Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors

TL;DR

This work develops a unifying, anomaly-based framework linking electromagnetic, gravitational, and dipole responses to the topological phases of insulators and superconductors in three dimensions across the full ten-fold classification. By deriving bulk topological terms (Chern-Simons and theta terms) from gauge, gravitational, and mixed anomalies, the authors connect surface quantized responses (charge, spin, and thermal Hall effects) to robust bulk invariants that persist under interactions. The analysis shows that many topological distinctions are encoded in anomaly structures, implying robustness to adiabatic deformations with interactions and providing a route to generalize to general dimensions via descent relations and the Atiyah-Singer index theorem. The results yield concrete predictions for surface and bulk responses across symmetry classes (AII, CI, CII, DIII, AIII) and lay groundwork for exploring higher-dimensional and fractional generalizations of topological phases through anomaly-inspired field theories and gravitational couplings.

Abstract

One of the defining properties of the conventional three-dimensional ("$\mathbb{Z}_2$-", or "spin-orbit"-) topological insulator is its characteristic magnetoelectric effect, as described by axion electrodynamics. In this paper, we discuss an analogue of such a magnetoelectric effect in the thermal (or gravitational) and the magnetic dipole responses in all symmetry classes which admit topologically non-trivial insulators or superconductors to exist in three dimensions. In particular, for topological superconductors (or superfluids) with time-reversal symmetry which lack SU(2) spin rotation symmetry (e.g. due to spin-orbit interactions), such as the B phase of $^3$He, the thermal response is the only probe which can detect the non-trivial topological character through transport. We show that, for such topological superconductors, applying a temperature gradient produces a thermal- (or mass-) surface current perpendicular to the thermal gradient. Such charge, thermal, or magnetic dipole responses provide a definition of topological insulators and superconductors beyond the single-particle picture. Moreover we find, for a significant part of the 'ten-fold' list of topological insulators found in previous work in the absence of interactions, that in general dimensions the effective field theory describing the space-time responses is governed by a field theory anomaly. Since anomalies are known to be insensitive to whether the underlying fermions are interacting or not, this shows that the classification of these topological insulators is robust to adiabatic deformations by interparticle interactions in general dimensionality. In particular, this applies to symmetry classes DIII, CI, and AIII in three spatial dimensions, and to symmetry classes D and C in two spatial dimensions.

Figures (2)

• Figure 1: Electric and thermal response of topological insulators, and thermal response of topological triplet superconductors, in a cylindrical geometry. (a): Electric ($\mathbf{j}$) or thermal ($\mathbf{j}^T$) current driven by applied electric field ($\mathbf{E}$) or thermal gradient ($\boldsymbol{\nabla} T/T$). (b): A response dual to (a) where an applied magnetic field in $z$-direction induces charge polarization.
• Figure 2: Surface of a spin chiral topological insulator (class CII) or topological singlet superconductor (class CI).