A non-semisimple non-invertible symmetry

TL;DR

This paper explores a novel non-invertible symmetry in spin chains, revealing unique junctions and ground state behaviors, including symmetry breaking and inequivalent transformations, through the construction of specific Hamiltonians.

Contribution

It introduces a new non-semisimple, non-invertible symmetry in spin chains with topological lines labeled by the Taft algebra, and analyzes its effects on ground states and symmetry breaking.

Findings

Identified a smooth path of gapped Hamiltonians with inequivalent ground state transformations.

Discovered a model where product and W states spontaneously break the symmetry.

Proposed an explanation for the indistinguishability of certain states in the infinite-volume limit.

Abstract

We investigate the action of a non-invertible symmetry on spins chains whose topological lines are labelled by representations of the four-dimensional Taft algebra. The main peculiarity of this symmetry is the existence of junctions between distinct indecomposable lines. Sacrificing Hermiticity, we construct several symmetric, frustration-free, gapped Hamiltonians with real spectra and analyse their ground state subspaces. Our study reveals two intriguing phenomena. First, we identify a smooth path of gapped symmetric Hamiltonians whose ground states transform inequivalently under the symmetry. Second, we find a model where a product state and the so-called W state spontaneously break the symmetry, and propose an explanation for the indistinguishability of these two states in the infinite-volume limit in terms of the symmetry category.