Effect of charge renormalization on the electric and thermoelectric transport along the vortex lattice of a Weyl superconductor

Building on the discovery that a Weyl superconductor in a magnetic field supports chiral Landau-level motion along the vortex lines, we investigate its transport properties out of equilibrium. We show that the vortex lattice carries an electric current $I=\frac{1}{2}(Q_{\mathrm{eff}}^2/h)(\mathrm{\Phi }/{\mathrm{\Phi }}_0)V$ between two normal-metal contacts at voltage difference $V$, with $\mathrm{\Phi }$ the magnetic flux through the system, ${\mathrm{\Phi }}_0$ the superconducting flux quantum, and $Q_{\mathrm{eff}}<e$ the renormalized charge of the Weyl fermions in the superconducting Landau level. Because the charge renormalization is energy dependent, a nonzero thermoelectric coefficient appears even in the absence of energy-dependent scattering processes.

Figure 1

(a) Vortex lattice in a Weyl superconductor sandwiched between metal electrodes; (b) circuit to measure the electrical transport along the vortex lines. The nonlocal conductance $G_{12}=d{I}_2/d{V}_1$ gives the current carried through the vortex lattice by nonequilibrium Weyl fermions in a chiral Landau level.

Figure 2

Left: The red solid curves show the dispersion of Landau levels in the $k_x\text{−}{k}_y$ plane perpendicular to the magnetic field (energy $E$ normalized by the energy $E_1$ of the first Landau level). The black dotted curves show the dispersion in zero magnetic field, with a Weyl cone at the $\mathrm{\Gamma }$ point of the magnetic Brillouin zone. Right: Particle density profile in the zeroth Landau level, in the $x\text{−}y$ plane perpendicular to the magnetic field, for a wave vector at the Weyl point $\left(\mathit{k}=K\widehat{z}\right)$. The magnetic unit cell is indicated by a white dashed rectangle. Both panels are calculated numerically for a Weyl superconductor with a triangular vortex lattice. The vortex cores are located at the bright points in the density profile. Similar plots for a square vortex lattice are in Ref. [7].

Figure 3

Dispersion relation of the zeroth Landau level in a superconducting vortex lattice, plotted from Eq. (2.1) for $\mu =0,{\mathrm{\Delta }}_0=0.5,\beta =1$. Only the dependence on the momentum $k_z$ along the magnetic field $B$ is shown; the dispersion is flat in the $x\text{−}y$ plane (see Fig. 2). The four branches are distinguished by the sign of the chirality (solid or dashed) and by the sign of the electric charge (red or blue). The zero-field Weyl points at $k_z=\pm K$ are indicated by arrows. Each branch has a degeneracy $N_{\mathrm{Landau}}=e\mathrm{\Phi }/h$ set by the enclosed flux $\mathrm{\Phi }=B{W}^2$.

Figure 4

Data points: Electrical conductance (top) and Fano factor (bottom) in the superconducting vortex lattice (lattice constant $d_0)$, as a function of the pair potential ${\mathrm{\Delta }}_0$ at fixed magnetization $\beta =1$, calculated from the tight-binding model (lattice constant $a_0)$ for different lattice constant ratios $N_0=d_0/a_0$. The black curves are the analytical predictions from the charge renormalization factor $\kappa$, both in the approximation of a linearized dispersion (black dashed curve, $\kappa ={\kappa }_0=\sqrt{1-{\mathrm{\Delta }}_0^2/{\beta }^2})$ and for the full nonlinear dispersion (black solid).

Figure 5

Dependence on ${\mathrm{\Delta }}_0$ for $\beta =0.5$ of the thermoelectric coefficient (4.10), calculated from the infinite-system analytics (black solid curve) or obtained from finite-size numerics (colored data points).

Figure 6

Same as Fig. 4, but for a magnetization $\beta$ that is perpendicular rather than parallel to the magnetic field $B$.