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Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space
by Nathan Seiberg, Sahand Seifnashri, Shu-Heng Shao
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Submission summary
| Authors (as registered SciPost users): | Nathan Seiberg · Sahand Seifnashri · Shu-Heng Shao |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2401.12281v2 (pdf) |
| Date accepted: | 2024-05-28 |
| Date submitted: | 2024-05-21 04:44 |
| Submitted by: | Shao, Shu-Heng |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.
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Author comments upon resubmission
We thank the two referees for their detailed and helpful comments on our manuscript. We have addressed them in the new version of the draft. We have also fixed the typos pointed out by the referees and a number of other minor mistakes. See below a summary of the more significant changes.
Report 2:
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The definition of simple is not specific to the model that we considered here. Indeed, for a commuting projector Hamiltonian, the identity defect is not simple since the total symmetry of the model is very large and contains local operators that commute with the Hamiltonian. Relatedly, in 1+1d continuum QFTs, symmetries implemented by local operators correspond to multifusion categories. In these cases, the identity defect is not simple.
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We revised the discussion on the "anomaly" involving parity/time-reversal. We only mention that it is reminiscent of anomalies, and we do not claim that these projective phases correspond to obstruction to gauging or additional LSM-type constraints.
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Same as above.
Report 1:
We have only explored the phase diagram of lattice systems where the continuum limit gives the Ising and the tricritical Ising CFTs. It would indeed be interesting to consider more general lattice models and continuum field theories with non-invertible symmetries. We note that there have been some recent papers studying non-invertible symmetries in the 3-state Potts lattice model and CFT.
List of changes
1. We have revised the discussion on the "anomaly" involving parity/time-reversal throughout the manuscript.
2. We expanded the discussion in section 3.4.
3. Various minor typos fixed and references added.
Published as SciPost Phys. 16, 154 (2024)