The Anderson-Josephson quantum dot -- A theory perspective

Recent progress in nanoscale manufacturing allowed to experimentally investigate quantum dots coupled to two superconducting leads in controlled and tunable setups. The equilibrium Josephson current was measured in on-chip SQUID devices and subgap states were investigated using weakly coupled metallic leads for spectroscopy. This put back two "classic" problems also on the agenda of theoretical condensed matter physics: the Josephson effect and quantum spins in superconductors. We discuss the status of the theoretical understanding of the Anderson-Josephson quantum dot in equilibrium mainly focusing on the Josephson current. We introduce a minimal model consisting of a dot which can only host one spin-up and one spin-down electron repelling each other by a local Coulomb interaction. The dot is tunnel-coupled to two superconducting leads described by the BCS Hamiltonian. This model was investigated using a variety of methods, some capturing aspects of Kondo physics others failing in this respect. We briefly review this. The model shows a first order level-crossing quantum phase transition when varying any parameter provided the others are within appropriate ranges. At vanishing temperature it leads to a jump of the Josephson current. We show that a quantitative agreement between accurate results obtained for the simple model and measurements of the current can be reached. This confirms that the experiments reveal the finite temperature signatures of the zero temperature transition. In addition, we consider two examples of more complex dot geometries which might be experimentally realized in the near future. The first is characterized by the interplay of the above level-crossing physics and the Fano effect, the second by the interplay of superconductivity and almost degenerate singlet and triplet two-body states.Comment: invited review for Journal of Physics: Condensed Matter, comments welcom

Spin-orbit coupling effects in one-dimensional ballistic quantum wires

We study the spin-dependent electronic transport through a one-dimensional ballistic quantum wire in the presence of Rashba spin-orbit interaction. In particular, we consider the effect of the spin-orbit interaction resulting from the lateral confinement of the two-dimensional electron gas to the one-dimensional wire geometry. We generalize a situation suggested earlier [P. Streda and P. Seba, Phys. Rev. Lett. 90, 256601 (2003)] which allows for spin-polarized electron transport. As a result of the lateral confinement, the spin is rotated out of the plane of the two-dimensional system. We furthermore investigate the spin-dependent transmission and the polarization of an electron current at a potential barrier. Finally, we construct a lattice model which shows similar low-energy physics. In the future, this lattice model will allow us to study how the electron-electron interaction affects the transport properties of the present setup.Comment: 7 pages, 6 figures, revised versio

Nonequilibrium functional renormalization group for interacting quantum systems

We propose a nonequilibrium version of functional renormalization within the Keldysh formalism by introducing a complex valued flow parameter in the Fermi or Bose functions of each reservoir. Our cutoff scheme provides a unified approach to equilibrium and nonequilibrium situations. We apply it to nonequilibrium transport through an interacting quantum wire coupled to two reservoirs and show that the nonequilibrium occupation induces new power law exponents for the conductance.Comment: 5 pages, 2 figures; published versio

The interacting resonant level model in nonequilibrium: finite temperature effects

We study the steady-state properties as well as the relaxation dynamics of the nonequilibrium interacting resonant level model at finite temperatures. It constitutes the prototype model of a correlated charge fluctuating quantum dot. The two reservoirs are held at different chemical potentials---the difference being the bias voltage---and different temperatures; we discuss the transport through as well as the occupancy of the single level dot. First, we show analytically that in the steady state the reservoir temperatures in competition with the other energy scales act as infrared cutoffs. This is rather intuitive but, depending on the parameter regime under consideration, leads to a surprisingly rich variety of power laws in the current as a function of the temperatures and the bias voltage with different interaction dependent exponents. Next we clarify how finite reservoir temperatures affect the dynamics. They allow to tune the interplay of the two frequencies characterizing the oscillatory part of the time evolution of the model at zero temperature. For the exponentially decaying part we disentangle the contributions of the level-lead hybridization and the temperatures to the decay rates. We identify a coherent-to-incoherent transition in the long time dynamics as the temperature is raised. It occurs at an interaction dependent critical temperature. Finally, taking different temperatures in the reservoirs we discuss the relaxation dynamics of a temperature gradient driven current.Comment: 12 pages, 6 figures, 1 tabl

Finite-temperature linear conductance from the Matsubara Green function without analytic continuation to the real axis

We illustrate how to calculate the finite-temperature linear-response conductance of quantum impurity models from the Matsubara Green function. A continued fraction expansion of the Fermi distribution is employed which was recently introduced by Ozaki [Phys. Rev. B 75, 035123 (2007)] and converges much faster than the usual Matsubara representation. We give a simplified derivation of Ozaki's idea using concepts from many-body condensed matter theory and present results for the rate of convergence. In case that the Green function of some model of interest is only known numerically, interpolating between Matsubara frequencies is much more stable than carrying out an analytic continuation to the real axis. We demonstrate this explicitly by considering an infinite tight-binding chain with a single site impurity as an exactly-solvable test system, showing that it is advantageous to calculate transport properties directly on the imaginary axis. The formalism is applied to the single impurity Anderson model, and the linear conductance at finite temperatures is calculated reliably at small to intermediate Coulomb interactions by virtue of the Matsubara functional renormalization group. Thus, this quantum many-body method combined with the continued fraction expansion of the Fermi function constitutes a promising tool to address more complex quantum dot geometries at finite temperatures.Comment: version accepted by Phys. Rev.

Correlation induced resonances in transport through coupled quantum dots

We investigate the effect of local electron correlations on transport through parallel quantum dots. The linear conductance as a function of gate voltage is strongly affected by the interplay of the interaction U and quantum interference. We find a pair of novel correlation induced resonances separated by an energy scale that depends exponentially on U. The effect is robust against a small detuning of the dot energy levels and occurs for arbitrary generic tunnel couplings. It should be observable in experiments on the basis of presently existing double-dot setups.Comment: 4+ pages, 5 figures included, version accepted for publication in PR

Kondo physics in transport through a quantum dot with Luttinger liquid leads

We study the gate voltage dependence of the linear conductance through a quantum dot coupled to one-dimensional leads. For interacting dot electrons but noninteracting leads Kondo physics implies broad plateau-like resonances. In the opposite case Luttinger liquid behavior leads to sharp resonances. In the presence of Kondo as well as Luttinger liquid physics and for experimentally relevant parameters, we find a line shape that resembles the one of the Kondo case.Comment: 4+ pages, 4 figures include

Nonequilibrium thermoelectric transport through vibrating molecular quantum dots

We employ the functional renormalization group to study the effects of phonon-assisted tunneling on the nonequilibrium steady-state transport through a single level molecular quantum dot coupled to electronic leads. Within the framework of the spinless Anderson-Holstein model, we focus on small to intermediate electron-phonon couplings, and we explore the evolution from the adiabatic to the antiadiabatic limit and also from the low-temperature non-perturbative regime to the high temperature perturbative one. We identify the phononic signatures in the bias-voltage dependence of the electrical current and the differential conductance. Considering a temperature gradient between the electronic leads, we further investigate the interplay between the transport of charge and heat. Within the linear response regime, we compare the temperature dependence of various thermoelectric coefficients to our earlier results obtained within the numerical renormalization group [Phys.~Rev.~B {\bf 96}, 195156 (2017)]. Beyond the linear response regime, in the context of thermoelectric generators, we discuss the influence of molecular vibrations on the output power and the efficiency. We find that the molecular energy dissipation, which is inevitable in the presence of phonons, is significantly suppressed in the antiadiabatic limit resulting in the enhancement of the thermoelectric efficiency.Comment: 11 pages, 7 figures, Published versio

Efficiency and power of a thermoelectric quantum dot device

We study linear response and nonequilibrium steady-state thermoelectric transport through a single-level quantum dot tunnel coupled to two reservoirs held at different temperatures as well as chemical potentials. A fermion occupying the dot interacts with those in the reservoirs by a short-ranged two-particle interaction. For parameters for which particles flow against a bias voltage from the hot to the cold reservoir this setup acts as an energy-conversion device with which electrical energy is gained out of waste heat. We investigate how correlations affect its efficiency and output power. In linear response the changes in the thermoelectric properties can be traced back to the interaction induced renormalization of the resonance line shape. In particular, small to intermediate repulsive interactions reduce the maximum efficiency. In nonequilibrium the situation is more complex and we identify a parameter regime in which for a fixed lower bound of the output power the efficiency increases.Comment: 6 pages, 6 figure