Quantization of the interacting Hall conductivity in the critical regime

TL;DR

This paper proves that in an interacting Haldane model, the integer quantum Hall conductivity remains quantized outside critical curves, which are modified by interactions, and describes the phase transition behavior across these curves.

Contribution

It extends previous perturbative results by providing a non-perturbative proof of quantization and explicitly constructing the interaction-modified critical curves in the model.

Findings

Hall conductivity remains quantized outside critical curves.

Critical curves are modified ('dressed') by interactions.

Phase transitions involve abrupt jumps in Hall coefficient.

Abstract

The Haldane model is a paradigmatic 2d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a `topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified (`dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.