SymTFT Approach for Mixed States with Non-Invertible Symmetries

TL;DR

This paper introduces a SymTFT-based framework for classifying phases of mixed states with both invertible and non-invertible symmetries, expanding the understanding of symmetry-protected topological phases.

Contribution

It develops a systematic classification method for mixed states with non-invertible symmetries using SymTFT, applicable to both continuum and lattice models.

Findings

Classified phases with non-invertible symmetries in 1+1d

Identified strong-to-weak symmetry breaking phases

Analyzed mixed symmetry-protected topological phases

Abstract

We develop a general framework for studying phases of mixed states with strong and weak symmetries, including non-invertible or categorical symmetries. The central idea is to consider a purification of the mixed state density matrix, which lives in a doubled Hilbert space. We propose a systematic classification of phases in this doubled Hilbert space, relying crucially on the Symmetry Topological Field Theory (SymTFT) approach. This framework applies not only to group symmetries but also, importantly, to non-invertible symmetries. We illustrate the approach in 1+1d to classify phases with strong (non-)invertible symmetries, which include strong-to-weak spontaneous symmetry breaking (SWSSB) phases and mixed strong/weak symmetry-protected topological phases (SPTs). We also develop an approach for studying symmetries that involve a combination of strong and weak symmetries. A noteworthy example of this has weak non-invertible Kramers-Wannier duality symmetry and strong $\mathbb{Z}_2$ symmetry. The continuum description is complemented by a lattice model analysis informed by the SymTFT framework.