Equivalent class of Emergent Single Weyl Fermion in 3d Topological States: gapless superconductors and superfluids Vs chiral fermions

TL;DR

The paper develops a practical, generic framework to realize a single Weyl cone in 3D lattice systems by spontaneously breaking U(1) charge symmetry, organized around three UV-completion paths that culminate in an equivalent class under Spin(4) symmetry. It leverages a real (Majorana) fermion formalism and Nambu/BdG mappings to show how gapless topological phases (tQCPs and nodal-point states) can UV-complete IR Weyl dynamics, with explicit lattice models (Path a, b, and c) illustrating the construction. A central result is that these models, across diverse parameter choices (δN_w=2,4,8) and TR-breaking fields, fall into two dual Spin(4) representations (1,1) and its dual, connected by σ–τ duality, and can be described by two copies of 3⊗3 (1,1) in the IR. The work further elucidates the non-compact, non-on-site symmetry charges that accompany these UV completions and shows how Schrieffer–Wolff projections can recover an effective single Weyl fermion in the IR. Overall, the study links gapless superconductors/superfluids to lattice chiral fermions via a unified, UV-complete symmetry framework with concrete lattice realizations and symmetry-charge structures, offering a path toward robust single-Weyl lattice models and insights into their UV-IR structure.

Abstract

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.

Figures (9)

• Figure 1: Top row from the left to right: shown on the left is one complex fermion degree of freedom at each k associated with the creation operator $c_{\bf k}^\dagger$. In the middle figure one further redefines the fermion creation operator as an annihilation operator for a hole-like fermion so that $c_{\bf k}^\dagger = d_{-{\bf k}}$. In the right figure, one rotates to the real fermion basis. The red line is for charge $+$e and the blue dotted line is for charge $-$e. The bottom figure represents the transformation from the Nambu space to a real fermion representation.
• Figure 2: The $T$-invariant tQCP (Right) as a extreme case of the nodal phase (Left). A $T$-invariant tQCP can be seen as a fine tuned phase of the Nodal phase, either by tuning the coupling parameter $\epsilon$ to 0 and then the T-breaking magnetic field (up path), or by first tuning B to 0, coming back to the gapped lattice model with T-symmetry, and then pushing it to a gapless phase.
• Figure 3: A T-symmetric tQCP in SPTs when the mass parameter is tuned to be zero, i.e. $\epsilon = 0$ (See Eq.\ref{['eq: tQCP0']},\ref{['eq: mass']}).
• Figure 4: (Nodal phase) Spectrum for $\mu = 4$ and, left $B=0$, the bands are 2 times degenerate; right $B=6.5$, the magnetic field parameter $B$ lifts the degeneracy and for a range of values gives only two crossings. This allows the pealing off of the excessive degrees of freedom (dotted bands) in this real fermion formalism, a crucial step towards the single Weyl fermion.
• Figure 5: Phase diagram for the number of Weyl cones in model Eq.\ref{['eq: NodalH']}. In the white region, there are no Weyl cones, in the green region, there is only one single Weyl cone, in the yellow region, there are two Weyl cones and in the red region, there are three Weyl cones.

Theorems & Definitions (2)

• Theorem 3.1: Nielsen & Ninomiya
• proof