Tetrads in solids: from elasticity theory to topological quantum Hall systems and Weyl fermions

TL;DR

This paper explores the role of elasticity tetrads in topological insulators and superfluid helium-3, revealing differences from Weyl fermion tetrads and their implications for topological responses and anomalies.

Contribution

It introduces the concept of elasticity tetrads in topological insulators, analyzing their unique topological responses and contrasting them with Weyl fermion tetrads.

Findings

Elasticity tetrads lead to unique topological terms in the action.

Differences between elasticity and Weyl fermion tetrads affect gauge invariance.

The principal difference impacts the formulation of topological responses.

Abstract

Theory of elasticity in topological insulators has many common features with relativistic quantum fields interacting with gravitational field in the tetrad form. Here we discuss several issues in the effective topological (pseudo)electromagnetic response in three-dimensional weak crystalline topological insulators with no time-reversal symmetry that feature elasticity tetrads, including a mixed "axial-gravitational" anomaly. This response has some resemblance to "quasitopological" terms proposed for massless Weyl quasiparticles with separate, emergent fermion tetrads. As an example, we discuss the chiral/axial anomaly in superfluid 3He-A. We demonstrate the principal difference between the elasticity tetrads and the Weyl fermion tetrads in the construction of the topological terms in the action. In particular, the topological action expressed in terms of the elasticity tetrads, cannot be expressed in terms of the Weyl fermion tetrads since in this case the gauge invariance is lost.