, Bhubaneswar 47. Mr. Naveen Singh, Banaras Hindu University, Varanasi

48. Mr. Subrata Sur, Harish-Chandra Research Institute, Allahabad

49. Mr. Naveen Surendran, Institute 0/ Mathematical Sciences, Chennai

50. Mr. Aseem Talukdar, Delhi University, New Delhi

51. Dr. Arghya Taraphdar, Indian Institute 0/ Technology, Kharagpur

52. Mr. Ashake Kumar Verma, Bhabha Atomic Research Centre, Mumbai