Electromagnetic and gravitational responses of two-dimensional non-interacting electrons in background magnetic field

TL;DR

The paper analyzes a non-interacting two-dimensional electron gas in a constant background magnetic field at integer filling $N$, coupling to external electromagnetic and metric fields to compute linear responses. Using an explicit quadratic expansion of the effective action, it identifies geometric Chern-Simons, Wen-Zee, and gravitational Chern-Simons contributions and derives expressions for the Hall conductivity and Hall viscosity, including higher-gradient corrections. It provides explicit density and current responses to curvature and inhomogeneous magnetic fields, such as a wavevector-dependent Hall conductivity $\sigma_H(k)$ and curvature-induced corrections to the Hall viscosity, as well as charge responses to conical defects. The results reproduce known IQHE limits and offer a robust geometric framework for QHE phenomena, with potential extensions to torsion and fractional quantum Hall contexts and connections to Ward identities.

Abstract

We compute electromagnetic, gravitational and mixed linear response functions of two- dimensional free fermions in external quantizing magnetic field at an integer filling factor. The results are presented in the form of the effective action and as an expansion of currents and stresses in wave-vectors and frequencies of the probing electromagnetic and metric fields. We identify the terms in linear response functions coming from geometric Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms in effective action. We derive the expressions for Hall conductivity, Hall viscosity and find the current and charge density responses to the spatial curvature as well as stresses caused by inhomogeneous electromagnetic fields.