Gravitational Anomalies and Thermal Hall effect in Topological Insulators

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.