M. Cheng

J. Wang

X. Yang

We show that Weyl fermions and anomalous topological order in 4 dimensions can live on the edge of the same 5-dimensional superconductor.

We construct (3+1)d fermionic topological orders with anomalous discrete symmetry of order-2N (fermion parity as a subgroup), related to the chiral anomaly of Weyl fermions. Using crystalline correspondence, we construct possible symmetric gapped boundaries for (4+1)d symmetry-protected topological states. Either a <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_4</annotation></semantics></math>Z4​ gauge theory works, or a non-TQFT state exists; otherwise, symmetric gapped boundaries are absent.

We investigate (3+1)d topological orders in fermionic systems with an anomalous ℤF2N symmetry, where its ℤF2 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. Guided by the crystalline correspondence principle, we construct (3+1)d symmetry-preserving gapped states on the boundary of a closely related (4+1)d CN×ℤF2 symmetry-protected topological (SPT) state (with CN 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 ℤ4 gauge theory description at low energy, and propose that a similar theory saturates the corresponding ℤF2N anomaly. For N∤ν, our construction cannot produce any topological quantum field theory (TQFT) symmetric gapped state; but for ν=N/2, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological orders inhomogeneously. For other values of ν, no symmetric gapped state is possible within our construction, which is consistent with the theorem by Cordova-Ohmori.

Topological boundary

(3+1)d boundary topological order of (4+1)d fermionic SPT state