Why is the HLR theory particle-hole symmetric?

TL;DR

This paper explains how the traditional HLR theory of the half-filled Landau level can exhibit particle-hole symmetry through inhomogeneous configurations that are tuned to a topological phase transition, reconciling theory with experimental observations.

Contribution

It demonstrates that the HLR mean-field theory can display particle-hole symmetry via inhomogeneous states linked to a topological phase transition.

Findings

HLR theory with disorder can show emergent PH symmetry.

Inhomogeneous configurations are tuned to a topological phase transition.

The dc Hall response of the system is consistent with PH symmetry.

Abstract

Long wavelength descriptions of a half-filled lowest Landau level ($\nu = 1/2$) must be consistent with the experimental observation of particle-hole (PH) symmetry. The traditional description of the $\nu=1/2$ state pioneered by Halperin, Lee and Read (HLR) naively appears to break PH symmetry. However, recent studies have shown that the HLR theory with weak quenched disorder can exhibit an emergent PH symmetry. We find that such inhomogeneous configurations of the $\nu=1/2$ fluid, when described by HLR mean-field theory, are tuned to a topological phase transition between an integer quantum Hall state and an insulator of composite fermions with a dc Hall conductivity $\sigma_{xy}^{\rm (cf)} = - {1 \over 2} {e^2 \over h}$. Our observations help explain why the HLR theory exhibits PH symmetric dc response.