Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems

TL;DR

This paper offers an algebraic framework to understand Nielsen-Ninomiya-type no-go theorems in quantum spin systems, highlighting the fundamental algebraic incompatibility between anomalies and local Hilbert space dimensions.

Contribution

It introduces an algebraic perspective on anomaly-based no-go theorems, moving beyond analytic proofs to emphasize symmetry algebra and local anomaly index computability.

Findings

Anomaly indices are incompatible with local Hilbert space dimensions.

Algebraic structure constrains lattice regularizations of quantum spin systems.

Gauge fixing trivializes determinants, imposing nontrivial constraints.

Abstract

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.