Phases with non-invertible symmetries in 1+1D $\unicode{x2013}$ symmetry protected topological orders as duality automorphisms

TL;DR

This paper investigates 1+1D gapped phases with non-invertible symmetries, revealing new SPT phases with distinct bulk excitations and developing a framework to classify both SPT and SSB phases using symmetry-topological-order concepts.

Contribution

It introduces a novel approach to classify gapped phases with non-invertible symmetries, including SPT and SSB phases, using symmetry-topological-order automorphisms, and uncovers new SPT phases with unique bulk excitations.

Findings

Discovery of new SPT phases with different bulk excitations

Development of a classification framework using symmetry-topological-order

Analysis of gapless phases with non-invertible symmetries

Abstract

We explore 1+1 dimensional (1+1D) gapped phases in systems with non-invertible symmetries, focusing on symmetry-protected topological (SPT) phases (defined as gapped phases with non-degenerate ground states), as well as SPT orders (defined as the differences between gapped/gapless phases with identical bulk excitations spectrum). For group-like symmetries, distinct SPT phases share identical bulk excitations and always differ by SPT orders. However, for certain non-invertible symmetries, we discover novel SPT phases that have different bulk excitations and thus do not differ by SPT orders. Additionally, we also study the spontaneous symmetry-breaking (SSB) phases of non-invertible symmetries. Unlike group-like symmetries, non-invertible symmetries lack the concept of subgroups, which complicates the definition of SSB phases as well as their identification. This challenge can be addressed by employing the symmetry-topological-order (symTO) framework for the symmetry. The Lagrangian condensable algebras and automorphisms of the symTO facilitate the classification of gapped phases in systems with such symmetries, enabling the analysis of both SPT and SSB phases (including those that differ by SPT orders). Finally, we apply this methodology to investigate gapless phases in symmetric systems and to study gapless phases differing by SPT orders.