Equivalence class of emergent single Weyl fermion lattice models in three dimensions: Gapless superconductors and superfluids versus chiral fermions

In this article, we put forward a practical but generic approach towards constructing a large family of (3+1) dimension lattice models which can naturally lead to a single Weyl cone in the infrared (IR) limit. Our proposal relies on spontaneous charge U (1) symmetry breaking to evade the usual no-go theorem of a single Weyl cone in a 3d lattice. We have explored three concrete paths in this approach, all involving fermionic topological symmetry protected states (SPTs). Path a) is to push a gapped SPT in a 3d lattice with time-reversal symmetry (or T-symmetry) to a gapless topological quantum critical point (tQCP) which involves a minimum change of topologies, i. e. δNw=2 where δNw is the change of winding numbers across the tQCP. Path b) is to peal off excessive degrees of freedom in the gapped SPT via applying T-symmetry breaking fields which naturally result in a pair of gapless nodal points of real fermions. Path c) is a hybrid of a) and b) where tQCPs, with δNw 2, are further subject to time-reversal-symmetry breaking actions. In the infrared limit, all the lattice models with single Weyl fermions studied here are isomorphic to either a tQCP in a DIII class topological superconductor with a protecting T-symmetry, or its dual, a T-symmetry breaking superconducting nodal point phase, and therefore form an equivalent class. For a generic T-symmetric tQCP along Path a), the conserved-charge operators span a six-dimensional linear space while for a T-symmetry breaking gapless state along Path b), c), charge operators typically span a two-dimensional linear space instead. Finally, we pinpoint connections between three spatial dimensional lattice chiral fermion models and gapless real fermions that can naturally appear in superfluids or superconductors studied previously.