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Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models

Ryohei Kobayashi, Ken Shiozaki, Yuta Kikuchi, Shinsei Ryu

TL;DR

This work extends the Lieb-Schultz-Mattis-Hastings framework to higher-form symmetries, deriving a 1-form U(1) LSMOH theorem and proving it via adiabatic 2-form flux insertion. It applies the theorem to pure U(1) lattice gauge theories that realize the quantum dimer model on bipartite lattices, clarifying how fractional 1-form filling enforces nontrivial ground-state structure or gaplessness, including at the RK point. The continuum perspective reveals a mixed 't Hooft anomaly between U(1)_{[1]} and lattice translation near RK, matching lattice observations of degenerate or gapless phases and providing a field-theoretic diagnostic via background fields. The results bridge lattice constraints and continuum anomalies, pointing to extensions to higher dimensions and broader crystal symmetries as fruitful directions for future work.

Abstract

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U(1)$ charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension $n > 0$. The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the $U(1)$ lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.

Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models

TL;DR

This work extends the Lieb-Schultz-Mattis-Hastings framework to higher-form symmetries, deriving a 1-form U(1) LSMOH theorem and proving it via adiabatic 2-form flux insertion. It applies the theorem to pure U(1) lattice gauge theories that realize the quantum dimer model on bipartite lattices, clarifying how fractional 1-form filling enforces nontrivial ground-state structure or gaplessness, including at the RK point. The continuum perspective reveals a mixed 't Hooft anomaly between U(1)_{[1]} and lattice translation near RK, matching lattice observations of degenerate or gapless phases and providing a field-theoretic diagnostic via background fields. The results bridge lattice constraints and continuum anomalies, pointing to extensions to higher dimensions and broader crystal symmetries as fruitful directions for future work.

Abstract

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension . The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.

Paper Structure

This paper contains 24 sections, 2 theorems, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results and outline
  3. Section \ref{['sec:1form']}:
  4. Section \ref{['sec:QDM']}:
  5. LSMOH theorem with 1-form symmetry
  6. 1-form $U(1)$ symmetry in the continuum
  7. 1-form $U(1)$ symmetry on a lattice
  8. LSMOH theorem with 1-form symmetry
  9. Application to the quantum dimer model
  10. Quantum dimer model and $U(1)$ lattice gauge theory
  11. 1-form symmetry and LSMOH theorem
  12. Continuum field theory description
  13. $U(1)_{[1]}\times \mathcal{T}_1$ symmetry
  14. Quantum anomaly in continuum description
  15. Gauge invariant operators
  16. ...and 9 more sections

Key Result

Theorem 2.1

(LSMOH theorem for 1-form symmetry) Consider a quantum many-body system defined on a $d$-dimensional periodic lattice, in the presence of a global 1-form $U(1)$ symmetry and a translation symmetry along the $l$-th primitive lattice vector, and assume that both symmetries are not broken. Then, if the

Figures (4)

  • Figure 1: ($a$) Configuration of 2-form field $B$ on $x_l x_m$-plane. ($b$) A large gauge transformation.
  • Figure 2: Cartesian coordinate on the honeycomb lattice. Lattice vectors connecting two neighboring vertices are labeled as $\mathbf{e}_\alpha, \mathbf{e}_\beta$ and $\mathbf{e}_\gamma$ respectively.
  • Figure 3: The schematic phase diagram of the QDM on the honeycomb lattice. The 1-form filling $\nu=1$ in the staggered phase. $\nu=1/3$ in the columnar and plaquette phase, which terminate at the RK critical point. There is a sequence of the incommensurate crystal and commensurate ordered phase between the RK point and the staggered phase, Fradkin04Vishwanath04 where $\nu$ increases continuously from 1/3 to 1.
  • Figure 4: ($a$) Configuration of 2-form field $B$ on $x_1 x_2$-plane at the end of the insertion process. ($b$) A large gauge transformation.

Theorems & Definitions (2)

  • Theorem 2.1
  • Theorem A.1