Equivalent class of Emergent Single Weyl Fermion in 3d Topological States: gapless superconductors and superfluids Vs 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. $δN_w=2$ where $δN_w$ 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 $δN_w \geq 2$, 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.

💡 Research Summary

This paper proposes a systematic and generic construction of three‑dimensional lattice models that host a single Weyl fermion cone in the infrared (IR) limit, thereby circumventing the Nielsen‑Ninomiya fermion‑doubling theorem. The key idea is to break the global charge‑U(1) symmetry spontaneously via a pair condensate, which converts the usual complex fermion description into a real‑fermion (BdG) framework. Real fermions are intrinsically charge‑conjugation symmetric, and this symmetry forces band‑crossing points to appear in ±p pairs. By interpreting these crossings as intersections of two three‑dimensional oriented sub‑manifolds within a six‑dimensional manifold, the authors show that the number of Weyl cones is directly linked to the parity of the intersection number. In the real‑fermion representation, an odd number of intersections (M = 1, 3, …) yields an odd number of Weyl cones, while an even number corresponds to Dirac cones (pairs of opposite chirality), reproducing the usual no‑go theorem for U(1)‑preserving systems.

Three concrete routes to achieve a single Weyl cone are explored:

• Path a (time‑reversal‑preserving) – Start from a three‑dimensional symmetry‑protected topological (SPT) phase with time‑reversal (T) symmetry. By tuning parameters to a topological quantum critical point (tQCP) belonging to class DIII, the system undergoes a minimal change of topological invariant δN_w = 2. The resulting gapless state retains T symmetry but has broken charge‑U(1). Two real‑fermion bands cross at ±p, and because of charge conjugation each crossing maps onto a single Weyl cone in the IR.

• Path b (time‑reversal‑breaking) – Apply a T‑breaking field (e.g., a magnetic field) to the same gapped SPT, “peeling off’’ excess degrees of freedom. The field lifts Kramers degeneracy, leaving only a pair of real‑fermion nodal points at ±p₀. The conserved charge operators now span a two‑dimensional linear space, and the low‑energy theory again reduces to a single Weyl cone.

• Path c (hybrid) – Combine the previous two ideas: start from a DIII tQCP with δN_w ≥ 2 and subsequently break T symmetry. The magnetic perturbation splits the Kramer doublets, yielding exactly two real‑fermion crossings at ±p₀. This hybrid construction reproduces the single‑Weyl‑cone physics while allowing richer topological changes (e.g., δN_w = 4).

All constructed Hamiltonians can be expressed in terms of two dual copies of the (1, 1) representation of Spin(4) ≅ SU(2)↑ × SU(2)↓. One SU(2) factor can be identified with the rotational subgroup of an emergent Lorentz SO(3,1) symmetry, while a σ–τ duality relates the T‑preserving and T‑breaking descriptions. Consequently, every model belongs to an equivalence class: either a DIII tQCP with protected T symmetry or its dual T‑breaking superconducting nodal‑point phase. The paper provides explicit lattice realizations (Models I–IV), computes their band structures over the full three‑torus Brillouin zone, and verifies the topological invariants and intersection counts numerically.

In summary, the work establishes that spontaneous charge‑U(1) breaking is a sufficient (and likely necessary) condition to evade the fermion‑doubling theorem in three dimensions. By reformulating the problem in a real‑fermion language and exploiting Spin(4) symmetry, the authors unify seemingly disparate gapless phases—topological quantum critical points in class DIII superconductors, magnetically polarized nodal superconductors, and lattice chiral fermion constructions—into a single equivalence class that naturally yields a single Weyl fermion. This provides a concrete pathway for realizing chiral fermions in lattice regularizations and highlights superconductors/superfluids as fertile platforms for emergent high‑energy‑like Weyl physics.