Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States

TL;DR

This paper introduces a tensor network method called LPDO to classify and analyze symmetry-protected topological phases in open quantum systems, extending concepts of injectivity and providing explicit constructions.

Contribution

It extends the concept of injectivity to LPDOs in multiple dimensions and develops a classification scheme for ASPT phases in open systems.

Findings

Extended injectivity concepts for LPDOs in 1+1D and 2+1D

Classification scheme for ASPT phases consistent with previous methods

Explicit construction of fixed-point LPDOs for ASPT phases

Abstract

We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological (SPT) phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in $(1+1)D$ and $(2+1)D$ systems, unveiling two distinct types of injectivity conditions inherent in short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. We first illustrate our framework with ASPT phases protected by fermion parity symmetry, then extend the classification of ASPT phases to a general group extension. We demonstrate examples of explicit construction of fixed-point LPDOs for ASPT phases including intrinsic ASPTs in both $(1+1)D$ and $(2+1)D$ systems.