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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
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/2401.12281v1 (pdf) |
| Date submitted: | 2024-04-01 06:08 |
| Submitted by: | Shao, Shu-Heng |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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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. We present a clear notion of an anomaly involving this non-invertible symmetry, parity/time-reversal symmetries, and lattice translations. 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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Report
This article investigated non-invertible symmetry associated with the Kramers-Wannier transformation on a tensor product Hilbert space of qubits on a periodic 1d chain. The authors pointed out that the lattice symmetry does not form a fusion category and that the Frobenius–Schur indicator of the continuum fusion category is not meaningful on the lattice. One of the main results is contained in Section 4.2 where an LSM-type constraint based on the non-invertible lattice symmetry was proposed. This article contains several important results, with remarkable clarity of exposition. The referee therefore recommends that the manuscript should be published in SciPost. Nevertheless, prior to publication, the authors are asked to address the following comments and questions.
Requested changes
1. The authors demonstrated that the LSM-type constraint discussed in Section 4.2 applied to the critical Ising Hamiltonian and deformations that preserve the non-invertible symmetry. The examples provided in the articles are Ising and tricritical Ising CFTs. The referee would like to ask the author to clarify whether there are any other CFTs in the continuum that can be described by such Hamiltonians. The referee also asks the authors to provide more examples if possible. (The referee is aware that the minimal models in the continuum were discussed in Appendix I. However, even the discussion there seemed to put emphasis on the tricritical Ising CFT.)
2. A similar comment applies to Figure 1. Can this or a similar phase diagram describe the flows between other CFTs? How about the other flows discussed in Section 7.2 of [arXiv:1802.04445]?
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)