Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors

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 analog of such a magnetoelectric effect in the thermal (or gravitational) and magnetic dipole responses in all symmetry classes that admit topologically nontrivial insulators or superconductors to exist in three dimensions. In particular, for topological superconductors (or superfluids) with time-reversal symmetry, which lack $\text{SU}\left(2\right)$ 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 that can detect the nontrivial 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 “tenfold” 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, 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.

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 ($\nabla T/T$). (b) A response dual to (a) where an applied magnetic field in the $z$ direction induces charge polarization.

Figure 2

Surface of a spin chiral topological insulator (class CII) or topological singlet superconductor (class CI).