# Elasticity tetrads, mixed axial-gravitational anomalies, and 3+1d quantum Hall effect

## TL;DR

This paper extends the topological quantum Hall effect framework to three-dimensional crystalline insulators, introducing elasticity tetrads and mixed axial-gravitational anomalies, revealing quantized responses linked to crystalline symmetries.

## Contribution

It introduces a (3+1)d mixed Chern-Simons term involving elasticity tetrads, connecting crystal deformations to topological invariants and anomalous responses in 3D topological insulators.

## Key findings

- Describes 3D topological insulators with mixed CS response.
- Links crystal deformations to quantized Hall conductance.
- Explores anomaly inflow in the presence of dislocations.

## Abstract

For two-dimensional topological insulators, the integer and intrinsic (without external magnetic field) quantum Hall effect is described by the gauge anomalous (2+1)-dimensional [2+1d] Chern-Simons (CS) response for the background gauge potential of the electromagnetic U(1) field. The Hall conductance is given by the quantized prefactor of the CS term, which is a momentum-space topological invariant. Here, we show that three-dimensional crystalline topological insulators with no other symmetries are described by a topological (3+1)-dimensional [3+1d] mixed CS term. In addition to the electromagnetic U(1) gauge field, this term contains elasticity tetrad fields $E^{\ a}_{\mu}({\bf r},t) = \partial_{\mu}X^a(\mathbf{r},t)$ which are gradients of crystalline U(1) phase fields $X^a(\mathbf{r},t)$ and describe the deformations of the crystal. For a crystal in three spatial dimensions $a=1,2,3$ and the mixed axial-gravitational response contains three parameters protected by crystalline symmetries: the weak momentum-space topological invariants. The response of the Hall conductance to the deformations of the crystal is quantized in terms of these invariants. In the presence of dislocations, the anomalous 3+1d CS term describes the Callan-Harvey anomaly inflow mechanism. The response can be extended to all odd spatial dimensions. The elasticity tetrads, being the gradients of the lattice U(1) fields, have canonical dimension of inverse length. Similarly, if such tetrad fields enter general relativity, the metric becomes dimensionful, but the physical parameters, such as Newton's constant, the cosmological constant, and masses of particles, become dimensionless.