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Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem

Anton Kapustin, Nikita Sopenko

Abstract

For any locality-preserving action of a group $G$ on a quantum spin chain one can define an anomaly index taking values in the group cohomology of $G$. The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any $G$-invariant Hamiltonian from having $G$-invariant gapped ground states. Lieb-Schultz-Mattis-type theorems are a special case of this result when $G$ involves translations. In the case when the symmetry group $G$ is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.

Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem

Abstract

For any locality-preserving action of a group on a quantum spin chain one can define an anomaly index taking values in the group cohomology of . The anomaly index is a kinematic quantity, it does not depend on the Hamiltonian. We prove that a nonzero anomaly index prohibits any -invariant Hamiltonian from having -invariant gapped ground states. Lieb-Schultz-Mattis-type theorems are a special case of this result when involves translations. In the case when the symmetry group is a Lie group, we define an anomaly index which takes values in the differentiable group cohomology as defined by J.-L. Brylinski and prove a similar result.
Paper Structure (28 sections, 22 theorems, 58 equations, 3 figures)

This paper contains 28 sections, 22 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. 1d quantum spin systems
  4. Locality-Preserving Automorphisms
  5. Group cohomology
  6. Anomaly index for symmetries of quantum spin chains
  7. Anomalous symmetries in quantum mechanics
  8. Anomalous symmetries of quantum spin chains
  9. Absence of symmetric gapped states for systems with an anomalous symmetry.
  10. Anomaly index for a Lie group symmetry
  11. Differentiable group cohomology of Lie groups
  12. Almost local observables
  13. Almost local derivations
  14. Almost Local Automorphisms
  15. Lie group symmetry of a quantum spin chain
  16. ...and 13 more sections

Key Result

Lemma 2.1

A locality-preserving automorphism $\alpha$ has a trivial GNVW index if and only if it admits a decomposition $\alpha = \alpha_{<0} \alpha_0 \alpha_{\geq 0}$ for some $\alpha_{<0} \in {{\mathcal{G}}^{\text{lp}}_{<0}}$, $\alpha_{0} \in {{\mathcal{G}}^{\text{lp}}_0}$, $\alpha_{\geq 0} \in {{\mathcal{G

Figures (3)

  • Figure 1: The block-partitioned unitary $S$ which swaps the original system (blue) and its copy (red).
  • Figure 2: The circuit $\tilde{S}$.
  • Figure 3: The circuit ${\tilde{S}}_+ S_+.$ The orange arrows are implemented first.

Theorems & Definitions (50)

  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.2
  • Corollary 2.1
  • Remark 2.3
  • Proposition 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 40 more