Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers

TL;DR

This paper introduces a new topological invariant called the topological Uhlmann number for classifying density matrices in 2D fermionic topological phases, enabling the study of thermal effects and revealing novel thermal-topological transitions.

Contribution

It constructs the topological Uhlmann number based on the Uhlmann phase, extending topological classification to finite-temperature density matrices in 2D fermionic systems.

Findings

The topological Uhlmann number classifies symmetry-protected topological orders at finite temperature.

Thermal topological phase diagrams are computed for various models, showing temperature effects.

Discovery of new thermal-topological transitions between non-trivial phases with high Chern numbers.

Abstract

We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number ${\rm n}_{\rm U}$. With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature $T$ is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two non-trivial phases in a model with high Chern numbers. At small temperature we recover the standard topological phases as the Uhlmann number approaches to the Chern number.