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Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems

Ruizhi Liu

TL;DR

The paper develops an algebraic framework to understand Nielsen–Ninomiya–type no-go theorems from group cohomological anomalies in quantum spin systems. By viewing symmetry actions as locality-preserving automorphisms and introducing an anomaly index valued in $H^{3}(G;U(1))$, it shows that, under local computability, a gauge fixing can trivialize determinants of quasi-local data, imposing strong lattice-regularization constraints. The main result identifies the anomaly class as an element of $H^{3}(G;\mathbb{Z}[n^{-1}]/\mathbb{Z})$ (and, upon stabilization, in $H^{3}(G;\mathbb{Q}/\mathbb{Z})$), explaining why certain anomalous symmetries cannot be realized in spin chains. The approach generalizes to Lie groups, differentiable group cohomology, and connects to operator $K$-theory, providing a broad, dimension-agnostic perspective on anomaly realizability and lattice regularizations with tails or non-onsite actions.

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.

Anomalies in quantum spin systems and Nielsen-Ninomiya type Theorems

TL;DR

The paper develops an algebraic framework to understand Nielsen–Ninomiya–type no-go theorems from group cohomological anomalies in quantum spin systems. By viewing symmetry actions as locality-preserving automorphisms and introducing an anomaly index valued in , it shows that, under local computability, a gauge fixing can trivialize determinants of quasi-local data, imposing strong lattice-regularization constraints. The main result identifies the anomaly class as an element of (and, upon stabilization, in ), explaining why certain anomalous symmetries cannot be realized in spin chains. The approach generalizes to Lie groups, differentiable group cohomology, and connects to operator -theory, providing a broad, dimension-agnostic perspective on anomaly realizability and lattice regularizations with tails or non-onsite actions.

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.
Paper Structure (19 sections, 7 theorems, 75 equations)

This paper contains 19 sections, 7 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Algebras and symmetries
  3. Warm up
  4. Projective representations in $(0+1)d$
  5. Finite group symmetries and QCA actions
  6. The proof to the main theorem
  7. de la Harpe-Skandanis determinant
  8. Comment on Lie group case
  9. The coefficient and operator $K$-theory
  10. Relating different coefficients
  11. Connection to operator $K$-theory
  12. Generality of our proof
  13. Discussion
  14. Acknowledgements
  15. Group cohomology and differentiable group cohomology
  16. ...and 4 more sections

Key Result

Theorem 1.1

Given a spin chain with local Hilbert space dimension $n$, then the 't Hooft anomaly of $G$-symmetry action on this spin chain is classified by $\mathrm{H}^{3}(G;\mathbb{Z}[n^{-1}]/\mathbb{Z})$, where is the (algebraic) localizationWe will write $P_{n}^{-1}$ for the localization functor for $P_{n}$, e.g. $\mathbb{Z}[n^{-1}]:=P_{n}^{-1}\mathbb{Z}$, see e.g. Sec. 4.7 of Ref. rotman2008introduction.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 5.1
  • Definition A.1
  • Definition A.2
  • Example A.1
  • Example A.2
  • Remark A.1
  • Example A.3
  • ...and 6 more