[論文レビュー] Non-invertible and higher-form symmetries in 2+1d lattice gauge theories

Yichul Choi, Yaman Sanghavi+2 more|arXiv (Cornell University)|May 21, 2024

Physics of Superconductivity and Magnetism被引用数 9

ひとこと要約

この論文は、非可換・高次元対称性を含む 2+1 次元格子モデルを構築・分析し、非可換なデュアリティ演算子を示し、関連する異常と非可換 SPT 相を論じる。

ABSTRACT

We explore exact generalized symmetries in the standard 2+1d lattice $\mathbb{Z}_2$ gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the "Higgs=SPT" proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.

研究の動機と目的

• Motivate how generalized symmetries constrain phase diagrams in 2+1d lattice models.
• Develop explicit lattice realizations of non-invertible and 1-form symmetries in Z2 gauge theories coupled to Ising matter.
• Analyze anomalies involving 1-form symmetries and their relation to Higgs=SPT scenarios.
• Identify and characterize non-invertible SPT phases in 2+1d, including cluster-like constructions.

提案手法

• Define lattice Hamiltonians for Z2 gauge theory coupled to Ising matter on a square/Lieb lattice.
• Construct a non-invertible duality operator D that swaps corresponding terms in the Hamiltonian under h=\tilde{h}, J=\tilde{J} (not implementable by invertible operators).
• Derive the operator algebra of D with other symmetry operators and show it matches a fusion 2-category 2-Rep((Z2(1)×Z2(1))⋊Z2(0)).
• Impose magnetic Gauss law to obtain a constrained Hilbert space and discuss topological vs non-topological 1-form symmetries η(γ).
• Show a mixed anomaly between 0-form U, 1-form η, and a swap symmetry, and relate to Higgs=SPT ideas.
• Present exact non-invertible SPT models, including cluster-like constructions, and discuss generalized Kennedy-Tasaki transformations.

実験結果

リサーチクエスチョン

• RQ1 How do non-invertible symmetries arise and act in 2+1d lattice gauge theories with tensor-product Hilbert spaces?
• RQ2 What is the explicit lattice realization and algebra of the non-invertible duality operator that exchanges Z2(0) and Z2(1) sectors?
• RQ3 How do 1-form symmetries and their anomalies manifest in these lattice models, and how do they relate to Higgs=SPT phenomena?
• RQ4 What are the characteristics and constructions of non-invertible SPT phases in 2+1d, and how do cluster-like models realize them?
• RQ5 Can the lattice realization of non-invertible symmetry be connected to continuum 2-representation theory and fusion categories?

主な発見

• A non-invertible duality operator D exists in a 2+1d lattice Z2 gauge theory coupled to Ising matter when h=\tilde{h} and J=\tilde{J}.
• D obeys the algebra D^2=C, C^2=4C, DC=CD=4D with ηD=Dη=UD=DU=D and C commuting similarly, matching the algebra of a fusion 2-category 2-Rep((Z2^(1)×Z2^(1))⋊Z2^(0)).
• The 1-form symmetry η(γ)=∏ℓ∈γ σ^z_ℓ is topological when the magnetic Gauss law is strictly enforced (g→∞) and becomes non-topological otherwise, enabling explicit breaking by local perturbations.
• There is a mixed anomaly involving the 0-form symmetry, the 1-form symmetry, and the non-invertible swap symmetry, which connects to the Higgs=SPT framework.
• The paper identifies two non-invertible SPT phases in 2+1d, including a cluster-like model, and discusses a generalized Kennedy-Tasaki transformation in this context.
• A lattice realization of non-invertible symmetry is demonstrated via the non-invertible swap operator in a coupled Ising–gauge system, representing a simple 2-Rep fusion structure.