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

TL;DR

This paper explores boundary topological orders of (4+1)d fermionic SPT states with anomalous $ ext{Z}_{2N}^F$ symmetry, constructing gapped boundary states and analyzing their topological properties and constraints.

Contribution

It provides a microscopic construction of symmetric gapped boundary states for (4+1)d fermionic SPTs with anomalous $ ext{Z}_{2N}^F$ symmetry, revealing their topological gauge theory descriptions and no-go conditions.

Findings

For $ u=N$, the boundary admits a $ ext{Z}_4$ gauge theory description.

For $ u=N/2$, a non-TQFT symmetric gapped state is constructed.

For other $ u$, symmetric gapped states are not possible within the construction.

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.