Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems

We provide an algebraic perspective on Nielsen--Ninomiya-type no-go theorems arising from group cohomological anomalies, revisiting in particular the version proved by Kapustin and Sopenko. Departing from their analytic proof, our approach emphasizes the algebraic structure of symmetry actions and the local computability of anomaly indices. We demonstrate that this no-go theorem is due to a fundamental algebraic incompatibility between anomaly data and the dimension of local Hilbert spaces. Specifically, when an anomaly index is locally computable via quasi-local unitary operators, a suitable gauge fixing trivializes their (generalized) determinants, imposing unexpected and nontrivial constraints on lattice regularizations.