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Non-invertible translation from Lieb-Schultz-Mattis anomaly

Tsubasa Oishi, Takuma Saito, Hiromi Ebisu

Abstract

Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.

Non-invertible translation from Lieb-Schultz-Mattis anomaly

Abstract

Symmetry provides powerful non-perturbative constraints in quantum many-body systems. A prominent example is the Lieb-Schultz-Mattis (LSM) anomaly -- a mixed 't Hooft anomaly between internal and translational symmetries that forbids a trivial symmetric gapped phase. In this work, we investigate lattice translation operators in systems with an LSM anomaly. We construct explicit lattice models in two and three spatial dimensions and show that, after gauging the full internal symmetry, translation becomes non-invertible and fuses into defects of the internal symmetry. The result is supported by the anomaly-inflow in view of topological field theory. Our work extends earlier one-dimensional observations to a unified higher-dimensional framework and clarifies their origin in mixed anomalies and higher-group structures, highlighting a coherent interplay between internal and crystalline symmetries.
Paper Structure (23 sections, 134 equations, 4 figures)

This paper contains 23 sections, 134 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of non-invertible translation symmetries in 1D
  3. Dipole symmetry from LSM anomaly
  4. Non-invertible translation symmetry from dipole symmetry
  5. Two dimensional lattice model
  6. Field theoretical interpretation
  7. Recap: LSM anomaly and modulated symmetry
  8. Non-invertible translation symmetry
  9. Outlook
  10. Non-invertible translation symmetry for $\mathbb{Z}_p$
  11. Construction of non-invertible translation operators
  12. Fusion rules
  13. Dipole symmetry in 2D
  14. Gauging $0$-form symmetry
  15. Gauging $1$-form symmetry
  16. ...and 8 more sections

Figures (4)

  • Figure 1: (a) Four terms that constitute the Hamiltonian \ref{['hamiltonian1']}. (b) $0$-form and $1$-form symmetries that the model respects corresponding to \ref{['symmetry']}. (c) Gauss's laws for $0$-form and $1$-form symmetries [first three terms, corresponding to \ref{['g1']}-\ref{['g3']}] and gauge flux operator (last term). (d) Some examples of how operator $W$ (the TST transformation) defined in \ref{['51']} acts on operators.
  • Figure 2: Schematic overview of how different gauging procedures of the same LSM anomaly \ref{['eq:3dbulkanomaly']} lead to qualitatively distinct symmetry structures. Partial gauging of internal symmetries produces spatially modulated (dipole) symmetries, whereas fully gauging all internal symmetries inevitably promotes lattice translation to a non-invertible symmetry defect. This correspondence is naturally understood from anomaly inflow, where the translation defect must carry additional topological degrees of freedom to cancel the mixed anomaly. Here, "gauge $(d-p-1)$- or $p$-form dipole symmetry" means gauging spatially modulated part of the symmetry.
  • Figure 3: Anomaly inflow from the right bulk SPT phase to symmetry defect.
  • Figure 4: (a) Configurations of terms that constitute the Hamiltonian \ref{['hami3']}. The first three configurations correspond to the first term in \ref{['hami3']}, whereas the fourth, fifth, sixth, seventh, eighth configurations do to the second, third, fourth, fifth, and sixth term in \ref{['hami3']}, respectively. (b) The Gauss's law when gauging one of the $1$-form symmetries \ref{['mem1']}, described by \ref{['gauss3d1']}. (c) Illustration for one of examples of the $1$-form dipole algebra \ref{['103']} with $ab=xy$; acting a translation operator $T_x$ on $\mathcal{M}^{Z(1)}_{xy,x}$ (left) gives $\mathcal{M}^{Z(1)}_{xy,0}$ (right).