Topological properties of spin-triplet superconductors and Fermi surface topology in the normal state

We report intimate relations between topological properties of full-gapped spin-triplet superconductors with time-reversal invariance and the Fermi surface topology in the normal states. An efficient method to calculate the ${\mathit{Z}}_2$ invariants and the winding number for the spin-triplet superconductors is developed and connections between these topological invariants and the Fermi surface structures in the normal states are pointed out. We also obtain a correspondence between the Fermi surface topology and gapless surface states in the superconducting states. The correspondence is inherent to spin-triplet superconductivity.

Figure 1

(Color online) The Fermi surfaces in the normal state and the edge states in 2D time-reversal-invariant spin-triplet superconducting state. (Top row) The Fermi surfaces and ${\pi }_a=\pm$ at the time-reversal-invariant momenta. $p_0\left[=p_0\left(S_F\right)\right]$ is the number of the connected components of the Fermi surface. (Bottom row) The energy spectra of the corresponding superconducting states described by Eq. (25) with edges at $i_x=0$ and $i_x=50$. Here $k_y$ denotes the momentum in the $y$ direction and $N_0$ the number of gapless helical edge states. We set the parameters of the lattice model [Eq. (25)] as (a) $t_x=0.4$, $t_y=1$, $\mu =-1$, and $d=0.5$, (b) $t_x=t_y=1$, $\mu =-1$, and $d=0.5$, and (c) $t_x=1$, $t_y=0.4$, $\mu =-1$, and $d=0.5$, respectively.

Figure 2

(Color online) Various Fermi surface topologies in three dimensions and the corresponding gapless surface states for the 3D time-reversal-invariant spin-triplet superconductor with $\mathit{d}\left(\mathit{k}\right)=\text{sin}{k}_x\hat{\mathit{x}}+\text{sin}{k}_y\hat{\mathit{y}}+\text{sin}{k}_z\hat{\mathit{z}}$. (Top row) Fermi surfaces in the first Brillouin zone and their topological invariants, $\chi \left(S_F\right);\left[p_0\left(C_1\right),p_0\left(C_2\right),p_0\left(C_3\right)\right]$. The green circles are $C_j$ $\left(j=1,2,3\right)$ for (d). (Middle and bottom rows) The corresponding surface states on the Brillouin zone for 001 surface (middle) and 100 surface (bottom) in the superconducting state. The blue solid circles symbolize the Dirac cones of the 2D gapless Majorana fermions and the energies of the surface states become zero at the time-reversal-invariant momenta enclosed by the blue circles. $N_0$ denotes the number of 2D gapless Majorana states on each surface.