(SPT-)LSM theorems from projective non-invertible symmetries

TL;DR

This paper explores the implications of non-invertible projective symmetries in a 1+1D quantum XY model, revealing new LSM anomalies, SPT phases, and enriched topological orders through a combination of algebraic and topological methods.

Contribution

It establishes a generalized LSM theorem for non-invertible symmetries, constructs explicit SPT ground states, and develops techniques for gauging non-invertible symmetries in lattice models.

Findings

Derived a condition for LSM anomalies with non-invertible symmetries.

Proved an SPT-LSM theorem for gapped ground states with non-invertible symmetries.

Constructed a fixed-point Hamiltonian for a non-invertible SPT phase.

Abstract

Projective symmetries are ubiquitous in quantum lattice models and can be leveraged to constrain their phase diagram and entanglement structure. In this paper, we investigate the consequences of projective algebras formed by non-invertible symmetries and lattice translations in a generalized $1+1$D quantum XY model based on group-valued qudits. This model is specified by a finite group $G$ and enjoys a projective $\mathsf{Rep}(G)\times Z(G)$ and translation symmetry, where symmetry operators obey a projective algebra in the presence of symmetry defects. For invertible symmetries, such projective algebras imply Lieb-Schultz-Mattis (LSM) anomalies. However, this is not generally true for non-invertible symmetries, and we derive a condition on $G$ for the existence of an LSM anomaly. When this condition is not met, we prove an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak symmetry protected topological (SPT) state with non-trivial entanglement, for which we construct an example fixed-point Hamiltonian. The projectivity also affects the dual symmetries after gauging $\mathsf{Rep}(G)\times Z(G)$ sub-symmetries, giving rise to non-Abelian and non-invertible dipole symmetries, as well as non-invertible translations. We complement our analysis with the SymTFT, where the projectivity causes it to be a topological order non-trivially enriched by translations. Throughout the paper, we develop techniques for gauging $\mathsf{Rep}(G)$ symmetry and inserting its symmetry defects on the lattice, which are applicable to other non-invertible symmetries.