From Koszul-Complex Stabilizer Models to Superselection Profiles: Topological Rigidity and Nonsplit Extensions

Abstract

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.