A non-semisimple non-invertible symmetry

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.