Majorana bands, Berry curvature, and thermal Hall conductivity in the vortex state of a chiral $p$-wave superconductor

Majorana quasiparticles localized in vortex cores of a chiral $p$-wave superconductor hybridize with one another to form bands in a vortex lattice. We begin by solving a fully microscopic theory describing all quasiparticle bands in a chiral $p$-wave superconductor in a magnetic field, then use this solution to build localized Wannier wave functions corresponding to Majorana quasiparticles. A low-energy tight-binding theory describing the intervortex hopping of these is then derived, and its topological properties—which depend crucially on the signs of the imaginary intervortex hopping parameters—are studied. We show that the energy gap between the Majorana bands may be either topologically trivial or nontrivial, depending on whether the Chern number contributions from the Majorana bands and those from the background superconducting condensate add constructively or destructively. This topology directly affects the temperature-dependent thermal Hall conductivity, which we also calculate.

Figure 1

Square vortex lattice, with arrows indicating the sign of the imaginary nearest-neighbor hopping of fermionic quasiparticles between vortices. The dotted lines indicate branch cuts connecting the two vortices within each magnetic unit cell.

Figure 2

(a) Quasiparticle energy bands of a $p$-wave superconductor with $p_x-i{p}_y$ pairing in a perpendicular magnetic field for a square vortex lattice with intervortex distance $L_x=L_y=12$, pairing strength $\mathrm{\Delta }/t=0.5$, chemical potential $\mu /t=-2$, and Zeeman shift $h_Z=0$. (b) Detailed view of the two Majorana bands, where the dashed lines are obtained by solving the full Hamiltonian (4), while the (virtually indistinguishable) solid lines come from the effective tight-binding theory with first- and second-neighbor hoppings. (c) Magnetic Brillouin zone, showing the path along which the energy bands are plotted. (d) The Majorana bands feature two Dirac cones along the line $k_y=-k_x$, which are gapped due to next-nearest-neighbor hopping.

Figure 3

Integrated Berry curvature, summed over bands up to energy $\xi$, with same parameters as in Fig. 2. The lower panel provides a detailed view near $\xi =0$, showing that the Chern number within the large energy gap is $\nu =-1$, while the Majorana bands carry Chern numbers $\nu =\pm 1$.

Figure 4

(a) Oscillations in the nearest- and next-nearest-neighbor hopping parameters for the Majorana bands described by the low-energy effective Hamiltonian (18). Solid (dashed) lines correspond to $t_0$ ($t_1$, multiplied $\times 10$), and blue (red) lines correspond to $p_x-i{p}_y$ ($p_x+i{p}_y$) pairing, with magnetic length $L_x=L_y=16$, chemical potential $\mu /t=-2$, and pairing strength $\mathrm{\Delta }/t=0.5$. (b) Energy difference per site between the two pairing states, showing that $p_x-i{p}_y$ pairing is favored for larger values of $\mu$. (c) Phase diagram showing the total Chern number in the gap between Majorana bands, which is determined by the type of pairing and the sign of $t_1$. The pairing is $p_x+i{p}_y$ to the left of the red line and $p_x-i{p}_y$ to the right.

Figure 5

Energy eigenvalues for a chiral $p$-wave superconductor (with $p_x-i{p}_y$ pairing and magnetic length $L_x=L_y=8$) in the vortex state on a cylinder with periodic boundary conditions along the $y$ direction and open boundary conditions along the $x$ direction. The right panels show the corresponding integrated Berry curvature summed over occupied bands. (a) With $\mu /t=-2$, the gap between Majorana bands is topologically trivial due to cancellation of the Berry curvature of the Majorana bands with that of the superconducting condensate, so that no edge modes are present. At energies above and below the Majorana bands, however, states localized at the right (red) and left (blue) edges are present due to the nontrivial topology of the background superconducting condensate. (b) With $\mu /t=-1.5$, the gap between Majorana bands is topologically nontrivial due to constructive addition of the Berry curvature of the Majorana bands with that of the condensate, so that two chiral edge modes are present.

Figure 6

Thermal Hall conductivity divided by temperature for $\mathrm{\Delta }/t=0.5, L_{x,y}=8$, and $p_x-i{p}_y$ pairing. The black dotted line corresponds to the quantized value for a system with Chern number $\nu =-1$. Solid red and dashed blue lines show the results for the cases where the gap around energy $\xi =0$ has Chern number $\nu =-2$ (at $\mu /t=-2$) and $\nu =0$ (at $\mu /t=-1.5$), respectively [see phase diagram in Fig. 4]. Insets show the corresponding ${\tilde{\sigma }}_{xy}\left(\xi \right)$, which determine ${\kappa }_{xy}\left(T\right)$ via Eq. (22). (a) With no Zeeman coupling, the low-temperature behavior is determined by ${\tilde{\sigma }}_{xy}\left(\xi \right)$ near $\xi \sim 0$. (b) With Zeeman coupling $h_Z=0.1t$, the low-temperature behavior is determined instead by ${\tilde{\sigma }}_{xy}\left(\xi \right)$ near $\xi \sim h_Z$.