与维特非平凡量子元胞自动机混合的不可逆对称性
Non-Invertible Symmetries Mixing with Witt Non-Trivial Quantum Cellular Automata
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中文总结 AI 辅助
研究3 + 1维\(\mathbb{Z}_p\) 1形式对称性相关操作，通过构建局部算子代数将其微观实现为量子元胞自动机，确定其中心扩张由维特群推广支配，给出不同\(p\)值下生成元及细化的晶格融合规则。
中文摘要 AI 辅助
自对偶性和对称保护拓扑（SPT）相的堆叠是量子多体系统的基本操作。对于3 + 1维中的\(\mathbb{Z}_p\) 1形式对称性，这些分别对应于Kramers - Wannier - Wegner对偶性\(S\)（非可逆对偶对称性的规范操作）和1形式对称性SPT \(T\)的堆叠。在连续统中，对于\(p = 2\)，它们形成\(PSL(2,\mathbb{Z}_4)\)的中心扩张，对于奇素数\(p\)，形成\(SL(2,\mathbb{Z}_p)\)的中心扩张，其中心元素是具有纯引力响应的可逆理论。这些中心扩张由阿贝尔任意子理论的维特群的扭曲、分级推广所支配。对于\(p = 2\)，所得群是单量子比特克利福德群。我们将此整个结构微观地实现为作用于与立方晶格上的自旋晶格希尔伯特空间相关的特定局部算子代数的量子元胞自动机（QCA）。具体而言，我们的局部算子代数通过从与\(\mathbb{Z}_p\) 1形式对称性对易的所有局部算子开始构建，并除以所有（局部）1形式对称性生成元来构建。中心元素总是可以通过唯一确定的QCA类扩展到完整的张量积代数。对于\(p = 2\)，它们由非平凡半子QCA生成，对于奇素数\(p\)，由非平凡\(\mathbb{Z}_p\)克利福德QCA生成。因此，晶格融合规则仅在这些QCA和晶格平移范围内再现连续统的融合规则，从而产生由QCA细化的融合规则。
英文摘要
Self-dualities and the stacking of symmetry-protected topological (SPT) phases are basic operations on quantum many-body systems. For a $\mathbb{Z}_p$ one-form symmetry in 3+1d these correspond to the Kramers-Wannier-Wegner duality $S$, which is the gauging operation underlying non-invertible duality symmetries, and the stacking of a 1-form symmetry SPT $T$. In the continuum, they form a central extension of $PSL(2,\mathbb{Z}_4)$ for $p=2$, and of $SL(2,\mathbb{Z}_p)$ for odd primes $p$, whose central elements are invertible theories with purely gravitational response. These central extensions are governed by a twisted, graded generalization of the Witt group of abelian anyon theories, which we determine. For $p=2$ the resulting group is the single-qubit Clifford group, with duality and entangler acting as the Hadamard and phase gates. We realize this entire structure microscopically as quantum cellular automata (QCA) acting on a certain local operator algebra associated with a spin lattice Hilbert space on a cubic lattice. Specifically, our local operator algebra is built by starting with all local operators commuting with a $\mathbb{Z}_p$ 1-form symmetry, and taking the quotient by all the (local) 1-form symmetry generators. The central elements can always be extended to the full tensor product algebra with a uniquely defined QCA class. For $p=2$ they are generated by the non-trivial semion QCA, and for odd prime $p$ they are generated by the non-trivial $\mathbb{Z}_p$ Clifford QCA. Consequently the lattice fusion rules reproduce the continuum ones only up to these QCAs and lattice translations, giving rise to fusion rules refined by QCAs.
