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arXiv:2007.09348 · Physical review. B./Physical review. B

Transport in spinless superconducting wires

Junaid Majeed Bhat, Abhishek Dhar

Published Dec 23, 2020• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.224512

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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.
We present a method for investigating the steady-state transport properties of one-dimensional correlated quantum systems. Using a procedure based on our analysis of finite-size effects in a related classical model (LCLCLC line) we show that stationary currents can be obtained from transient currents in finite systems driven out of equilibrium. The nonequilibrium dynamics of correlated quantum systems is calculated using the time-evolving block decimation method. To demonstrate our method we determine the full III-VVV characteristic of the spinless fermion model with nearest-neighbor hopping tH{t}_{\text{H}}tH​ and interaction VH{V}_{\text{H}}VH​ using two different setups to generate currents (turning on/off a potential bias). Our numerical results agree with exact results for noninteracting fermions (VH=0{V}_{\text{H}}=0VH​=0). For interacting fermions we find that in the linear regime $eV\ensuremath{\ll}4{t}_{\text{H}}$ the current III is independent from the setup and our numerical data agree with the predictions of the Luttinger liquid theory combined with the Bethe Ansatz solution. For larger potentials VVV the steady-state current depends on the current-generating setup and as VVV increases we find a negative differential conductance with one setup while the currents saturate at finite values in the other one. Both effects are due to finite renormalized bandwidths.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
A fermion system with a simple attractive interaction is discussed with the aid of time-dependent correlation functions. Although perturbation theory is inapplicable, a sequence of correlation approximations described in the first paper of this series can be employed. The lowest approximation in the sequence expresses the two-particle correlation function in terms of single-particle functions and leads to the Hartree approximation; the second expresses three-particle correlation function in terms of one- and two-particle correlation functions and leads to the time-dependent correlation functions that characterize the super-conducting model of Bardeen, Cooper, and Schrieffer. In the second section of this paper these correlation functions are determined and the thermodynamic properties of the superconductor are calculated from them.In the third section of the paper, the electromagnetic effects of the superconductor predicted by the Bardeen-Cooper-Schrieffer time-dependent correlation functions are considered. Their unsatisfactory description of current conservation is indicated and overcome in the fourth section by a more accurate solution valid at nonvanishing temperature. This solution predicts different diffusive properties but the same Meissner effect and superconductive behavior, since the longitudinal current correlation function is modified while the transverse current correlation functions is not.The fifth section of the paper is devoted to the properties of a pure superconductor which depend on the lifetimes of the single-particle excitations. The effect of these lifetimes on the static electrical conductivity is determined, and it is shown that they do not destroy supercurrents although they eliminate a gap in the single-particle excitation spectrum. Their effect on the thermal conductivity is also calculated using heat current correlation functions. It is shown that a model which treats the lifetime of the single-particle excitation due to lattice interactions as constant yields results in agreement with observed thermal conductivities.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.
Andreev scattering and the Josephson current through a one-dimensional interacting electron liquid sandwiched between two superconductors are reexamined. We first present some apparently new results on the noninteracting case by studying an exactly solvable tight-binding model rather than the usual continuum model. We show that perfect Andreev scattering (i.e., zero normal scattering) at the Fermi energy can only be achieved by fine-tuning junction parameters, a fine-tuning which is possible even with bandwidth mismatch between superconductor and normal metal. We also obtain exact results for the Josephson current, which is generally a smooth function of the superconducting phase difference except when the junction parameters are adjusted to give perfect Andreev scattering, in which case it becomes a sawtooth function. We then observe that, even when interactions are included, all low-energy properties of a junction $(E\ensuremath{\ll}\ensuremath{\Delta},$ the superconducting gap) can be obtained by ``integrating out'' the superconducting electrons to obtain an effective Hamiltonian describing the metallic electrons only with a boundary pairing interaction. This boundary model provides a suitable starting point for bosonization-renormalization group-boundary conformal field theory analysis. We argue that total normal reflection and total Andreev reflection correspond to two fixed points of the boundary renormalization group. For repulsive bulk interactions the Andreev fixed point is unstable and the normal one stable. However, the reverse is true for attractive interactions. This implies that a generic junction Hamiltonian (without fine-tuned junction parameters) will renormalize to the normal fixed point for repulsive interations but to the Andreev one for attractive interations. An exact mapping of our tight-binding model to the Hubbard model with a transverse magnetic field is used to help understand this behavior. We calculate the critical exponents, which are different at these two different fixed points.
Proceeding from a microscopic model Hamiltonian for a system of Fermi particles with attraction, we have derived an effective Lagrangian, which allows analysis of the superconducting properties of the system for arbitrary fermion concentrations. Exact solution of the equations for the gap and the chemical potential enables us to examine the crossover from the regime of local pairs and superfluidity to the Cooper pairing and the BCS-like superconductivity. It is shown that the physical parameter that demarcates the regions of exotic and normal behavior is the fermion bound state energy. The expressions derived suggest that in the region of relatively low concentrations the parameters of a 2D metal correspond to a type-II superconductor.