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Symmetry Spans and Enforced Gaplessness

Takamasa Ando, Kantaro Ohmori

TL;DR

The paper introduces the symmetry-span mechanism, wherein a common sub-symmetry $\mathcal{E}$ embeds into two larger symmetries $\mathcal{C}$ and $\mathcal{D}$, so a gapped phase must restrict to compatible pullbacks; gaplessness is forced when the common pullback intersection is empty. This general criterion is applied in 1+1D to construct explicit symmetry spans using non-invertible and ordinary symmetries (e.g., $\mathsf{TY}(\mathbb{Z}_N)$ and $\mathrm{Rep}(D_8)$), and to realize them on lattices via on-site Hamiltonians that flow to gapless CFTs such as $U(1)_{2k}$ WZW and Spin$(4)_1$; the lattices realize dualities and gapped phases consistent with the pullback constraints. The continuum examples show that the IR theories acquire anomalous continuous symmetries that underlie the gaplessness, while lattice realizations use only non-anomalous UV symmetries to generate emergent anomalies in the IR. Overall, the symmetry-span framework provides a controlled route to enforce gaplessness through symmetry-compatible embeddings, with potential extensions to higher dimensions and connections to LSM-type constraints and fracton-like phases.

Abstract

Anomaly matching for continuous symmetries has been the primary tool for establishing symmetry enforced gaplessness - the phenomenon where global symmetry alone forces a quantum system to be gapless in the infrared. We introduce a new mechanism based on \textit{symmetry spans}: configurations in which a global symmetry $\mathcal{E}$ is simultaneously embedded into two larger symmetries, as $\mathcal{D}\hookleftarrow\mathcal{E}\hookrightarrow\mathcal{C}$. Any gapped phase with the full symmetry must, upon restriction to $\mathcal{E}$, arise as the restriction of both a gapped $\mathcal{C}$-symmetric phase and a gapped $\mathcal{D}$-symmetric phase. When no such compatible phase exists, gaplessness is enforced. This mechanism can operate with only discrete and non-anomalous continuous symmetries in the UV, both of which admit well-understood lattice realizations. We construct explicit symmetry spans enforcing gaplessness in 1+1 dimensions, exhibit their realization in conformal field theories, and provide lattice Hamiltonians with the relevant symmetry embeddings.

Symmetry Spans and Enforced Gaplessness

TL;DR

The paper introduces the symmetry-span mechanism, wherein a common sub-symmetry embeds into two larger symmetries and , so a gapped phase must restrict to compatible pullbacks; gaplessness is forced when the common pullback intersection is empty. This general criterion is applied in 1+1D to construct explicit symmetry spans using non-invertible and ordinary symmetries (e.g., and ), and to realize them on lattices via on-site Hamiltonians that flow to gapless CFTs such as WZW and Spin; the lattices realize dualities and gapped phases consistent with the pullback constraints. The continuum examples show that the IR theories acquire anomalous continuous symmetries that underlie the gaplessness, while lattice realizations use only non-anomalous UV symmetries to generate emergent anomalies in the IR. Overall, the symmetry-span framework provides a controlled route to enforce gaplessness through symmetry-compatible embeddings, with potential extensions to higher dimensions and connections to LSM-type constraints and fracton-like phases.

Abstract

Anomaly matching for continuous symmetries has been the primary tool for establishing symmetry enforced gaplessness - the phenomenon where global symmetry alone forces a quantum system to be gapless in the infrared. We introduce a new mechanism based on \textit{symmetry spans}: configurations in which a global symmetry is simultaneously embedded into two larger symmetries, as . Any gapped phase with the full symmetry must, upon restriction to , arise as the restriction of both a gapped -symmetric phase and a gapped -symmetric phase. When no such compatible phase exists, gaplessness is enforced. This mechanism can operate with only discrete and non-anomalous continuous symmetries in the UV, both of which admit well-understood lattice realizations. We construct explicit symmetry spans enforcing gaplessness in 1+1 dimensions, exhibit their realization in conformal field theories, and provide lattice Hamiltonians with the relevant symmetry embeddings.
Paper Structure (22 sections, 75 equations, 2 figures, 1 table)

This paper contains 22 sections, 75 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-invertible symmetries in 1+1D
  3. Symmetry embedding and pullback of module categories
  4. Finite group symmetry
  5. Anomalous finite group symmetry
  6. Embedding into a continuous symmetry
  7. ZN embedded into Tambara--Yamagami category
  8. Rep(D8) symmetry and its pullback to Z2xZ2
  9. Symmetry enforced gaplessness
  10. Examples
  11. Continuum Examples
  12. U(1)2k WZW CFT
  13. T2 CFT
  14. Commuting triple
  15. Lattice realization
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Auto-equivalence of $\mathop{\mathrm{Vect}}\nolimits_H$ by an SPT phase $\beta\in \mathrm{H}^2(H,\mathop{\mathrm{U}}\nolimits(1))$. A symmetry operator $U_g$ is dressed by a strip of the SPT phase $\beta$, whose boundaries are the symmetry-breaking boundary $\mathcal{B}$ and its orientation reversal $\overline{\mathcal{B}}$.
  • Figure 2: Relations of several CFTs and gauging operations.