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We study the nonlinear steady-state transport of spinless fermions through a quantum dot with a local two-particle interaction. The dot degree of freedom is in addition coupled to a phonon mode. This setup combines the nonequilibrium physics of the interacting resonant level model and that of the Anderson-Holstein model. The fermion-fermion interaction defies a perturbative treatment. We mainly focus on the antiadiabatic limit, with the phonon frequency being larger than the lead-dot tunneling rate. In this regime also the fermion-boson coupling cannot be treated perturbatively. Our goal is two-fold. We investigate the competing roles of the fermion-fermion and fermion-boson interactions on the emergent low-energy scale $T_{\rm K}$ and show how $T_{\rm K}$ manifests in the transport coefficients as well as the current-voltage characteristics. For small to intermediate interactions, the latter is in addition directly affected by both interactions independently. With increasing fermion-boson interaction the Franck-Condon blockade suppresses the current at small voltages and the emission of phonons leads to shoulders or steps at multiples of the phonon frequency, while the local fermion-fermion interaction implies a negative differential conductance at voltages larger than $T_{\rm K}$. We, in addition, use the model to investigate the limitations of our low-order truncated functional renormalization group approach on the Keldysh contour. In particular, we quantify the role of the broken current conservation., Comment: 13 pages, 9 figures

We present a quantitative analysis of two-particle interaction effects in generalized, one-dimensional Aubry-Andr\'e-Harper models with the Fermi energy placed in one of the band gaps. We investigate systems with periodic as well as open boundary conditions; for the latter focusing on the number of edge states and the boundary charge. Both these observables are important for the classification of noninteracting topological systems. In our first class of models the unit cell structure stems from periodically modulated single-particle parameters. In the second it results from the spatial modulation of the two-particle interaction. For both types of models, we find that the single-particle band gaps are renormalized by the interaction in accordance with expectations employing general field theoretical arguments. While interaction induced effective edge states can be found in the local single-particle spectral function close to a boundary, the characteristics of the boundary charge are not modified by the interaction. This indicates that our results for the Rice-Mele and Su-Schriefer-Heeger model [Phys. Rev. B 102, 085122 (2020)] are generic and can be found in lattice models with more complex unit cells as well.

We investigate the interacting, one-dimensional Rice-Mele model, a prototypical fermionic model of topological properties. To set the stage, we firstly compute the single-particle spectral function, the local density, and the boundary charge in the absence of interactions. The boundary charge is fully determined by bulk properties indicating a bulk-boundary correspondence. In a large parameter regime it agrees with the one obtained from an effective low-energy theory (arXiv:2004.00463). Secondly, we investigate the robustness of our results towards two-particle interactions. To resum the series of leading logarithms for small gaps, which dismantle plain perturbation theory in the interaction, we use an essentially analytical renormalization group approach. It is controlled for small interactions and can directly be applied to the microscopic lattice model. We benchmark the results against numerical density matrix renormalization group data. The main interaction effect in the bulk is a power-law renormalization of the gap with an interaction dependent exponent. The important characteristics of the boundary charge are unaltered and can be understood from the renormalized bulk properties, elevating the bulk-boundary correspondence to the interacting regime. This requires a consistent treatment not only of the low-energy gap renormalization but also of the high-energy band width one. In contrast to low-energy field theories our renormalization group approach also provides the latter. We show that the interaction spoils the relation between the bulk properties and the number of edge states, consistent with the observation that the Rice-Mele model with finite potential modulation does not reveal any zero-energy edge states.

We review the intriguing many-body physics resulting out of the interplay of a single, local impurity and the two-particle interaction in a one-dimensional Fermi system. Even if the underlying homogeneous correlated system is taken to be metallic, this interplay leads to an emergent quantum phase transition between metallic and insulating states. We show that the zero temperature critical point and the universal low-energy physics associated to it, is realized in two different models, the field theoretical local sine-Gordon model and spinless fermions on a lattice with nearest-neighbor hopping and two-particle interaction, as well as in an experimental setup consisting of a highly tunable quantum circuit. Despite the different high-energy physics of the three systems the universal low-energy scaling curves of the conductance as a function of temperature agree up to a very high precision without any free parameter. Overall this provides a convincing example of how emergent universality in complex systems originating from a common underlying quantum critical point establishes a bridge between different fields of physics. In our case between field theory, quantum many-body theory of correlated Fermi systems, and experimental circuit quantum electrodynamics., Comment: version as accepted for publication in EPJST

The nonlinear Luttinger liquid phenomenology of one-dimensional correlated Fermi systems is an attempt to describe the effect of the band curvature beyond the Tomonaga-Luttinger liquid paradigm. It relies on the observation that the dynamical structure factor of the interacting electron gas shows a logarithmic threshold singularity when evaluated to first order perturbation theory in the two-particle interaction. This term was interpreted as the linear one in an expansion which was conjectured to resum to a power law. A field theory, the mobile impurity model, which is constructed such that it provides the power law in the structure factor, was suggested to be the proper effective model and used to compute the single-particle spectral function. This forms the basis of the nonlinear Luttinger liquid phenomenology. Surprisingly, the second order perturbative contribution to the structure factor was so far not studied. We first close this gap and show that it is consistent with the conjectured power law. Secondly, we critically assess the steps leading to the mobile impurity Hamiltonian. We show that the model does not allow to include the effect of the momentum dependence of the (bulk) two-particle potential. This dependence was recently shown to spoil power laws in the single-particle spectral function which previously were believed to be part of the Tomonaga-Luttinger liquid universality. Although our second order results for the structure factor are consistent with power-law scaling, this raises doubts that the conjectured nonlinear Luttinger liquid phenomenology can be considered as universal. We conclude that more work is required to clarify this., Comment: version as published in SciPost

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 welcome

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 version

We investigate the effect of vibrational degrees of freedom on the linear thermoelectric transport through a single-level quantum dot described by the spinless Anderson-Holstein impurity model. To study the effects of strong electron-phonon coupling, we use the nonperturbative numerical renormalization group approach. We also compare our results, at weak to intermediate coupling, with those obtained by employing the functional renormalization group method, finding good agreement in this parameter regime. When applying a gate voltage at finite temperatures, the inelastic scattering processes, induced by phonon-assisted tunneling, result in an interesting interplay between electrical and thermal transport. We explore different parameter regimes and identify situations for which the thermoelectric power as well as the dimensionless figure of merit are significantly enhanced via a Mahan-Sofo type of mechanism. We show, in particular, that this occurs at strong electron-phonon coupling and in the antiadiabatic regime., Comment: 14 pages, 8 figures, Published version

We investigate the spinless Anderson-Holstein model routinely employed to describe the basic physics of phonon-assisted tunneling in molecular devices. Our focus is on small to intermediate electron-phonon coupling; we complement a recent strong coupling study [Phys.~Rev.~B {87}, 075319 (2013)]. The entire crossover from the antiadiabatic regime to the adiabatic one is considered. Our analysis using the essentially analytical functional renormalization group approach backed-up by numerical renormalization group calculations goes beyond lowest order perturbation theory in the electron-phonon coupling. In particular, we provide an analytic expression for the effective tunneling coupling at particle-hole symmetry valid for all ratios of the bare tunnel coupling and the phonon frequency. It contains the exponential polaronic as well as the power-law renormalization; the latter can be traced back to x-ray edge-like physics. In the antiadiabatic and the adiabatic limit this expression agrees with the known ones obtained by mapping to an effective interacting resonant level model and lowest order perturbation theory, respectively. Away from particle-hole symmetry, we discuss and compare results from several approaches for the zero temperature electrical conductance of the model., Comment: 11 pages, 6 figures, Published version

We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (`gapless') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter g and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in g. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in g. The more local a given quantity, the longer the linear behavior prevails when increasing g. Typical local correlation functions and observables for which the term O(g) vanishes thermalize even to order g^2. Our results show that lowest order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion., Comment: version as published

We investigate the thermodynamics and transient dynamics of the (unbiased) Ohmic two-state system by exploiting the equivalence of this model to the interacting resonant level model. For the thermodynamics, we show, by using the numerical renormalization group (NRG) method, how the universal specific heat and susceptibility curves evolve with increasing dissipation strength, $\alpha$, from those of an isolated two-level system at vanishingly small dissipation strength, with the characteristic activated-like behavior in this limit, to those of the isotropic Kondo model in the limit $\alpha\to 1^{-}$. For the transient dynamics of the two-level system, $P(t)=\langle \sigma_{z}(t)\rangle$, with initial-state preparation $P(t\leq0)=+1$, we apply the time-dependent extension of the NRG (TDNRG) to the interacting resonant level model, and compare the results obtained with those from the noninteracting-blip approximation (NIBA), the functional renormalization group (FRG), and the time-dependent density matrix renormalization group (TD-DMRG). We demonstrate excellent agreement on short to intermediate time scales between TDNRG and TD-DMRG for $0\lesssim\alpha \lesssim 0.9$ for $P(t)$, and between TDNRG and FRG in the vicinity of $\alpha=1/2$. Furthermore, we quantify the error in the NIBA for a range of $\alpha$, finding significant errors in the latter even for $0.1\leq \alpha\leq 0.4$. We also briefly discuss why the long-time errors in the present formulation of the TDNRG prevents an investigation of the crossover between coherent and incoherent dynamics. Our results for $P(t)$ at short to intermediate times could act as useful benchmarks for the development of new techniques to simulate the transient dynamics of spin-boson problems., Comment: 27 pages, 22 figures; published version

We study the zero-frequency current noise of the interacting resonant level model for arbitrary bias voltages using a functional renormalization group approach. For this we extend the existing nonequilibrium scheme by deriving and solving flow equations for the current-vertex functions. On-resonance artificial divergences of the latter found in lowest-order perturbation theory in the two-particle interaction are consistently removed. Away from resonance they are shifted to higher orders. This allows us to gain a comprehensive picture of the current noise in the scaling limit. At high bias voltages, the current noise exhibits a universal power-law decay, whose exponent is to leading order in the interaction identical to that of the current. The effective charge on resonance is analyzed in detail employing properties of the vertex correction. We find that it is only modified to second or higher order in the two-particle interaction., Comment: 15 pages, 16 figures

We reinvestigate the momentum-resolved single-particle spectral function of the Tomonaga-Luttinger model. In particular, we focus on the role of the momentum-dependence of the two-particle interaction V(q). Usually, V(q) is assumed to be a constant and integrals are regularized in the ultraviolet `by hand' employing an ad hoc procedure. As the momentum dependence of the interaction is irrelevant in the renormalization group sense this does not affect the universal low-energy properties of the model, e.g. exponents of power laws, if all energy scales are sent to zero. If, however, the momentum k is fixed away from the Fermi momentum k_F, with |k-k_F| setting a nonvanishing energy scale, the details of V(q) start to matter. We provide strong evidence that any curvature of the two-particle interaction at small transferred momentum q destroys power-law scaling of the momentum resolved spectral function as a function of energy. Even for |k-k_F| much smaller than the momentum space range of the interaction the spectral line shape depends on the details of V(q). The significance of our results for universality in the Luttinger liquid sense, for experiments on quasi one-dimensional metals, and for recent attempts to compute the spectral function of one-dimensional correlated systems taking effects of the curvature of the single-particle dispersion into account (nonlinear Luttinger liquid phenomenology) is discussed., Comment: 17 pages, 13 figures

We report on strong renormalization encountered in periodically driven interacting quantum dots in the non-adiabatic regime. Correlations between lead and dot electrons enhance or suppress the amplitude of driving depending on the sign of the interaction. Employing a newly developed flexible renormalization group based approach for periodic driving to an interacting resonant level we show analytically that the magnitude of this effect follows a power law. Our setup can act as a non-Markovian, single-parameter quantum pump., Comment: 6 pages, 3 figures

The magnetic state of a quantum dot attached to superconducting leads is experimentally shown to be controlled by the superconducting phase difference across the dot. This is done by probing the relation between the Josephson current and the superconducting phase difference of a carbon nanotube junction whose Kondo energy and superconducting gap are of comparable size. It exhibits distinctively anharmonic behavior, revealing a phase mediated singlet to doublet transition. We obtain an excellent quantitative agreement with numerically exact quantum Monte Carlo calculations. This provides strong support that we indeed observed the finite temperature signatures of the phase controlled zero temperature level-crossing transition originating from strong local electronic correlations., Comment: 5 pages, 4 figures + supp. materials

We show that to understand the orthogonality catastrophe in the half-filled lattice model of spinless fermions with repulsive nearest neighbor interaction and a local impurity in its Luttinger liquid phase one has to take into account (i) the impurity scaling, (ii) unusual finite size $L$ corrections of the form $\ln(L)/L$, as well as (iii) the renormalization group flow of the umklapp scattering. The latter defines a length scale $L_u$ which becomes exceedingly large the closer the system is to its transition into the charge-density wave phase. Beyond this transition umklapp scattering is relevant in the renormalization group sense. Field theory can only be employed for length scales larger than $L_u$. For small to intermediate two-particle interactions, for which the regime $L > L_u$ can be accessed, and taking into account the finite size corrections resulting from (i) and (ii) we provide strong evidence that the impurity backscattering contribution to the orthogonality exponent is asymptotically given by $1/16$. While further increasing the two-particle interaction leads to a faster renormalization group flow of the impurity towards the cut chain fixed point, the increased bare amplitude of the umklapp scattering renders it virtually impossible to confirm the expected asymptotic value of $1/16$ given the accessible system sizes. We employ the density matrix renormalization group., Comment: 12 pages, 9 figures

The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale $T_K$ (Kondo temperature) that breaks scale invariance. For the non-interacting resonant level model, the non-equilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitely [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable $T_K t$. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two non-interacting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field theory predictions for the universal scaling as well as for the large time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models., Comment: 14 pages, 10 figures

We show that the single-particle spectral properties of gapless one-dimensional Fermi systems in the Luttinger liquid state reached at intermediate times after an abrupt quench of the two-particle interaction are highly indicative of the unusual nonequilibrium nature of this state. The line shapes of the momentum integrated and resolved spectral functions strongly differ from their ground state as well as finite temperature equilibrium counterparts. Using an energy resolution improved version of radio-frequency spectroscopy of quasi one-dimensional cold Fermi gases it should be possible to experimentally identify this nonequilibrium state by its pronounced spectral signatures., Comment: 5 pages, 3 figures

We reveal several distinct regimes of the relaxation dynamics of a small quantum system coupled to an environment within the plane of the dissipation strength and the reservoir temperature. This is achieved by discriminating between coherent dynamics with damped oscillatory behavior on all time scales, partially coherent behavior being nonmonotonic at intermediate times but monotonic at large ones, and purely monotonic incoherent decay. Surprisingly, elevated temperature can render the system `more coherent' by inducing a transition from the partially coherent to the coherent regime. This provides a refined view on the relaxation dynamics of open quantum systems., Comment: 5 pages, 3 figures

We present a comprehensive study of the spectral and transport properties in the Anderson--Holstein model both in and out of equilibrium using the functional renormalization group (FRG). We show how the previously established machinery of Matsubara and Keldysh FRG can be extended to include the local phonon mode. Based on the analysis of spectral properties in equilibrium we identify different regimes depending on the strength of the electron--phonon interaction and the frequency of the phonon mode. We supplement these considerations with analytical results from the Kondo model. We also calculate the non-linear differential conductance through the Anderson--Holstein quantum dot and find clear signatures of the presence of the phonon mode., Comment: 19 pages, 8 figures

We study the dynamics resulting out of an abrupt change of the two-particle interaction in two models of closed one-dimensional Fermi systems: (a) the field theoretical Tomonaga-Luttinger model and (b) a microscopic lattice model. Using a nonperturbative approach which is controlled for small two-particle interactions we are able to reach large times allowing us to access the properties of the steady state of the lattice model. Comparing those to the exact solution of the full dynamics in the Tomonaga-Luttinger model we provide evidence for universal Luttinger liquid behavior., Comment: 9 pages, 5 figures, 1 table as accepted for publication by PRB

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 figures

The nonequilibrium dynamics of a small quantum system coupled to a dissipative environment is studied. We show that (1) the oscillatory dynamics close to a coherent-to-incoherent transition is surprisingly different from the one of the classical damped harmonic oscillator and that (2) non-markovian memory plays a prominent role in the time evolution after a quantum quench., Comment: 5 pages, 3 figures

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 table

We provide strong evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: The leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density matrix renormalization group.

By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong- and weak-tunnel couplings, leading to gate-dependent level renormalization. By comparison to a one- and two-shell model, this is shown to be a consequence of disorder-induced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization., Comment: 5 pages, 3 figures; revised version as published

We study electron transport through a multi-level quantum dot with Rashba spin-orbit interaction in the presence of local Coulomb repulsion. Motivated by recent experiments, we compute the level splitting induced by the spin-orbit interaction at finite Zeeman fields $B$, which provides a measure of the renormalized spin-orbit energy. This level splitting is responsible for the suppression of the Kondo ridges at finite $B$ characteristic for the multi-level structure. In addition, the dependence of renormalized $g$-factors on the relative orientation of the applied $B$ field and the spin-orbit direction following two different protocols used in experiments is investigated., Comment: 11 pages, 13 figures

We study the relaxation dynamics of the one-dimensional Tomonaga-Luttinger model after an interaction quench paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical form are investigated all leading to universal Luttinger liquid physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the Luttinger liquid phenomenology. For generic regular potentials the large time decay of the momentum distribution function towards the steady-state value is characterized by a power law with a universal exponent which only depends on the potential at zero momentum transfer. A commonly employed ad hoc procedure fails to give this exponent. Besides quenches from zero to positive interactions we also consider abrupt changes of the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and the one obtained in equilibrium at equal two-particle interaction., Comment: 21 pages, 4 figures

Motivated by recent scanning tunneling and photoemission spectroscopy measurements on self-organized gold chains on a germanium surface we reinvestigate the local single-particle spectral properties of Luttinger liquids. In the first part we use the bosonization approach to exactly compute the local spectral function of a simplified field theoretical low-energy model and take a closer look at scaling properties as a function of the ratio of energy and temperature. Translational invariant Luttinger liquids as well as those with an open boundary (cut chain geometry) are considered. We explicitly show that the scaling functions of both setups have the same analytic form. The scaling behavior suggests a variety of consistency checks which can be performed on measured data to experimentally verify Luttinger liquid behavior. In a second part we approximately compute the local spectral function of a microscopic lattice model---the extended Hubbard model---close to an open boundary using the functional renormalization group. We show that as a function of energy and temperature it follows the field theoretical prediction in the low-energy regime and point out the importance of nonuniversal energy scales inherent to any microscopic model. The spatial dependence of this spectral function is characterized by oscillatory behavior and an envelope function which follows a power law both in accordance with the field theoretical continuum model. Interestingly, for the lattice model we find a phase shift which is proportional to the two-particle interaction and not accounted for in the standard bosonization approach to Luttinger liquids with an open boundary. We briefly comment on the effects of several one-dimensional branches cutting the Fermi energy and Rashba spin-orbit interaction., Comment: 19 pages, 5 figures, version as accepted for publication in J. Phys.:Condensed Matter

We introduce a real time version of the functional renormalization group which allows to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation dynamics of the interacting resonant level model which describes a fermionic quantum dot dominated by charge fluctuations. Moreover, we study decoherence and relaxation phenomena within the ohmic spin-boson model by mapping the latter to the interacting resonant level model.

We investigate two serially-aligned quantum dots in the molecular regime of large tunnel couplings t. A Zeeman field B is used to tune the energy difference of singlet and triplet spin configurations. Attaching this geometry to BCS source and drain leads with gap Delta and phase difference phi gives rise to an equilibrium supercurrent J. To compute J in presence of Coulomb interactions U between the dot electrons, we employ the functional renormalization group (FRG). For B\simt -- where the singlet and lowest-lying triplet spin states are equal in energy -- the current exhibits characteristics of a 0-pi transition similar to a single impurity. Its magnitude in the pi phase, however, jumps discontinuously at B=t, being smaller on the triplet side. Exploiting the flexibility of the FRG, we demonstrate that this effect is generic and calculate J for realistic experimental parameters Delta, U, and gate voltages epsilon. To obtain a more thorough understanding of the discontinuity, we analytically treat the limit Delta=infty where one can access the exact many-particle states. Finally, carrying out perturbation theory in the dot-lead couplings substantiates the intuitive picture that Cooper pair tunneling is favored by a singlet spin configuration while inhibited by a triplet one.

Using low-temperature scanning tunneling spectroscopy applied to the Cs-induced two-dimensional electron system (2DES) on p-type InSb(110), we probe electron-electron interaction effects in the quantum Hall regime. The 2DES is decoupled from p-doped bulk states and exhibits spreading resistance within the insulating quantum Hall phases. In quantitative agreement with calculations we find an exchange enhancement of the spin splitting. Moreover, we observe that both the spatially averaged as well as the local density of states feature a characteristic Coulomb gap at the Fermi level. These results show that electron-electron interaction effects can be probed down to a resolution below all relevant length scales., Comment: supplementary movie in ancillary files

We study electron transport through a serial double quantum dot with Rashba spin-orbit interaction (SOI) and Zeeman field of amplitude B in presence of local Coulomb repulsion. The linear conductance as a function of a gate voltage Vg equally shifting the levels on both dots shows two B=0 Kondo ridges which are robust against SOI as time-reversal symmetry is preserved. Resulting from the crossing of a spin-up and a spin-down level at vanishing SOI two additional Kondo plateaus appear at finite B. They are not protected by symmetry and rapidly vanish if the SOI is turned on. Left-right asymmetric level-lead couplings and detuned on-site energies lead to a simultaneous breaking of left-right and bonding-anti-bonding state symmetry. In this case the finite-B Kondo ridges in the Vg-B plane are bent with respect to the Vg-axis. For the Kondo ridge to develop different level renormalizations must be compensated by adjusting B., Comment: 8 pages, 5 figures, revised version as published

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. B

We address the question whether observables of an exactly solvable model of electrons coupled to (optical) phonons relax into large time stationary state values and investigate if the asymptotic expectation values can be computed using a stationary density matrix. Two initial nonequilibrium situations are considered. A sudden quench of the electron-phonon coupling, starting from the noninteracting canonical equilibrium at temperature T in the electron as well as in the phonon subsystems, leads to a rather simple dynamics. A richer time evolution emerges if the initial state is taken as the product of the phonon vacuum and the filled Fermi sea supplemented by a highly excited additional electron. Our model has a natural set of constants of motion, with as many elements as degrees of freedom. In accordance with earlier studies of such type of models we find that expectation values which become stationary can be described by the density matrix of a generalized Gibbs ensemble which differs from that of a canonical ensemble. For the model at hand it appears to be evident that the eigenmode occupancy operators should be used in the construction of the stationary density matrix., Comment: 15 pages, 11 figures, published version

We review recent progresses in the theoretical description of correlation and quantum fluctuation phenomena in charge transport through single molecules, quantum dots, and quantum wires. A variety of physical phenomena is addressed, relating to co-tunneling, pair-tunneling, adiabatic quantum pumping, charge and spin fluctuations, and inhomogeneous Luttinger liquids. We review theoretical many-body methods to treat correlation effects, quantum fluctuations, nonequilibrium physics, and the time evolution into the stationary state of complex nanoelectronic systems., Comment: 48 pages, 14 figures, Topical Review for Nanotechnology

We study the steady-state current in a minimal model for a quantum dot dominated by charge fluctuations and analytically describe the time evolution into this state. The current is driven by a finite bias voltage V across the dot, and two different renormalization group methods are used to treat small to intermediate local Coulomb interactions. The corresponding flow equations can be solved analytically which allows to identify all microscopic cutoff scales. Exploring the entire parameter space we find rich non-equilibrium physics which cannot be understood by simply considering the bias voltage as an infrared cutoff. For the experimentally relevant case of left-right asymmetric couplings, the current generically shows a power-law suppression for large V. The relaxation dynamics towards the steady state features characteristic oscillations as well as an interplay of exponential and power-law decay.

We investigate equilibrium and steady-state non-equilibrium transport properties of a spinless resonant level locally coupled to two conduction bands of width ~\Gamma via a Coulomb interaction U and a hybridization t'. In order to study the effects of finite bias voltages beyond linear response, a generalization of the functional renormalization group to Keldysh frequency space is employed. Being mostly unexplored in the context of quantum impurity systems out of equilibrium, we benchmark this method against recently-published time-dependent density matrix renormalization group data. We thoroughly investigate the scaling limit \Gamma\to\infty characterized by the appearance of power laws. Most importantly, at the particle-hole symmetric point the steady-state current decays like J ~ V^{-\alpha_J} as a function of the bias voltage V>>t', with an exponent \alpha_J(U) that we calculate to leading order in the Coulomb interaction strength. In contrast, we do not observe a pure power-law (but more complex) current-voltage-relation if the energy \epsilon of the resonant level is pinned close to either one of the chemical potentials \pm V/2.

We use Wilsons weak coupling ``momentum'' shell renormalization group method to show that two-particle interaction terms commonly neglected in bosonization of one-dimensional correlated electron systems with open boundaries are indeed irrelevant in the renormalization group sense. Our study provides a more solid ground for many investigations of Luttinger liquids with open boundaries., Comment: version as published

We study the performance of a quantum wire spin filter that is based on the Rashba spin-orbit interaction in the presence of the electron-electron interaction. The finite length wire is attached to two semi-infinite nonmagnetic leads. Analyzing the spin polarization of the linear conductance at zero temperature, we show that spin-filtering is possible by adequate tuning of the system parameters first considering noninteracting electrons. Next, the functional renormalization group method is used to capture correlation effects induced by the Coulomb interaction. For short wires we show that the energy regime in which spin polarization is found is strongly affected by the Coulomb interaction. For long wires we find the power-law suppression of the total conductance on low energy scales typical for inhomogeneous Luttinger liquids while the degree of spin polarization stays constant.

We study a quantum dot Josephson junction inside an Aharonov-Bohm environment. The geometry is modeled by an Anderson impurity coupled to two directly-linked BCS leads. We illustrate that the well-established picture of the low-energy physics being governed by an interplay of two distinct (singlet and doublet) phases is still valid for this interferometric setup. The phase boundary depends, however, non-monotonically on the coupling strength between the superconductors, causing the system to exhibit re-entrance behavior and multiple phase transitions. We compute the zero-temperature Josephson current and demonstrate that it can become negative in the singlet phase by virtue of the Coulomb interaction U. As a starting point, the limit of large superconducting energy gaps \Delta=\infty is solved analytically. In order to tackle arbitrary \Delta<\infty and U>0, we employ a truncated functional renormalization group scheme which was previously demonstrated to give quantitatively reliable results for the quantum dot Josephson problem.

Understanding the charging of exceptionally narrow levels in quantum dots in the presence of interactions remains a challenge within mesoscopic physics. We address this fundamental question in the generic model of a narrow level capacitively coupled to a broad one. Using bosonization we show that for arbitrary capacitive coupling charging can be described by an analogy to the magnetization in the anisotropic Kondo model, featuring a low-energy crossover scale that depends in a power-law fashion on the tunneling amplitude to the level. Explicit analytical expressions for the exponent are derived and confirmed by detailed numerical and functional renormalization-group calculations., Comment: 4+ pages, 3 figures, PRL version

We investigate the Josephson current in a single wall carbon nanotube connected to superconducting electrodes. We focus on the parameter regime in which transport is dominated by Kondo physics. A sizeable supercurrent is observed for odd number of electrons on the nanotube when the Kondo temperature Tk is sufficiently large compared to the superconducting gap. On the other hand when, in the center of the Kondo ridge, Tk is slightly smaller than the superconducting gap, the supercurrent is found to be extremely sensitive to the gate voltage Vbg. Whereas it is largely suppressed at the center of the ridge, it shows a sharp increase at a finite value of Vbg. This increase can be attributed to a doublet-singlet transition of the spin state of the nanotube island leading to a pi shift in the current phase relation. This transition is very sensitive to the asymmetry of the contacts and is in good agreement with theoretical predictions., Comment: 5 pages, 4 figures

We use the Matsubara functional renormalization group (FRG) to describe electronic correlations within the single impurity Anderson model. In contrast to standard FRG calculations, we account for the frequency-dependence of the two-particle vertex in order to address finite-energy properties (e.g, spectral functions). By comparing with data obtained from the numerical renormalization group (NRG) framework, the FRG approximation is shown to work well for arbitrary parameters (particularly finite temperatures) provided that the electron-electron interaction U is not too large. We demonstrate that aspects of (large U) Kondo physics which are described well by a simpler frequency-independent truncation scheme are no longer captured by the 'higher-order' frequency-dependent approximation. In contrast, at small to intermediate U the results obtained by the more elaborate scheme agree better with NRG data. We suggest to parametrize the two-particle vertex not by three independent energy variables but by introducing three functions each of a single frequency. This considerably reduces the numerical effort to integrate the FRG flow equations., Comment: accepted by J. Phys.: Condensed Matter

We study electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems closer to setups realized in experiments. In particular, we compute the temperature dependence of the linear conductance $G$ of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of overhanging parts of the leads and the wire implies a $G(T)$ which is much more complex than the power-law behavior described so far for end-coupled wires. Depending on the precise setup the wide temperature regime of power-law scaling found in the end-coupled case is broken up in up to five fairly narrow regimes interrupted by extended crossover regions. Our results can be used to optimize the experimental setups designed for a verification of Luttinger liquid power-law scaling., Comment: accepted by J. Phys.: Condens. Matter

We investigate how the different velocities characterizing the low-energy spectral properties and the low-temperature thermodynamics of one-dimensional correlated electron systems (Luttinger liquids) affect the transport properties of ring-like conductors. The Luttinger liquid ring is coupled to two noninteracting leads and pierced by a magnetic flux. We study the flux dependence of the linear conductance. It shows a dip structure which is governed by the interaction dependent velocities. Our work extends an earlier study which was restricted to rather specific choices of the interaction parameters. We show that for generic repulsive two-particle interactions the number of dips can be estimated from the ratio of the charge current velocity and the spin velocity. In addition, we clarify the range of validity of the central approximation underlying the earlier study., Comment: 10 pages including figures

We compare two fermionic renormalization group methods which have been used to investigate the electronic transport properties of one-dimensional metals with two-particle interaction (Luttinger liquids) and local inhomogeneities. The first one is a poor man's method setup to resum ``leading-log'' divergences of the effective transmission at the Fermi momentum. Generically the resulting equations can be solved analytically. The second approach is based on the functional renormalization group method and leads to a set of differential equations which can only for certain setups and in limiting cases be solved analytically, while in general it must be integrated numerically. Both methods are claimed to be applicable for inhomogeneities of arbitrary strength and to capture effects of the two-particle interaction, such as interaction dependent exponents, up to leading order. We critically review this for the simplest case of a single impurity. While on first glance the poor man's approach seems to describe the crossover from the ``perfect'' to the ``open chain fixed point'' we collect evidence that difficulties may arise close to the ``perfect chain fixed point''. Due to a subtle relation between the scaling dimensions of the two fixed points this becomes apparent only in a detailed analysis. In the functional renormalization group method the coupling of the different scattering channels is kept which leads to a better description of the underlying physics., Comment: 25 pages, accepted for publication in NJP, remarks added on the poor man's RG treatment of the Y-junction and the Breit-Wigner line shapes

We investigate the Josephson current J(\phi) through a quantum dot embedded between two superconductors showing a phase difference \phi. The system is modeled as a single Anderson impurity coupled to BCS leads, and the functional and the numerical renormalization group frameworks are employed to treat the local Coulomb interaction U. We reestablish the picture of a quantum phase transition occurring if the ratio between the Kondo temperature T_K and the superconducting energy gap \Delta or, at appropriate T_K/\Delta, the phase difference \phi or the impurity energy is varied. We present accurate zero- as well as finite-temperature T data for the current itself, thereby settling a dispute raised about its magnitude. For small to intermediate U and at T=0 the truncated functional renormalization group is demonstrated to produce reliable results without the need to implement demanding numerics. It thus provides a tool to extract characteristics from experimental current-voltage measurements., Comment: version accepted for publication in PRB

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 version