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Majorana chain and Ising model -- (non-invertible) translations, anomalies, and emanant symmetries
by Nathan Seiberg, Shu-Heng Shao
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Shu-Heng Shao |
| Submission information | |
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| Preprint Link: | scipost_202402_00026v1 (pdf) |
| Date accepted: | 2024-02-21 |
| Date submitted: | 2024-02-16 19:33 |
| 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 study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity $(-1)^{F_\text{L}}$ emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 't Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.
Author comments upon resubmission
We are grateful to the two referees for their constructive feedback and valuable suggestions.
Following the suggestion of the first referee, we corrected the typos and edited the paragraph below (1.8).
Regarding the first referee's question on the mod 8 anomaly, we agree that it is a challenging problem that deserves further investigation. We do not have a general criterion for the Hamiltonian to realize a given anomaly. All we can infer from the lattice model is the mod 2 part of the said anomaly.
We also added a paragraph at the end of the Conclusions section including the suggestion and reference pointed out by the second referee.
Sincerely,
Nathan Seiberg and Shu-Heng Shao
Published as SciPost Phys. 16, 064 (2024)