Structured Weyl Points in Spin-Orbit Coupled Fermionic Superfluids

Abstract

We demonstrate that a Weyl point, widely examined in 3D Weyl semimetals and superfluids, can develop a pair of non-degenerate gapless spheres. Such a bouquet of two spheres is characterized by three distinct topological invariants of manifolds with full energy gaps, i.e., the Chern number of a 0D point inside one developed sphere, the winding number of a 1D loop around the original Weyl point, and the Chern number of a 2D surface enclosing the whole bouquet. We show that such structured Weyl points can be realized in the superfluid quasiparticle spectrum of a 3D degenerate Fermi gas subject to spin-orbit couplings and Zeeman fields, which supports Fulde-Ferrell superfluids as the ground state. [17,18]. A Weyl point can also be regarded as a monopole in 3D momentum space that exhibits an integer topological charge, i.e., the quantized first Chern number of a surface enclosing the singularity. Weyl points were also suggested to exist in the quasiparticle spectrum of superfluid 3 He A phase [2]. In contrast to traditional fully gapped superfluids, the Weyl superfluids bear doubly degenerate nodes pinned to zero energy, around which the quasiparticle energies disperse linearly in all directions. Most recently, the existence of such Weyl nodes has also been generalized to various coldatom superfluids and solid-state superconductors [19][20][21][22][23][24][25][26][27][28].In this Letter, we investigate whether a Weyl point can develop a nontrivial structure at zero energy and whether there exist any topological property protecting the developed structure. (i) We first consider a toy model to examine the possibility for a Weyl point to develop a gapless structure. Mathematically, the structured Weyl point can be viewed as a bouquet of two spheres (or wedge of two spheres) [29], which is a new class of topological state that has not been explored previously. Amazingly, the zero-energy bouquet is characterized by three distinct topological invariants: the first Chern number of a surface enclosing the whole bouquet, the zeroth Chern numbers of the interiors of the two spheres, and the winding number of a loop enclosing the touching point in the plane of symmetry. (ii) We further show that the structured Weyl points can be physically realized in the quasiparticle excitation spectrum of a 3D spin-orbit-coupled (SOC) fermionic cold-atom superfluid with the FuldeFerrell (FF) ground state. FF superfluids [30][31][32][33][34][35][36][37][38][39][40][41] have been studied recently in SOC degenerate Fermi gases subject to Zeeman fields, which yield asymmetries of the Fermi surface and induce the FF Cooper pairing with nonzero total momenta. We obtain a rich phase diagram in the gapless region of 3D FF superfluids, where not only the featureless Weyl points [19][20][21][22][23][24][25][26][27][28] but also the structured Weyl points emerge. Note that the featureless Weyl points have been well studied in SOC FF superfluids [24], and here we focus only on the novel structured Weyl points. (iii) We also discuss how the structured W...

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