Particle-hole symmetry and electromagnetic response of a half-filled Landau level

TL;DR

This paper explores the implications of particle-hole symmetry at half-filled Landau levels, deriving exact relationships for electromagnetic responses and analyzing their validity within different theoretical frameworks.

Contribution

It establishes exact relationships between Hall conductivity and susceptibility under particle-hole symmetry and examines their validity in the Dirac composite fermion theory versus the HLR theory.

Findings

Exact relationship between Hall conductivity and susceptibility derived.

Relationship holds for clean and symmetric disordered systems.

Breakdown of the relationship in the HLR theory demonstrated.

Abstract

We derive exact physical consequences of particle-hole symmetry of the $\nu=1/2$ state of electrons in a strong magnetic field. We show that if the symmetry is not spontaneously broken, the Hall conductivity and the susceptibility satisfy an exact relationship, valid at any wave numbers and any frequencies much below the cyclotron frequency. The relationship holds for clean systems and also for systems with statistically particle-hole symmetric disorder. We work out the constraints this relationship imposes on the theory of the Dirac composite fermion. We also argue that that the exact relationship is violated in the Halperin-Lee-Read (HLR) field theory and present an explicit calculation within a Galilean invariant mean-field approximation to the HLR theory to illustrate the breakdown.