Low-Dimensional Quantum Field Theories for Condensed Matter Physicists

The following sections are included:

• PREFACE

• Contents

The following sections are included:

• INTRODUCTION

• HOMOTOPY AND TOPOLOGICAL INVARIANTS

• THE DUALITY BETWEEN 𝕄 AND

• DIFFERENTIAL CALCULUS ON LIE GROUPS AND COSET SPACES

• FIBER SPACES AND PARALLEL TRANSPORT

• DIFFERENTIAL CALCULUS ON FIBER BUNDLES

• SEQUENCES ON PRINCIPAL BUNDLES. CHERN-SIMONS TERMS

• CONCLUSIONS

• ACKNOWLEDGEMENTS

• APPENDIX

• GLOSSARIES. I. TOPOLOGICAL TERMS

• GLOSSARIES. II. ALGEBRAIC TERMS

• REFERENCES

The following two loosely connected sets of topics are reviewed in these lecture notes: 1) Gauge invariance, its treatment in field theories and its implications for internal symmetries and edge states such as those in the quantum Hall effect. 2) Quantisation on multiply connected spaces and a topological proof the spin-statistics theorem which avoids quantum field theory and relativity. Under 1), after explaining the meaning of gauge invariance and the theory of constraints, we discuss boundary conditions on gauge transformations and the definition of internal symmetries in gauge field theories. We then show how the edge states in the quantum Hall effect can be derived from the Chern-Simons action using the preceding ideas. Under 2), after explaining the significance of fibre bundles for quantum physics, we review quantisation on multiply connected spaces in detail, explaining also mathematical ideas such as those of the universal covering space and the fundamental group. These ideas are then used to prove the aforementioned topological spin-statistics theorem.

This paper gives a systematic review of a field theoretical approach to the fractional quantum Hall effect (FQHE) that has been developed in the past few years. We first illustrate some simple physical ideas to motivate such an approach and then present a systematic derivation of the Chern-Simons-Landau-Ginzburg (CSLG) action for the FQHE, starting from the microscopic Hamiltonian. It is demonstrated that all the phenomenological aspects of the FQHE can be derived from the mean field solution and the small fluctuations of the CSLG action. Although this formalism is logically independent of Laughlin's wave function approach, their physical consequences are equivalent. The CSLG theory demonstrates a deep connection between the phenomena of superfluidity and the FQHE, and can provide a simple and direct formalism to address many new macroscopic phenomena of the FQHE.

In this set of lectures we review the fermion field theoretic approach to the Fractional Quantum Hall Effect and use it to discuss the origin of its remarkable universality. We discuss the semiclassical expansion around the average field approximation (AFA). We reexamine the AFA and the role of fluctuations. We argue that, order-by-order in the semiclassical expansion, the response functions obey the correct symmetry properties required by Galilean and Gauge Invariance and by the incompressibility of the fluid. In particular, we find that the low-momentum limit of the semiclassical approximation to the response functions is exact and that it saturates the f-sum rule. We discuss the nature of the spectrum of collective excitations of FQHE systems in the low-momentum limit. We applied these results to the problem of the screening of external charges and fluxes by the electron fluid, and obtained asymptotic expressions of the charge and current density profiles, for different types of interactions. The universality of the FQHE is demonstrated by deriving the form of the wave function of the ground state at long distances. We show that the wave functions of the fluid ground states of Fractional Quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equaltime density correlation functions at long distances.

Anyons (particles obeying fractional statistics) are studied in successively improved approximations, each of which exhibits superconductivity of this two-dimensional system. The random-phase approximation is developed in analogy with the three-dimensional electron gas. Gauge invariance is ensured by the use of conserving approximations, illustrated by the Hartree and Hartree-Fock descriptions of single-particle properties and the associated self-consistent collective modes.

An elementary introduction to abelian bosonization is provided here. It is shown that although it is applicable to relativistic Dirac fermions, there are many examples of problems in condensed matter theory which are described (with respect to some characteristics) by just such objects. Examples considered here are the uniform and random bond Ising models, and the Hubbard model.

An elementary introduction is given to the basic concepts of abelian and nonabelian Bosonization, and Kac-Moody algebras, with an emphasis on applications in Condensed Matter Physics. The subject is developed using free Fermions. No previous knowledge of conformal field theory is required from the reader.

The following sections are included:

• INTRODUCTION
  • The Hubbard Model
  • The Kondo Model

• REFERENCES
  • Footnote

• The Quantum Inverse Method

• REFERENCES

• The Kondo Model

• REFERENCES

• The Hubbard Model
  • The repulsive Hubbard model
  • The attractive Hubbard model

• REFERENCES

I review recent work, performed in collaboration primarily with N. Read and Jinwu Ye, on the properties of quantum antiferromagnets in two dimensions. The emphasis is on the properties of the antiferromagnet in states which do not have any long-range magnetic order. The universal spin dynamics in the quantum critical region of number of frustrated and random antiferromagnets is studied; implications for neutron scattering experiments in the lightly-doped cuprates are noted. The nature of the quantum-disordered phase of non-random frustrated antiferromagnets is examined in some detail: the states found have (i) collinear spin correlations, spin-Peierls or valence-bond-solid order, and confined spinons, order and confined spinons or (ii) coplanar spin correlations, no spin-Peierls order and deconfined bosonic spinons.