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Lieb-Schultz-Mattis Anomalies and Anomaly Matching

Liujun Zou, Meng Cheng

Abstract

Lieb-Schultz-Mattis (LSM) anomalies are powerful symmetry-based constraints on the correlation, entanglement and dynamics of quantum many-body systems. In this review, we discuss various LSM anomalies and anomaly matching. We start with a pedagogical introduction to these subjects in quantum spin chains, and then generalize the discussion to higher dimensions and other systems. Besides covering the topics related to the standard LSM anomalies, we also review LSM anomalies in disordered systems where the lattice symmetries are only preserved on average, fermionic systems, and systems where the symmetric short-range entangled states are possible but must be nontrivial symmetry-protected topological phases.

Lieb-Schultz-Mattis Anomalies and Anomaly Matching

Abstract

Lieb-Schultz-Mattis (LSM) anomalies are powerful symmetry-based constraints on the correlation, entanglement and dynamics of quantum many-body systems. In this review, we discuss various LSM anomalies and anomaly matching. We start with a pedagogical introduction to these subjects in quantum spin chains, and then generalize the discussion to higher dimensions and other systems. Besides covering the topics related to the standard LSM anomalies, we also review LSM anomalies in disordered systems where the lattice symmetries are only preserved on average, fermionic systems, and systems where the symmetric short-range entangled states are possible but must be nontrivial symmetry-protected topological phases.

Paper Structure

This paper contains 19 sections, 6 theorems, 20 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. LSM ANOMALY IN QUANTUM SPIN CHAINS
  3. The LSM Theorem for Spin Chains
  4. Rigorous Approaches
  5. LSM Anomaly Matching in Low-Energy Theory
  6. Filling Anomaly
  7. LSM IN HIGHER DIMENSIONS
  8. ANOMALY MATCHING
  9. General Theory of UV-IR Matching
  10. LSM Anomalies for Various Lattice Systems
  11. Dirac Spin Liquids
  12. Topological Quantum Spin Liquids
  13. Compressible Phases
  14. OTHER GENERALIZATIONS
  15. Disordered LSM and Average Anomaly
  16. ...and 4 more sections

Key Result

Theorem 2.1

The Hamiltonian in Eq. eq:Heisenberg-chain cannot have both a gapped spectrum and a unique ground state, if $S\in\mathbb{Z}+\frac{1}{2}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Panel (a) shows the generators of the $p6m$ group, including translations $T_1$ and $T_2$, a 6-fold rotation $C_6$ and a mirror reflection $M$. The two translation vectors have the same length, and their angle is $2\pi/3$. The reflection axis of $M$ bisects these two translation vectors. In panel (b), the hexagon is a translation unit cell of the $p6m$ lattice symmetry. There are three irreducible Wyckoff positions, labelled by $a$, $b$ and $c$, and they form the sites of the triangular, honeycomb and kagome lattices, respectively. The $C_6$ rotation center in panel (a) is at a type-$a$ point.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1