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Oxford University Press eBooks

Many-Body Methods for Chern-Simons Systems

Lucjan Jacak, Piotr Sitko, Konrad Wieczorek, Arkadiusz Wójs

Published Sep 25, 2003• 0 citations
Publisher

Abstract

This chapter addresses the quantum theory of many-body systems with Chern–Simons field in order to describe the change of statistics in two-dimensional systems. The random phase approximation is used for an anyon gas. Within this approach the correlation energy of an anyon gas is found. The Hartree–Fock approximation for Chern–Simons systems and diagram analysis for a gas of anyons for self-consistent Hartree and Hartree–Fock approximations is also given.

DOI
10.1093/acprof:oso/9780198528708.003.0005

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This chapter addresses the quantum theory of many-body systems with Chern–Simons field in order to describe the change of statistics in two-dimensional systems. The random phase approximation is used for an anyon gas. Within this approach the correlation energy of an anyon gas is found. The Hartree–Fock approximation for Chern–Simons systems and diagram analysis for a gas of anyons for self-consistent Hartree and Hartree–Fock approximations is also given.
The present state of the gauge theory of high-temperature superconductivity in strongly-correlated two-dimensional spin systems is reviewed. Basic ideas on the statistics of elementary excitations in spatially two-dimensional systems are presented and are used in an analysis of the structure of the energy spectrum and the form of the wave function of a set of anyon quasiparticles in both the long-wave continuous and lattice limits. The continuous and lattice theories are used to classify the phase states, and the hierarchy of phase transitions is described in terms of the topological quantum field theory. Thermodynamic and electrodynamic properties of anyon systems are described, and the experimental consequences of the Chern–Simons theory of high-Tc superconductivity are discussed.
The energy density is computed for a U(2) Chern-Simons theory coupled to a non-relativistic fermion field (a theory of ``non-Abelian anyons'') under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This suggests that there is no spontaneous breaking of the SU(2) subgroup of the symmetry, at least in this mean-field approximation. For spin-1/2 fermions, we find self-consistent mean-field states with a small non-Abelian charge density, which vanishes as the theory of free fermions is approached.