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We explore exact generalized symmetries in the standard 2+1d lattice\mathbb{Z}_2 gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation.
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This content will become publicly available on December 3, 2025
Intertwined order of generalized global symmetries
We investigate the interplay of generalized global symmetries in 2+1 dimensions by introducing a lattice model that couples a ZN clock model to a ZN gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed ’t Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend our lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules.
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- Award ID(s):
- 2225920
- PAR ID:
- 10565212
- Publisher / Repository:
- 2412.02748
- Date Published:
- Edition / Version:
- v1
- Format(s):
- Medium: X
- Institution:
- University of Illinois at Urbana-Champaign
- Sponsoring Org:
- National Science Foundation
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