Topological Nodal Cooper Pairing in Doped Weyl Metals

Abstract

We generalize the concept of Berry connection of the single-electron band structure to the twoparticle Cooper pair states between two Fermi surfaces with opposite Chern numbers. Because of underlying Fermi surface topology, the pairing Berry phase acquires non-trivial monopole structure. Consequently, pairing gap functions have the topologically-protected nodal structure as vortices in the momentum space with the total vorticity solely determined by the monopole charge qp. The pairing nodes behave as the Weyl-Majorana points of the Bogoliubov-de Gennes pairing Hamiltonian. Their relation with the connection patterns of the surface modes from the Weyl band structure and the Majorana surface modes inside the pairing gap is also discussed. Under the approximation of spherical Fermi surfaces, the pairing symmetry are represented by monopole harmonic functions. The lowest possible pairing channel carries angular momentum number j = |qp|, and the corresponding gap functions are holomorphic or anti-holomorphic functions on Fermi surfaces. The study of topological states has renewed our understanding of condensed matter physics. The discovery of two-dimensional integer quantum Hall states 1,2 initiated the exploration of novel states characterized by band topology rather than symmetry 3-8 , with magnetic band structures that possess non-trivial Chern numbers arising from broken time-reversal (TR) symmetry. The study of Berry curvature of Bloch bands in such lattice structures has led to to many results in anomalous Hall and quantum anomalous Hall physics 9-16 . The band structure topology has also been generalized to systems of topological insulators with TR symmetry [17][18][19][20][21][22][23][24][25][26][27] . The stable gapless surface modes which appear at the boundary of gapped topological systems have analogs in gapless semimetallic systems, which can also have non-trivial band topology. For example, topological Weyl semi-metals have been proposed and realized in three-dimensional (3D) systems in the absence of either TR or inversion symmetry . Their band structure is characterized by degenerate Weyl points in the Brillouin-zone (BZ), which can be understood as monopole sources and sinks of Berry-curvature flux in k-space.Topological phenomena are usually understood in terms of contributions from all the filled electronic states rather than the states in the vicinity of Fermi surfaces. The apparent disagreement with the central tenet of Fermi-liquid theory that all conduction processes can be understood at the Fermi level can be resolved by introducing the Berry phase of quasiparticles on the Fermi surface 13 . So far, the study of the Fermi surface topology and the associated Berry phase structure has mainly been discussed at the single-particle level 11-14 .Here we study a novel class of exotic superconductivity which can be realized in doped Weyl metals, and more generally in systems with topologically non-trivial Fermi surfaces. In superconductivity with pairing between states on two disjoint Fermi surface s...

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Topological spinor vortex matter on spherical surface induced by non-Abelian spin-orbital-angular-momentum coupling

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