Gravitational Anomalies and Thermal Hall effect in Topological Insulators

TL;DR

This work critically examines the proposed gravitational Hall effect for Majorana surface modes on DIII topological insulators and clarifies that a uniform gravitational field cannot drive energy flow on the 2D surface. By invoking anomaly inflow and gravitational Chern–Simons terms, the authors show that any surface energy-momentum flux must be sourced by tidal (curvature-gradient) fields and absorbed by 1+1D edge modes, yielding a covariant gravitational anomaly. The analysis demonstrates the equivalence of formulations based on the Christoffel symbols and the spin connection, with the bulk inflow reliably reproducing the covariant anomaly and linking it to the domain-wall edge. The results refine the understanding of thermal responses in topological phases and guide interpretations of potential experiments to detect gravitationally induced thermal transport through edge modes. Overall, the paper clarifies the proper bulk–boundary framework for gravitational responses in topological superconductors and emphasizes the role of edge-domain-wall anomalies over naive uniform-field analogies.

Abstract

It has been suggested that a temperature gradient will induce a Leduc-Righi, or thermal Hall, current in the Majorana quasiparticles localized on the surface of class DIII topological insulators, and that the magnitude of this current can be related {\it via} an Einstein argument to a Hall-like energy flux induced by gravity. We critically examine this idea, and argue that the gravitational Hall effect is more complicated than its familiar analogue. A conventional Hall current is generated by a {\it uniform} electric field, but computing the flux from the gravitational Chern-Simons functional shows that gravitational field {\it gradients} - i.e. tidal forces - are needed to induce a energy-momentum flow. We relate the surface energy-momentum flux to a domain-wall gravitational anomaly {\it via} the Callan-Harvey inflow mechanism. We stress that the gauge invariance of the combined bulk-plus-boundary theory ensures that the current in the domain wall always experiences a "covariant" rather than "consistent" anomaly. We use this observation to confirm that the tidally induced energy-momentum current exactly accounts for the covariant gravitational anomaly in $(1+1)$ dimensional domain-wall fermions. The same anomaly arises whether we write the Chern-Simons functional in terms of the Christofflel symbol or in terms of the the spin connection.