Field Theories in Condensed Matter Physics. Edited by. Sumathi Rao. Harish-Chandra Research Institute Allahabad. lop

Transcription

1 Field Theories in Condensed Matter Physics Edited by Sumathi Rao Harish-Chandra Research Institute Allahabad lop Institute of Physics Publishing Bristol and Philadelphia

2 Contents Preface xiii Introduction 1 1 Quantum Many Particle Physics 7 Pinaki Majumdar 1.1 Preamble Introduction Introduction to many particle physics Phases of many particle systems Quantities of physical interest Fermi and Bose liquids Phase transitions and broken symmetry Phase transitions and symmetry breaking Symmetry breaking and interactions in ВЕС Normal Fermi systems: model problems Neutral fermions: dilute hardcore Fermi gas Charged fermions: the electron gas Electrons and phonons: Migdal-Eliashberg theory Weak coupling theory: BCS The normal state: Migdal theory BCS theory: Greens function approach Superconductivity: Eliashberg theory Conclusion: 'field theory' and many particle physics 63

3 viii CONTENTS 2 Critical Phenomena 69 Somendra M. Bhattacharjee 2.1 Preamble Large system: Thermodynamic limit Where is the problem? Recapitulation - A few formal stuff Extensivity Convexity: Stability Consequences of divergence Generalized scaling One variable: Temperature Solidarity with thermodynamics More variables: Temperature and field On exponent relations Relevance, irrelevance and universality Digression A first-order transition: a=l Example: Polymers : no "ordering" Exponents and correlations Correlation function Relations among the exponents Length-scale dependent parameters Models as examples: Gaussian and ф Specific heat for the Gaussian model Cut-off and anomalous dimensions Through correlations Epilogue Phase Transitions and Critical Phenomena 119 Deepak Kumar 3.1 Introduction Thermodynamic stability Lattice gas : mean field approximation Landau theory 134

4 CONTENTS ix 3.5 Spatial correlations Breakdown of mean field theory Ginzburg-Landau free energy functional Renormalisation group (RG) RG for a one dimensional Ising chain RG for a two-dimensional Ising model General features of RG Irrelevant variables RG scaling for correlation functions RG for Ginzburg-Landau model Tree-level approximation Critical exponents for d > Anomalous dimensions Perturbation series for d < Generalisation to a n-component model Topological Defects 189 Ajit M. Srivastava 4.1 The subject of topological defect What is a topological defect? Meaning of order parameter Spontaneous symmetry breakdown(ssb) SSB in particle physics Order parameter space The domain wall Why defect? Why topological? Energy considerations Examples of topological defects Condensed matter versus particle physics Detailed understanding of a topological defect Free homotopy of maps Based homotopy and the fundamental group Classification of defects using homotopy groups Defect structure in liquid crystals 227

5 CONTENTS Defects in nematics Non abelian ж\ - biaxial nematics Formation of topological defects 231 Introduction to Bosonization 239 Sumathi Rao and Diptiman Sen 5.1 Fermi and Luttinger liquids Bosonization Bosonization of a fermion with one chirality Bosonisation with two chiralities Field theory near the Fermi "momenta Correlation functions and dimensions of operators RG analysis of perturbed models Applications of bosonization Quantum antiferromagnetic spin 1/2 chain Hubbard model Transport in a Luttinger liquid - clean wire Transport in the presence of isolated impurities Concluding remarks 328 Quantum Hall Effect 335 R. Rajaraman 6.1 Classical Hall effect Quantized Hall effect Landau problem Degeneracy counting Laughlin wavefunction Plasma analogy Quasi-holes and their Laughlin wavefunction Localization physics and the QH plateaux Chern-Simons theory Vortices in the CS field and quasiholes Jain's theory of composite fermions 355

6 CONTENTS XI 7 Low-dimensional Quantum Spin Systems 359 Indrani Böse 7.1 Introduction Ground and excited states Theorems and rigorous results for antiferromagnets Lieb-Mattis theorem Marshall's sign rule Lieb, Schultz and Mattis theorem Mermin-Wagner theorem Possible ground states and excitation spectra The Bethe Ansatz 387