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In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) andp -ality (orderp ) mapping on the\mathbb{Z}_p× \mathbb{Z}_p symmetric Hamiltonians, wherep is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.
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Levin-Wen is a Gauge Theory: Entanglement from Topology
Abstract We show that the Levin-Wen model of a unitary fusion category$${\mathcal {C}}$$ is a gauge theory with gauge symmetry given by the tube algebra$${\text {Tube}}({\mathcal {C}})$$ . In particular, we define a model corresponding to a$${\text {Tube}}({\mathcal {C}})$$ symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case$${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ , we show how our procedure reduces to the twisted gauging of a trivalG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.
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- PAR ID:
- 10618293
- Publisher / Repository:
- Springer Nature Link
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 405
- Issue:
- 11
- ISSN:
- 0010-3616
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s00220-024-05144-x
