Viscoelastic response of topological tight-binding models in two and three dimensions

TL;DR

This paper investigates the Hall viscosity in topological insulators on lattices, develops two methods to implement lattice deformations, and explores their effects on physical properties, including potential experimental signatures.

Contribution

It introduces two novel methods for implementing lattice deformations in topological models and computes the Hall viscosity using these approaches, extending understanding beyond the continuum limit.

Findings

Hall viscosity calculations agree with previous results in continuum regimes

Lattice effects introduce new length scales affecting topological responses

Potential experimental signatures identified through phononic properties

Abstract

The topological response to external perturbations is an effective probe to characterize different topological phases of matter. Besides the Hall conductance, the Hall viscosity is another example of such a response that measures how electronic wave functions respond to changes in the underlying geometry. Topological (Chern) insulators are known to have a quantized Hall conductance. A natural question is how the Hall viscosity behaves for these materials. So far, most of studies on the Hall viscosity of Chern insulators have focused on the continuum limit. The presence of lattice breaks the continuous translational symmetry to a discrete group and this causes two complications: it introduces a new length scale associated with the lattice constant, and makes the momentum periodic. We develop two different methods of how to implement a lattice deformation: (1) a lattice distortion is encoded as a shift in the lattice momentum, and (2) a lattice deformation is treated microscopically in the gradient expansion of the hopping matrix elements. After establishing the method of deformation we can compute the Hall viscosity through a linear response (Kubo) formula. We examine these methods for three models: the Hofstadter model, the Chern insulator, and the surface of a 3D topological insulator. Our results in certain regimes of parameters, where the continuum limit is relevant, are in agreement with previous calculations. We also provide possible experimental signatures of the Hall viscosity by studying the phononic properties of a single crystal 3D topological insulator.