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Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases

Gil Young Cho, Chang-Tse Hsieh, Shinsei Ryu

TL;DR

The paper investigates how the Lieb-Schultz-Mattis-Hastings (LSMOH) no-go constraints and the boundaries of symmetry-protected topological (SPT) phases relate through quantum anomalies. It shows that the same low-energy effective field theory (EFT) can describe both a fractionally filled 1D lattice model and the boundary of a 2D SPT, but the anomalies that protect these gapless states differ: a chiral anomaly corresponds to the LSMOH constraint, while an additional Z_N anomaly is intrinsic to SPT boundaries. The authors extend the LSMOH framework to multi-species, multi-charge systems to construct novel symmetric insulators and discuss the perturbative versus non-perturbative nature of stability, including a (3+1)D chiral anomaly analysis in Weyl semimetals that confirms locality of stability near the EFT. Overall, the work provides a unified anomaly-based lens to compare gaplessness enforced by lattice translations and SPT boundaries, with implications for designing exotic symmetric insulators and understanding higher-dimensional topological semimetals.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that, they can be both described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it is a local symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, We find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly, which is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gapless-ness local in the parameter space.

Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases

TL;DR

The paper investigates how the Lieb-Schultz-Mattis-Hastings (LSMOH) no-go constraints and the boundaries of symmetry-protected topological (SPT) phases relate through quantum anomalies. It shows that the same low-energy effective field theory (EFT) can describe both a fractionally filled 1D lattice model and the boundary of a 2D SPT, but the anomalies that protect these gapless states differ: a chiral anomaly corresponds to the LSMOH constraint, while an additional Z_N anomaly is intrinsic to SPT boundaries. The authors extend the LSMOH framework to multi-species, multi-charge systems to construct novel symmetric insulators and discuss the perturbative versus non-perturbative nature of stability, including a (3+1)D chiral anomaly analysis in Weyl semimetals that confirms locality of stability near the EFT. Overall, the work provides a unified anomaly-based lens to compare gaplessness enforced by lattice translations and SPT boundaries, with implications for designing exotic symmetric insulators and understanding higher-dimensional topological semimetals.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem dictates that emergent low-energy states from a lattice model cannot be a trivial symmetric insulator if the filling per unit cell is not integral and if the lattice translation symmetry and particle number conservation are strictly imposed. In this paper, we compare the one-dimensional gapless states enforced by the LSM theorem and the boundaries of one-higher dimensional strong symmetry-protected topological (SPT) phases from the perspective of quantum anomalies. We first note that, they can be both described by the same low-energy effective field theory with the same effective symmetry realizations on low-energy modes, wherein non-on-site lattice translation symmetry is encoded as if it is a local symmetry. In spite of the identical form of the low-energy effective field theories, we show that the quantum anomalies of the theories play different roles in the two systems. In particular, We find that the chiral anomaly is equivalent to the LSM theorem, whereas there is another anomaly, which is not related to the LSM theorem but is intrinsic to the SPT states. As an application, we extend the conventional LSM theorem to multiple-charge multiple-species problems and construct several exotic symmetric insulators. We also find that the (3+1)d chiral anomaly provides only the perturbative stability of the gapless-ness local in the parameter space.

Paper Structure

This paper contains 25 sections, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. LSMOH and no-go theorem; On-site v.s. non-on-site symmetries
  3. LSMOH v.s. SPT anomalies
  4. Effective symmetry and perturbative stability
  5. Summary
  6. 2d SPTs and 1d lattice models
  7. (2+1)d QSHE
  8. (1+1)d lattice spinless fermions
  9. Effective symmetry
  10. Other examples
  11. Anomaly and LSMOH theorem
  12. Chiral anomaly and the LSMOH theorem
  13. Chiral Anomaly: Field Theory
  14. Chiral Anomaly: Lattice formulation
  15. Extension of the LSMOH theorem
  16. ...and 10 more sections

Figures (1)

  • Figure 1: Semi-classical illustrations of Anomaly. (A) (1+1)d metallic state. (B) On adiabatic insertion of the flux by $2\pi$, one state at the left is pumped to the right. Equivalently, the momentum labeling each state is shifted by $\frac{2\pi}{L}$. (C) Spectrum of the Wely semimetal in cubic lattice. Number of the state below the chemical potential $\mu=0$ is precisely $L_x \times L_y \times L_z$, which is equivalent to the number of the electrons. (D) On applying the mangetic field, the band structure is changed.