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General Lieb-Schultz-Mattis type theorems for quantum spin chains

Yoshiko Ogata, Yuji Tachikawa, Hal Tasaki

TL;DR

The paper develops an operator-algebraic framework to prove generalized Lieb-Schultz-Mattis no-go theorems for 1D quantum spin chains with discrete on-site symmetry, incorporating translation and reflection invariance. It introduces edge-based indices from half-infinite chains and ties them to degree-2 cohomology class data of on-site projective representations, yielding a translation-invariant result that forces the site cohomology to vanish and a reflection-invariant result that imposes c0=2c. The approach provides a unified, rigorous treatment of LSM-type obstructions and clarifies the role of symmetry fractionalization and edge states in SPT-related physics, with extensions discussed to compact Lie groups. Overall, it offers a broadly applicable framework for understanding when a unique gapped ground state is forbidden by symmetry in 1D quantum spin systems.

Abstract

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.

General Lieb-Schultz-Mattis type theorems for quantum spin chains

TL;DR

The paper develops an operator-algebraic framework to prove generalized Lieb-Schultz-Mattis no-go theorems for 1D quantum spin chains with discrete on-site symmetry, incorporating translation and reflection invariance. It introduces edge-based indices from half-infinite chains and ties them to degree-2 cohomology class data of on-site projective representations, yielding a translation-invariant result that forces the site cohomology to vanish and a reflection-invariant result that imposes c0=2c. The approach provides a unified, rigorous treatment of LSM-type obstructions and clarifies the role of symmetry fractionalization and edge states in SPT-related physics, with extensions discussed to compact Lie groups. Overall, it offers a broadly applicable framework for understanding when a unique gapped ground state is forbidden by symmetry in 1D quantum spin systems.

Abstract

We develop a general operator algebraic method which focuses on projective representations of symmetry group for proving Lieb-Schultz-Mattis type theorems, i.e., no-go theorems that rule out the existence of a unique gapped ground state (or, more generally, a pure split state), for quantum spin chains with on-site symmetry. We first prove a theorem for translation invariant spin chains that unifies and extends two theorems proved by two of the authors in [OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that are invariant under the reflection about the origin and not necessarily translation invariant.

Paper Structure

This paper contains 15 sections, 13 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two classes of examples
  3. Outline of the argument
  4. Setting and main results
  5. Examples in section \ref{['s:intro']} in this language
  6. Indices for half-infinite chains and the proofs of theorems
  7. Proof of Lemmas
  8. Discussion
  9. Generalization to compact Lie groups
  10. When $G$ is a compact connected semisimple group
  11. Proof of the theorem \ref{['T:trans']}'
  12. Proof of the lemma \ref{['Le:trick']}
  13. When $G$ is a general compact Lie group
  14. Comment on the case $G=U(1)$
  15. A theorem for reflection invariant models with U(1) symmetry

Key Result

Corollary 1

If $\bar{S}$ is a half-odd integer, then it is never the case that the above translation-invariant model (with $\mathbb{Z}_2\times\mathbb{Z}_2$ or time-reversal invariance) has a unique gapped ground state.

Figures (2)

  • Figure 1: One can associate a unique index $\sigma\in H^2(G,{\rm U}(1)_{\mathfrak{p}})$ with the pure split state $\rho$ restricted onto a half-infinite chain. (a) The indices $\sigma^\mathrm{L}_{x-1}$ and $\sigma^\mathrm{R}_x$ describe the transformation properties of "edges states" that emerge when the infinite chain is decomposed into two half-infinite chains $\{\ldots,x-2,x-1\}$ and $\{x,x+1,\ldots\}$. (b) The half infinite chain $\{x,x+1,\ldots\}$ may be regarded as consisting of the site $x$ and $\{x+1,x+2,\ldots\}$. We have the corresponding identity \ref{['e:2']}, which is a key ingredient of the present work.
  • Figure 2: The lattice $\Lambda_m$ consists of the central site (black dot) and $m$ semi-infinite chains attached to it. The figure is for $m=5$.

Theorems & Definitions (14)

  • Corollary 1
  • Corollary 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7: =Ogata2
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 4 more