Gravitational anomalies and thermal Hall effect in topological insulators

Abstract

It has been suggested that, after being gapped by small symmetry breaking field, the Majorana quasiparticles localized on the surface of a class DIII topological insulator will exhibit a thermal Hall effect that arises from a gravitational Chern-Simons term. We critically examine this idea, and argue that the thermo-gravitational Hall effect is more complicated than its familiar analogue. A conventional Hall current is generated by a uniform electric field, but computing the flux from the gravitational Chern-Simons functional shows that gravitational field gradients -i.e. tidal forcesare needed to induce a energy-momentum flow. We relate the resulting surface energy-momentum flux to a domain-wall gravitational anomaly 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 Christoffel symbol or in terms of the the spin connection.

Gravitational anomalies on the Newton-Cartan background

“…We also know that the low energy effective action for Quantum Hall systems involves the Schrödinger field coupled to, in general, a background gravitational field in d + 1 dimensions; where d = 2n. Anomalies play a crucial role in Hall phenomenology [18][19][20][21][22][23][24][26][27][28], with the guiding principle in the presence of boundaries being that the bulk and boundary contributions collectively should be non-anomalous [19]. In taking the NR limit for these systems, we have a bulk gravitational anomaly and no boundary anomaly.…”

“…In considering relativistic systems with a symmetric stress-energy tensor, the trace anomaly arises when the quantum stress-energy tensor is not traceless, while its failure to be conserved results in the diffeomorphism anomaly. These anomalies have important consequences in black holes physics and cosmology [4][5][6][7][8][9][10][11][12][13][14][15][16][17], as well as in the computation of transport coefficients and response functions of condensed matter systems [18][19][20][21][22][23][24][25][26][27][28]. Gravitational anomalies are in addition background dependent, as evident from the difference of Lifshitz anomalies from those of relativistic backgrounds.…”

Multipole expansion in the quantum hall effect

“…The system has been considered on spatial metric backgrounds for studying the heat transport [2][3][4] and the response of the fluid to strain. In particular, the Hall viscosity has been identified as a new universal quantity describing the non-dissipative transport [5][6][7][8].…”

Classical version of the non-Abelian gauge anomaly

“…The anomaly is then accounted for by the inflow of gauge current from the bulk, and this inflowing current can obtained by functionally differentiating a bulk Chern-Simons action. The boundary variation of the Chern-Simons term is then precisely the BardeenZumino polynomial [24] that converts the consistent gauge current to the covariant current (see for example [25]). A similar argument shows that in an anomalous theory the current that appears in the Lorentz-force contribution to the energy-momentum conservation law is the covariant current [26,27].…”