In this article, we put forward a practical but generic approach towards constructing a large family of $(3+1)$-dimensional (3D) lattice models which can naturally lead to a single Weyl cone in the infrared limit. Our proposal relies on spontaneous charge $\text{U}\left(1\right)$ symmetry breaking via a pair condensate 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., $\delta N_w=2$ where $\delta N_w$ is the change of winding numbers across the tQCP. Path (b) is to peel 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 $\delta N_w\ge 2$, are further subject to time-reversal symmetry breaking actions. We identify detailed phase boundaries of emergent single Weyl fermions in these lattice models. The whole family of lattice models with single Weyl fermions here can be effectively encoded in two dual copies of (1, 1) representations of a Spin(4) group. 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 equivalence 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 paths (b) and (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.

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• Critical phenomena
• Quantum phase transitions
• Superconductivity
• Superfluidity
• Symmetry protected topological states
• Topological superconductors
• Topological materials
• Bogoliubov-de Gennes equations
• Projective symmetries
• Symmetries in condensed matter