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

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...

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Main Author: Salvatore D. Pace, Ho Tat Lam, and Ömer M. Aksoy
Format: Article
Language:English
Published: SciPost 2025-01-01
Series:SciPost Physics
Online Access:https://scipost.org/SciPostPhys.18.1.028
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author Salvatore D. Pace, Ho Tat Lam, and Ömer M. Aksoy
author_facet Salvatore D. Pace, Ho Tat Lam, and Ömer M. Aksoy
author_sort Salvatore D. Pace, Ho Tat Lam, and Ömer M. Aksoy
collection DOAJ
description 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 $\mathsf{Rep}(G)× Z(G)$ 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 $\mathsf{Rep}(G)× Z(G)$ 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.
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spelling doaj-art-9500adaf92294e1da31cd541b2d1db312025-01-22T11:54:32ZengSciPostSciPost Physics2542-46532025-01-0118102810.21468/SciPostPhys.18.1.028(SPT-)LSM theorems from projective non-invertible symmetriesSalvatore D. Pace, Ho Tat Lam, and Ömer M. AksoyProjective 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 $\mathsf{Rep}(G)× Z(G)$ 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 $\mathsf{Rep}(G)× Z(G)$ 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.https://scipost.org/SciPostPhys.18.1.028
spellingShingle Salvatore D. Pace, Ho Tat Lam, and Ömer M. Aksoy
(SPT-)LSM theorems from projective non-invertible symmetries
SciPost Physics
title (SPT-)LSM theorems from projective non-invertible symmetries
title_full (SPT-)LSM theorems from projective non-invertible symmetries
title_fullStr (SPT-)LSM theorems from projective non-invertible symmetries
title_full_unstemmed (SPT-)LSM theorems from projective non-invertible symmetries
title_short (SPT-)LSM theorems from projective non-invertible symmetries
title_sort spt lsm theorems from projective non invertible symmetries
url https://scipost.org/SciPostPhys.18.1.028
work_keys_str_mv AT salvatoredpacehotatlamandomermaksoy sptlsmtheoremsfromprojectivenoninvertiblesymmetries