Density Matrix Topological Insulators

TL;DR

This paper introduces a framework for stabilizing topological insulators in dissipative systems using a generalized Chern number for mixed states, demonstrated on the Haldane model with thermal noise.

Contribution

It defines the band Liouvillian and the density matrix Chern value, extending topological classification to dissipative, noisy environments, and relates it to finite-temperature conductivity.

Findings

Density matrix Chern value witnesses topological order in mixed states.

Topological properties can be preserved under dissipation using the proposed formalism.

Application to the Haldane model confirms the approach's validity.

Abstract

Thermal noise can destroy topological insulators (TI). However we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices.