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arXiv:2411.05786 · Physical review. B./Physical review. B

Boundary topological orders of ( 4 + 1 )d fermionic Z 2 N F symmetry protected topological states

Meng Cheng, Juven Wang, Xinping Yang

Published Feb 4, 2026• 1 citation
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Abstract

We investigate (3+1)d topological orders in fermionic systems with an anomalous $\mathbb{Z}_{2N}^{\mathrm{F}}$ symmetry, where its $\mathbb{Z}_2^{\mathrm{F}}$ subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of $ν$ copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous $\mathbb{Z}_{2N}^\mathrm{F}$ symmetry of (3+1)d Weyl fermion to the anomalous $C_N \times \mathbb{Z}_2^\mathrm{F}$ symmetry. Then we microscopically construct symmetry-preserving gapped boundary states of the closely-related (4+1)d $C_N\times \mathbb{Z}_2^{\mathrm{F}}$ symmetry-protected topological (SPT) state (with $C_N$ being the $N$-fold rotation), whenever it is possible. In particular, for $ν=N$, we show that the (3+1)d symmetric gapped state admits a topological $\mathbb{Z}_4$ gauge theory description at low energy, and propose that a similar theory saturates the corresponding $\mathbb{Z}_{2N}^\mathrm{F}$ anomaly. For $N \nmid ν$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $ν=N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological order inhomogeneously. For other values of $ν$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

DOI
10.1103/bjvz-tmj2

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