Comment on "Galilean invariance at quantum Hall edge"

TL;DR

This paper challenges previous claims by showing that the $q^2$ contribution to edge conductivity in quantum Hall systems is non-universal and depends on edge details, contrasting with earlier universal predictions.

Contribution

It provides a calculation demonstrating the non-universality of the $q^2$ term in edge conductivity for non-interacting electrons, refining the understanding of edge responses in quantum Hall systems.

Findings

The $q^2$ term is non-universal and depends on edge potential details.

In a linear potential, the $q^2$ term resembles previous predictions.

The results may extend to fractional quantum Hall states with specific interactions.

Abstract

In a recent paper by S. Moroz, C. Hoyos, and L. Radzihovsky [Phys. Rev. B 91, 195409 (2015)], it is claimed that the conductivity at low frequency $\omega$ and small wavevector $q$ along the edge of a quantum Hall (QH) system (that possesses Galilean invariance along the edge) contains a universal contribution of order $q^2$ that is determined by the orbital spin per particle in the bulk of the system, or alternatively by the shift of the ground state. (These quantities are known to be related to the Hall viscosity of the bulk.) In this Comment we calculate the real part of the conductivity, integrated over $\omega$, in this regime for the edge of a system of non-interacting electrons filling either the lowest, or the lowest $\nu$ ($\nu=1$, $2$, . . .), Landau level(s), and show that the $q^2$ term is non-universal and depends on details of the confining potential at the edge. In the special case of a linear potential, a form similar to the prediction is obtained, it is possible that this corrected form of the prediction may also hold for fractional QH states in systems with special forms of interactions between electrons.