Transport in spinless superconducting wires

Abstract

We consider electron transport in a model of a spinless superconductor described by a Kitaev type lattice Hamiltonian where the electron interactions are modelled through a superconducting pairing term. The superconductor is sandwiched between two normal metals kept at different temperatures and chemical potentials and are themselves modelled as non-interacting spinless fermions. For this set-up we compute the exact steady state properties of the system using the quantum Langevin equation approach. The closed form exact expressions for current and other two-point correlations are obtained in the Landauer-type forms and involve two nonequilibrium Green's functions. 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 and to comment on the role of bound states. Our results are quite general and applicable to arbitrary lattices in any dimensions.

Nonequilibrium electrical, thermal and spin transport in open quantum systems of topological superconductors, semiconductors and metals

“…We have earlier argued that the generalized LEGF technique applies to those systems, which reach a unique NESS in the long-time limit t → ∞. For example, the LEGF can easily be applied to an N-TS-N device because such a junction satisfies the requirement of unique NESS [26,35]. However, an X-Y -Z configuration with a superconducting wire for X lead does not attain a unique NESS unless the Z wire is an N/SM or a topological superconductor at TP.…”

“…Here, H Y is the Hamiltonian of the finite Majorana/SM wire. The structure of Σ+ X/Z (ω) is determined by the tunnelling Hamiltonian (35) and the Hamiltonian of the lead made of either a Majorana or an SM wire (13). The self-energy correction term associated with the X lead can be written as:…”

“…On the theoretical side, while there is a vast number of theoretical studies on electrical transport in systems with TSs [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35], the thermal [8,9,36,2,[37][38][39] and mostly spin transport [40,41] in these systems are much less explored. One of this paper's primary goals is a detailed and unified description of electrical, thermal, and spin transport in different devices made of TSs.…”

“…One of this paper's primary goals is a detailed and unified description of electrical, thermal, and spin transport in different devices made of TSs. The quantum transport in such devices are generally studied utilizing the generalized scattering theory [42,43,8,9], the Keldysh formalism [44,20,22,29,38], and the quantum Langevin equations & Green's function (LEGF) method [26,28,35]. These theoretical techniques are mainly employed to calculate electrical currents and differential conductances, which are supposed to show non-trivial behavior in superconductors' topological phases.…”

“…The LEGF method is an open-quantum system formulation within the Heisenberg representation of quantum mechanics to study nonequilibrium quantum transport in mesoscale and nanoscale devices [45][46][47][48]. This method, based on a direct solution of the Heisenberg equations of system and bath variables, has been applied extensively in calculating steady-state [47,48] and time-dependent [46] electrical, thermal, and optical transport properties in devices consisting of noninteracting baths (e.g., metals, harmonic oscillators) [49,26,28,50,51,35]. An extension of this method for superconducting baths lacks to date.…”