Superconductivity in doped inversion-symmetric Weyl semimetals

We study theoretically the superconductivity in doped Weyl semimetals with an inversion symmetry based on the Bogoliubov-de Gennes equations. In principle, the two superconducting states, i.e., the zero momentum BCS-like pairing and the finite momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing, are competing in this kind of systems. From the self-consistent calculation and the free-energy analysis, we propose that the BCS-type state may be the ground state. The competition between these two pairing states is studied in detail through normal state Fermi surface and the finite-energy spectral functions. We also study the physical properties and address the Majorana Fermions excitation in these two superconducting states, respectively.

Figure 1

Intensity plots of normal-state zero-energy spectral functions with open-boundary condition along $y$ direction ($1\le y\le 200$). (a) The spectral function in the bulk with $y=100$. The arrow indicates the Fermi surface nesting vector. (b) The spectral function at the system edge with $y=1$. (c) The spectral function of spin up particles with $y=100$. (d) The spectral function of spin down particles with $y=100$.

Figure 2

The gap magnitude and the superconducting condensation energy in the BCS state and FFLO state, respectively.

Figure 3

(a) Schematic illustration of the BCS pairing and FFLO pairing, respectively. (b)–(d) The spin-dependent spectral functions as functions of the energy and momentum with $k_z=\pi /4-k_x$ [along the (red) solid line cut shown in Fig. 1].

Figure 4

Numerical results of the BCS-I state. (a) The eigenvalues of the Hamiltonian along $k_x=0$. (b)–(d) Intensity plots of zero energy spectral functions.

Figure 5

Numerical results of the BCS-II state. (a) The eigenvalues of the Hamiltonian along $k_x=0$. Panels (b) and (c) are intensity plots of zero-energy spectral functions at the system bulk and edge, respectively. (d) The spatial distributions of the two MFs at the momentum $(k_x,k_z)=(0,0)$.

Figure 6

Numerical results of the FFLO-type pairing. (a) The eigenvalues of the Hamiltonian along $k_x=0$. (b) and (c) Intensity plots of zero energy spectral functions at two boundaries, respectively. (d) The spatial distributions of the two MFs at the momentum ${\mathbf{k}}_{\mathbf{0}}$.