Transport in spinless superconducting wires

We consider electron transport in a model of a spinless superconductor described by a Kitaev-type lattice Hamiltonian where the electron interactions are modeled through a superconducting pairing term. The superconductor is sandwiched between two normal metals that are kept at different temperatures and chemical potentials and that are themselves modeled as noninteracting spinless fermions. For this setup we compute the exact steady-state properties of the system using the quantum-Langevin-equation approach. Closed-form exact expressions for particle current, energy current, and other two-point correlations are obtained in the Landauer-type forms and involve two nonequilibrium Green's functions. The current expressions are found to be the sum of three terms having simple physical interpretations. We then discuss a numerical approach where we construct the time evolution of the two-point correlators of the system from the eigenspectrum of the complete quadratic Hamiltonian describing the system and leads. By starting from an initial state corresponding to the leads in thermal equilibrium and the system in an arbitrary state, the long-time solution for the correlations, before recurrence times, gives us steady-state properties. We use this independent numerical method for verifying the results of the exact solution. We also investigate analytically the presence of high-energy bound states and obtain expressions for their contributions to two-point correlators. As applications of our general formalism we present results on thermal conductance and on the conductance of a Kitaev chain with next-nearest-neighbor interactions which allows topological phases with different winding numbers.

Figure 1

Comparison of numerical time evolution and analytical steady-state results: Parameter values are $N=2$, $N_b=100$, ${\mu }_R=1$, ${\beta }_R=10$, ${\mu }_L=0$, ${\beta }_L=0$, $V_L=V_R={\eta }_w=1$, ${\mu }_w=0$, ${\eta }_b=1.5$, and $\mathrm{\Delta }=0.40$. (a) Comparison of the numerically calculated particle current at the left, $J_L\left(t\right)$, and the right, $J_R\left(t\right)$, ends of the wire with the corresponding value, $J_{th}$, given by the expression in Eq. (39). (b) Comparison of the numerically calculated densities, $N_1\left(t\right)=\langle a_1^{†}\left(t\right)a_1\left(t\right)\rangle$ and $N_2\left(t\right)=\langle a_2^{†}\left(t\right)a_2\left(t\right)\rangle$, on the two sites of the wire with the corresponding value, $N_{th}$, given by the expression in Eq. (52). Similarly, (c) and (d) show the comparison of the energy current and the energy density from direct numerics with the values obtained from steady-state expressions. Note that the left and the right heat currents have the same magnitude, unlike the particle currents. The initial oscillations seen in the plots correspond to the transient phase, while the behavior near $t=60$ is due to the finite size of the baths. In the intermediate region we see perfect agreement between the numerical solution and the steady-state value.

Figure 2

Spectrum of the entire system at parameter values $N=2$, $N_b=100$, $V^L=V^R={\eta }_w=1$, ${\mu }_w=0$, and ${\eta }_b=2.5$ for two values of $\mathrm{\Delta }$. In (a) we do not see any discrete energy level, while in (b) a discrete energy level outside the main band can be seen. As discussed in the text, the nonexistence of a steady state, indicated in a nonvanishing $I_{ij}$, is related to the existence of the discrete level, which corresponds to a bound state (see Fig. 3).

Figure 3

(a) The variation of ${\sum }_{ij}{\left|I_{ij}\right|}^2$ over different parameters of the Hamiltonian in Eq. (66) with ${\eta }_s=V_L=V_R=1$ and $N=2$. Physically, this quantity should identically vanish. We see that this happens only for certain parameter regimes of the Hamiltonian. In (b) we verify that the nonzero values are associated with the presence of high-energy bound states in the spectrum of the full system seen in Fig. 2. This plot shows the gap between the bound-state energy and the edge of the band, $E_{\mathrm{bound}}$, for the same parameters as in (a). We see that the value of $\mathrm{\Delta }$ at which the bound state appears is exactly the same value where the corresponding curves in (a) start taking nonzero values. (c) and (d) demonstrate that the numerical simulation for the two-point correlators agrees with the analytic results obtained by adding the bound state contributions to the steady-state values. While (c) shows that the commutation relations are satisfied when we add the bound-state contribution, (d), on the other hand, depicts the persistent oscillations in the particle densities due to the bound states. Parameter values for these two plots are $N=2$ for (c) and $N=3$ for (d), $N_b=100$, ${\mu }_R=1$, ${\beta }_R=10$, ${\mu }_L=0$, ${\beta }_L=0$, $V_L=V_R={\eta }_w=1$, ${\mu }_w=0$, ${\eta }_b=1$, and $\mathrm{\Delta }=0.80$.

Figure 4

(a)–(d) Variation of zero-temperature conductance and the terms contributing to it with the left-bath chemical potential, ${\mu }_L$. Parameter values are $N=100$, $V_L=V_R=0.2$, ${\eta }_w={\mu }_w=1$, ${\eta }_b=1$, and $\mathrm{\Delta }=0.25$.

Figure 5

(a) Variation of thermal conductance at ${\mu }_L=0$ in units of ${\pi }^2{k}_B^2{T}_L/6$ with the chemical potential, ${\mu }_w$, on the wire for different wire sizes. (b) shows the wave function of the Majorana zero mode at ${\mu }_w$ far away and close to the topological-phase-transition point for $N=100$. The extended nature of the Majorana wave functions spreading across the wire can be clearly seen when ${\mu }_w$ is close to the phase-transition point. Other parameter values are $V_L=V_R=0.25$, ${\eta }_w={\eta }_b=1$, and $\mathrm{\Delta }=0.3$.

Figure 6

(a) and (b) show the spectrum of an isolated next-nearest-neighbor Kitaev chain at the parameter values ${\eta }_1=1$, ${\mu }_w=-2$, $N=100$, $V_L=V_R=0.3$, and ${\eta }_b=1.5$ at zero and nonzero values of $\theta$ and for the values of ${\eta }_2=-1.5,-0.5$, and 1 corresponding to phase 2, phase 0, and phase 1, respectively. The topological edge modes can be seen in the spectral plots within the superconducting gap. The corresponding conductance results obtained from Eq. (46) are shown in (c) and (d). The conductance plots for $\theta =0$ show a single zero-bias peak for the two topologically nontrivial phases, while they show two distinct peaks for a nonzero $\theta$ value for phase 2.