Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors

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.

Electromagnetic and Gravitational Responses in Topological Insulators and Superconductors

The study of topological insulators and superconductors has advanced significantly, with a focus on how these intriguing quantum phases respond to electromagnetic and gravitational fields. In the paper by Ryu, Moore, and Ludwig, a detailed analysis is provided of the response functions such as charge, thermal, and magnetic dipole in various topological phases of matter, highlighting the implications these have on both theoretical and experimental fronts.

One of the key aspects of three-dimensional topological insulators, especially the $\mathbb{Z}_2$ type which includes materials like Bi$_2$Se$_3$, is their characteristic response to electromagnetic fields, forming a quantized magnetoelectric effect akin to "axion electrodynamics." This phenomenon allows such materials to exhibit a topological term in their electromagnetic response, adding an ${\bf E} \cdot {\bf B}$ term to the Lagrangian, connected with a topological "theta" term. This theoretical framework not only provides a deeper understanding of the intrinsic properties of these materials but also points to practical applications, such as in the field of spintronics where the manipulation of electron spins is crucial.

The paper extends the analysis of responses to gravitational fields, drawing analogies to thermal responses, particularly in systems such as the B phase of $^3$He. This phase of matter, when placed in a temperature gradient, can produce observable surface thermal currents, reflective of its topological nature. The notion that gravitational fields can couple to these quantum phases serves as both a theoretical curiosity and a potential tool for better understanding the robustness of these phases under various perturbations.

In symmetry classes DIII, CI, and AIII, robust topological terms arise in their respective effective field theories, elucidating their stability even under interactions. A significant finding is that the anomalies in field theories, such as gauge and gravitational anomalies, persist regardless of inter-particle interactions, suggesting the indelible nature of these topological classifications across dimensions. This insight aligns with the conjecture that anomalies, known from quantum field theory to signal breakdowns of classical symmetries due to quantum effects, might illuminate the classification of systems as topologically nontrivial.

The paper also speculates on future developments, notably the potential identification and exploitation of these topological features in new materials. The understanding of gravitational and dipole responses in topological superconductors will particularly benefit from further material advances, possibly leading to emergent technologies that could harness these robust quantum effects.

Overall, the insights from Ryu, Moore, and Ludwig's analysis contribute significantly to the theoretical landscape of topological phases, bridging the gap between abstract mathematical descriptions and tangible physical phenomena, and steering future experimental efforts towards novel applications and materials in this dynamically evolving field.