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 $({\phi }_X,{\phi }_Z)$, we introduce the associated topological superselection profile ${\mathcal{S}}_{\sigma }:={\tau }_{\ge 1}\mathbf{R}{\mathrm{Hom}}_R(\overline{\mathrm{coker}{\phi }_{\sigma }},R),\sigma =X,Z$. Its cohomology layers $E_{\sigma }^{\ell }:=H^{\ell }({\mathcal{S}}_{\sigma }){\cong }_R{\mathrm{Ext}}_R^{\ell }(\overline{\mathrm{coker}{\phi }_{\sigma }},R)$ encode sectors, fusion, and translation action. When finite, the $\ell =1,2,\dots$ layers describe pointlike, looplike, and higher-dimensional excitations; ${\mathcal{S}}_{\sigma }$ retains inter-layer gluing. For regular Koszul models, we compute $E_{\sigma }^{\ell }$ explicitly, find ${\mathcal{S}}_{\sigma }$ single-layer, and obtain finite-size $Z$-logicals from $\mathrm{Tor}$. Single-layer means that at most one positive-degree layer $E_{\sigma }^{\ell }$ 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 $E_{\sigma }^{\ell }$ 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.

34 pages, 3 figures, 2 tables