Mixed-State Topological Phase: Quantized Topological Order Parameter and Lieb-Schultz-Mattis Theorem

TL;DR

This paper extends the concept of topological phases to mixed states in one-dimensional spin systems, introducing a quantized order parameter and generalizing the Lieb-Schultz-Mattis theorem without relying on spectral gaps.

Contribution

It proposes a quantized topological order parameter for mixed states and generalizes the Lieb-Schultz-Mattis theorem to mixed states without spectral gap assumptions.

Findings

Quantized topological order parameter for mixed states

Concrete models realizing distinct phases with disorder

Generalization of Lieb-Schultz-Mattis theorem to mixed states

Abstract

We investigate the extension of pure-state symmetry protected topological phases to mixed-state regime with a strong U(1) and a weak $\mathbb{Z}_2$ symmetries in one-dimensional spin systems by the concept of quantum channels. We propose a corresponding topological phase order parameter for short-range entangled mixed states by showing that it is quantized and its distinct values can be realized by concrete spin systems with disorders, sharply signaling phase transitions among them. We also give a model-independent way to generate two distinct phases by various types of translation and reflection transformations. These results on the short-range entangled mixed states further enable us to generalize the conventional Lieb-Schultz-Mattis theorem to mixed states, even without the concept of spectral gaps and lattice Hamiltonians.