AI 中文总结
该研究引入拓扑超选择剖面刻画平移不变CSS码的SET相，证明正则Koszul模型的单层刚性，并发现层间粘合携带单一层无法观测的SET相信息。
AI 中文摘要
Koszul复稳定子模型将环面码层级与双变量自行车码统一在同一个同调框架下。为研究这些模型中、以及更普遍的具有稳定子映射$(φ_X,φ_Z)$的平移不变Calderbank-Shor-Steane（CSS）码中的平移对称性富集拓扑（SET）相，我们引入了相关的拓扑超选择剖面$\mathscr S_σ:=τ_{\u22651}\mathbf R\\!\operatorname{Hom}_R(\overline{\operatorname{coker}φ_σ},R),\\ σ=X,Z$。其同调层$E_σ^\ell:=H^\ell(\mathscr S_σ)\cong_R\operatorname{Ext}_R^\ell(\overline{\operatorname{coker}φ_σ},R)$编码了扇区、融合和平移作用。当有限时，$\ell=1,2,\ldots$层分别描述类点、类环及更高维激发；$\text{\mathscr S}_σ$保留了层间粘合关系。对于正则Koszul模型，我们显式计算了$E_σ^\ell$，发现$\text{\mathscr S}_σ$为单层结构，并从$\text{Tor}$得到有限尺寸$Z$-逻辑量子比特。单层意味着至多有一个正次数层$E_σ^\ell$非零。我们的矩阵级Schanuel引理方法证明了刚性：在有限分辨率假设下，非零层和次数唯一确定了拓扑CSS码在稳定平移不变Clifford层面的平移SET序。对于素数维量子比特，该假设自动成立，有限层的平移核给出了到环面码堆叠的精确最小粗粒化。我们实现了可容许的单层数据。在此范畴之外，3D量子比特环面码的分裂与非分裂扩张产生了8个模型，它们所有的$E_σ^\ell$层都相同且具有平凡平移作用，但具有不同的尺寸相关基态简并度，因此属于不同的平移SET相。由此可见层间粘合携带了单个层无法观测到的SET数据。
英文摘要
Koszul-complex stabilizer models unify the toric-code hierarchy and bivariate-bicycle codes in one homological framework. To study translation-symmetry-enriched topological (SET) phases in these models and, more generally, in translation-invariant Calderbank-Shor-Steane (CSS) codes with stabilizer maps $(φ_X,φ_Z)$, we introduce the associated topological superselection profile $\mathscr S_σ:=τ_{\geq1}\mathbf R\!\operatorname{Hom}_R(\overline{\operatorname{coker}φ_σ},R),\ σ=X,Z$. Its cohomology layers $E_σ^\ell:=H^\ell(\mathscr S_σ)\cong_R\operatorname{Ext}_R^\ell(\overline{\operatorname{coker}φ_σ},R)$ encode sectors, fusion, and translation action. When finite, the $\ell=1,2,\ldots$ layers describe pointlike, looplike, and higher-dimensional excitations; $\mathscr S_σ$ retains inter-layer gluing. For regular Koszul models, we compute $E_σ^\ell$ explicitly, find $\mathscr S_σ$ single-layer, and obtain finite-size $Z$-logicals from $\operatorname{Tor}$. Single-layer means that at most one positive-degree layer $E_σ^\ell$ is nonzero. Our matrix-level Schanuel-lemma method proves rigidity: under a finite-resolution hypothesis, the nonzero layer and degree uniquely determine a topological CSS code's translation SET order at the stable translation-invariant Clifford level. For prime qudits the hypothesis is automatic, and a finite layer's translation kernel gives the exact minimal coarse graining to a toric-code stack. We realize admissible single-layer data. Beyond this regime, split and nonsplit extensions of 3D qubit toric codes yield eight models sharing all identical $E_σ^\ell$ layers with trivial translation action but having distinct size-dependent ground-state degeneracies, hence distinct translation SET phases. Thus inter-layer gluing carries SET data invisible to individual layers.
Comments34 pages, 3 figures, 2 tables
