Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states

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 $\nu$ 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 $\nu =N$, 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 $N\nmid \nu$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $\nu =N/2$, 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 $\nu$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

16 pages, 3 figures

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Cited by in corpus (1)

• Anomaly of 4d Weyl fermions with discrete symmetries