Conformal Symmetry and Universal Properties of Quantum Hall States

TL;DR

This paper derives the conformal field theory describing edge excitations in quantum Hall states from microscopic electron dynamics, connecting it to effective theories and highlighting universal finite-size effects.

Contribution

It provides a detailed derivation of the conformal field theory for quantum Hall edge states from microscopic principles, extending to fractional states via chiral bosonization.

Findings

Derivation of conformal field theory from microscopic electron dynamics.

Connection between microscopic theory and effective Chern-Simons approach.

Prediction of universal finite-size effects in energy spectra.

Abstract

The low-lying excitations of a quantum Hall state on a disk geometry are edge excitations. Their dynamics is governed by a conformal field theory on the cylinder defined by the disk boundary and the time variable. We give a simple and detailed derivation of this conformal field theory for integer filling, starting from the microscopic dynamics of $(2+1)$-dimensional non-relativistic electrons in Landau levels. This construction can be generalized to describe Laughlin's fractional Hall states via chiral bosonization, thereby making contact with the effective Chern-Simons theory approach. The conformal field theory dictates the finite-size effects in the energy spectrum. An experimental or numerical verification of these universal effects would provide a further confirmation of Laughlin's theory of incompressible quantum fluids.