Symmetry Spans and Enforced Gaplessness

TL;DR

The paper introduces a new mechanism using symmetry spans to enforce gaplessness in quantum systems, expanding beyond traditional anomaly matching methods, with explicit constructions in 1+1 dimensions.

Contribution

It develops the concept of symmetry spans involving embeddings into larger symmetries to establish gaplessness, applicable with discrete and non-anomalous continuous symmetries.

Findings

Constructed explicit symmetry spans enforcing gaplessness in 1+1 dimensions.

Demonstrated realization of symmetry spans in conformal field theories.

Provided lattice Hamiltonians with the relevant symmetry embeddings.

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 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.