Topological indices for open and thermal systems via Uhlmann's phase

TL;DR

This paper introduces a topological classification scheme for open and thermal quantum systems using Uhlmann's phase, extending the TKNN integers to finite temperature and dephasing scenarios, with applications to well-known models.

Contribution

It develops a new integer classification of topological phases for mixed states using Uhlmann's phase, generalizing TKNN invariants to finite temperature and dephasing conditions.

Findings

Topological indices are well-defined below a critical temperature T_c, become trivial above another T_c, and are ill-defined in an intermediate regime.

Application to Haldane and BHZ models demonstrates the scheme's effectiveness.

The approach generalizes to multi-band Chern insulators.

Abstract

Two-dimensional topological phases are characterized by TKNN integers, which classify Bloch energy bands or groups of Bloch bands. However, quantization does not survive thermal averaging or dephasing to mixed states. We show that using Uhlmann's parallel transport for density matrices (Rep. Math. Phys. 24, 229 (1986)), an integer classification of topological phases can be defined for a finite generalized temperature $T$ or dephasing Lindbladian. This scheme reduces to the familiar TKNN classification for $T<T_{{\rm c},1}$, becomes trivial for $T>T_{{\rm c},2}$, and exhibits a `gapless' intermediate regime where topological indices are not well-defined. We demonstrate these ideas in detail, applying them to Haldane's honeycomb lattice model and the Bernevig-Hughes-Zhang model, and we comment on their generalization to multi-band Chern insulators.