[论文解读] (SPT-)LSM theorems from projective non-invertible symmetries

Salvatore D. Pace, Ho Tat Lam+1 more|arXiv (Cornell University)|Sep 26, 2024

Quantum chaos and dynamical systems被引用 6

一句话总结

本论文分析由非可逆 Rep(G)×Z(G) 对称与在1+1D G-qudit XY模型中的平移形成的投影代数，推导出LSM异常或非可逆弱SPT纠缠基态，并发展 gauging 与 SymTFT 框架。

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 $\mathsf{Rep}(G) imes Z(G)$ 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 $\mathsf{Rep}(G) imes Z(G)$ 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.

研究动机与目标

• Motivate how projective algebras of non-invertible internal symmetries with translation constrain quantum phases and entanglement.
• Investigate a generalized 1+1D XY model built from a finite group G and its center Z(G).
• Determine when the Rep(G)×Z(G) symmetry with translations yields an LSM anomaly versus an SPT-LSM constraint.
• Characterize the entanglement structure of ground states under the identified constraints.
• Provide explicit constructions and examples illustrating non-invertible weak SPTs and their gauging.
• Outline how the SymTFT framework captures the enriched topological order arising from translations.

提出的方法

• Define G-qudits and group based Pauli operators to realize the Hilbert space and symmetry actions.
• Insert Rep(G) symmetry defects to derive the projective algebra with Z(G) and translations.
• Derive conditions on G, notably the role of the center and commutator subgroup, that determine LSM anomalies or SPT-LSM constraints.
• Construct exactly solvable models realizing non-invertible weak SPTs, exemplified by G = D8.
• Demonstrate gauging procedures for Z(G) and Rep(G) symmetries to obtain non-invertible dipole and translation symmetries.
• Employ SymTFT to interpret the gauging web as a topological order enriched by translations.]
• research_questions:[
• What projective algebras arise from inserting Rep(G)×Z(G)×Z_L symmetry defects in a G-based XY model?
• Under what conditions on G does the Rep(G)×Z(G)×Z_L symmetry lead to an LSM anomaly versus a non-invertible weak SPT constraint?
• How does non-invertibility of Rep(G) affect dual symmetries after gauging, and what new dipole or translation symmetries emerge?
• Can one realize and classify non-invertible weak SPTs protected by Rep(G)×Z(G)×Z_L and relate them to crystalline equivalence principles?
• How can gauging procedures and SymTFT capture the full gauging web and the resulting topological order enriched by translations?

实验结果

研究问题

• RQ1 What projective algebras arise from inserting Rep(G)×Z(G)×Z_L symmetry defects in a G-based XY model?
• RQ2 Under what conditions on G does the Rep(G)×Z(G)×Z_L symmetry lead to an LSM anomaly versus a non-invertible weak SPT constraint?
• RQ3 How does non-invertibility of Rep(G) affect dual symmetries after gauging, and what new dipole or translation symmetries emerge?
• RQ4 Can one realize and classify non-invertible weak SPTs protected by Rep(G)×Z(G)×Z_L and relate them to crystalline equivalence principles?
• RQ5 How can gauging procedures and SymTFT capture the full gauging web and the resulting topological order enriched by translations?

主要发现

• The center of G controls whether a non-invertible Rep(G)×Z(G)×Z_L symmetry yields an LSM anomaly or a non-invertible weak SPT constraint.
• If Z(G) is not contained in the commutator subgroup, an LSM anomaly enforces long-range entanglement and excludes symmetric SPTs.
• If Z(G) lies in the commutator subgroup, all allowed SPTs must be non-invertible weak SPTs with nontrivial entanglement and translation-dressed symmetry charges.
• An exact solvable model for G = D8 realizes the non-invertible weak SPTs and connects to SPTs protected by Rep(D8)×Z2×Z2 symmetry.
• Gauging the Z(G) and Rep(G) symmetries yields a gauging web with non-Abelian dipole and non-invertible translation symmetries.
• The SymTFT description shows a symmetry-enriched topological order where translations act nontrivially on anyons.]