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Proceedings of the Ninth Chris Engelbrecht Summer School in Held at Storms River Mouth, Tsitsikamma National Park, South Africa, 17-28 January 1994

~ Springer Editor Hendrik B. Geyer Institute of Theoretical Physics University of Stellenbosch 7599 Stellenbosch, South Africa

Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Field theory, topology and condensed matter physics : proceedings of the Ninth Chris Engelbrecht Summer School in Theoretical Physics, held at Storms River Mouth, Tsitsikamma National Park, South Africa, 17-z8 January 1994 / Hendrik B. Geyer (ed.). Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 1995 (Lecture notes in physics ; Vol. 456) ISBN 3-540-60138-4 NE: Geyer, Hendrik B. [Hrsg.]; Chris Engelbrecht Summer School in Theoretical Phys- ics <9, 1994, Stellenbosch>; GT

ISBN 3-540-60138-4 Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright.All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustra- tions, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this gublication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag.Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready by the authors SPIN: 10508107 55/3142-543210 - Printed on acid-free paper Preface

Although cross-fertilisation between field theory and condensed matter physics can claim a rather long history, it is probably fair to say that recent interest and developments in topics such as the quantum Hall and fractional quantum Hall effects, anyons, two-dimensional systems in general, bosonisation and, of course, the recurring theme of high-Tc superconductivity, have focused renewed attention on the fruitful symbiosis of these two major fields in physics.

That the above theme for the 9th in the series of South African Summer Schools in Theoretical Physics indeed proved to be a happy choice, can first and foremost be attributed to the expertise and enthusiastic lecturing of the distinguished panel of lecturers who came together for this occasion. We were quite fortunate to have Francois David (Saclay), Antal Jevicki (Brown University), Xiao-Gang Wen (MIT), Paul Wiegmann (University of Chicago & Landau Institute) and Anthony Zee (ITP, Santa Barbara) present the series of lectures reflected in these notes.

I am sure readers will appreciate the lucid and exceptionally well-coordinated presentations as much as did the 40-odd participants. They may even share in some of the excitement that went along with experiencing the presentation of these lectures first-hand. Unfortunately there is no way to recapture the working atmosphere and many fruitful discussions in the majestic setting of the Tsi- tsikamma lecture room, only 30 meters away from where the Atlantic Ocean pounds this rocky stretch of South African coastline. It indeed testifies to the skills of the lecturers that they managed to win the battle for attention up to the last lecture! This would not have been possible without the considerable time, effort and enthusiasm put into this venture by all the lecturers. We are indeed indebted to them, not least for finding two weeks in an already tight schedule and coming to South Africa at what may have been perceived to be an uncertain time, shortly before the first countrywide democratic elections. It is now history that South Africans rose to the occasion in no uncertain terms.

The Tsitsikamma Coastal National Park proved to be as stimulating an envi- ronment as one could wish for and many discussions were carried further on the various coastal and forest trails. The personnel of the National Parks Board also vI lived up to their reputation and their friendly, efficient service and hospitality certainly contributed much to the success of the School.

The Organising Committee wishes to acknowledge the continued financial sup- port of the Foundation for Research Development, without which it would not be possible to organise the series of Summer Schools on an internationally compet- itive level. We also once again benefited from the expert administrative support of Erna Pheiffer.

It is a pleasure to thank all members of the Organising Committee for their active support throughout the many months of preparation that went into organising the School, as well as Mrs Juliana de Waal and Mr Rudolf Visser for their assistance with the editorial process. Finally, we are grateful that the editors of Springer-Verlag once again agreed to publish the School's lectures in their Lecture Notes in Physics series.

Stellenbosch, South Africa Hendrik B Geyer April 1995 Co~e~s

Introduction to Conformal Invariance in Statistical Mechanics and to Random Surface Models F. David

1 Introduction 1 2 Introduction to Conformal Invariance in Statistical Mechanics 2 2.1 Renormalization , Scale and Conformal Invariance 2 2.2 Conformal Invariance in Two Dimensions 9 2.3 The Trace Anomaly and Finite Size Effects 16 2.4 Minimal Models and the FQS Classification 19 2.5 Massive Theories, RG Flows and Zamolodchikov's c-Theorem 26 3 Introduction to Statistical Mechanics of Random Surfaces 29 3.1 Polymers and Self-avoiding Walks 29 3.2 Random Surfaces and Membranes 34 3.3 Self-avoiding Polymerized Surfaces 43

Introduction to Path Integrals, Matrix Models and Strings A. Jevicki 55

1 Introduction 55 2 Path Integrals 56 3 Strings 61 3.1 Two-Dimensional Strings 64 4 Yang Mills Theory and Loop Space 67 4.1 Yang-Mills Theory on a Circle 68 5 Matrix Models, Field Theory and Strings 73 5.1 One-Dimensional Matrix Model 75 5.2 Collective Field Theory 77 5.3 Field Theory of Strings 81 5.4 Exact States and Woo Symmetry 86 6 Field Theory and Schwinger-Dyson Equations 91 VIII

Quantum Hall Fluids A. Zee 99

1 Topological Quantum Fluids 99 2 Quantum Phase and Fractional Statistics 102 3 Fractional Hall Fluids 106 4 A General Approach to Hall Fluids 107 5 Detailed Construction of Field Theory of Hall Fluids 113 6 Duality and Vortices 114 7 Multitude of Gauge Fields 120 8 Topological Order 122 9 On the Edge 126 10 More General K Matrices 129 11 Hierarchy 132 12 Magnetic Monopoles and Multi-layered Hall Systems 134 13 Shift and Spin 139 14 The Even Integers 141 15 Concluding Remarks and a Puzzle 149

Topological Orders and Edge Excitations in Fractional Quantum Hall States Xiao-Gaag Wen 155

1 Introduction 155 1.1 Characterization of Topological Orders 156 1.2 Classification of Topological Orders 157 1.3 Edge Excitations - a Practical Way to Measure Topological Orders in Experiments 158 2 Effective Theory of FQH Liquids 158 3 Hydrodynamical Approach to the Edge Excitations 163 4 Effective Theory and the Edge States 166 5 Charged Excitations and Electron Propagator on the Edges of Generic FQH States 171 ix

Topological Mechanism of Superconductivity P. Wiegmann 177

1 Criterion for Superconductivity 177 1.1 London Equations as Hydrodynamics of an Ideal Liquid 177 1.2 Landau Criterion 178 1.3 Topological Order 179 2 Frohlich Superconductivity - One Dimension 181 3 Fermionic Number and Solitons 184 3.1 Level Crossing 184 3.2 Models and Solitons 185 3.3 Anomalous Calculus 187 4 Topological Fluid as a Superfluid 191 4.1 The Basic Model - Holons and Spinons - Zero Current Theory 191 4.2 Topological Instability 191 4.3 Topological Field Theory Coupled with the Matter Field and Superfluidity 192 4.4 Superconductivity 193 4.5 Topological Superconductivity and Bosonization in Higher Dimensions 195 5 A Gauge Model for Correlated Electronic Systems 196 5.1 A Model for the Doped Mott Insulator - Resonant Valence Bonds and Chirality 196 5.2 Flux Phase 200 5.3 Continuum Limit 200 5.4 Lattice Adopted Basis 202 6 Conclusion 205

Lecturers

Francois David, Saclay Antal Jevicki, Brown University Xiao-Gang Wen, MIT Paul B Wiegmann, University of Chicago & Landau Institute Anthony Zee, ITP, Santa Barbara

Organising Committee

Hendrik B Geyer (Chairman), University of Stellenbosch W Dieter Heiss, University of the Witwatersrand Joao P Rodrigues, University of the Witwatersrand Frederik G Scholtz, University of Stellenbosch Wynand S Verwoerd, UNISA Erna Pheiffer (Secretariat), FRD

Participants

C A I Anghel, University of the Witwatersrand I Barashenkov, University of Cape Town N Bilic, University of Cape Town M Bondila, University of Cape Town R M Carter, University of Pretoria O L de Lange, University of Natal R S de Mello Koch, University of the Witwatersrand /l A de Wet, Plettenberg Bay H B Geyer, University of Stellenbosch G R Gluckman, University of Cape Town W D Heiss, University of the Witwatersrand J J Henning, Pretoria G J Hill, University of the Witwatersrand D P :loubert, University of the Witwatersrand R B Kassman, University of the Witwatersrand Xll

A Mattson, Chalmers University of Technology, GSteborg M Molisana, National University of Lesotho A Moncada, University of Stellenbosch P Navr&til, University of Stellenbosch R W N Nilen, University of the Witwatersrand S Radu, University of the Witwatersrand R A Ritchie, University of Pretoria J A P Rodrigues, University of the Witwatersrand J S Rozowsky, University of Cape Town F G Scholtz, University of Stellenbosch A N Theron, University of Stellenbosch G P Tshelane, University of the Witwatersrand T F Tshepe, University of the Witwatersrand P du T van der Merwe, Atomic Energy Corporation W S Verwoerd, University of South Africa J von Delft, Cornell University A Welter, University of Natal E J H Wessels, Cape Town