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We present a complete microscopic Fermi-liquid description for next-to-leading order transport through an Anderson impurity under a finite bias voltage $V$. It is applicable to multilevel quantum dots without particle-hole or time-reversal symmetry, and is constructed based on the nonequilibrium Keldysh formalism, taking into account the current conservation between electrons in the impurity levels and the conduction bands. Specifically, we derive the formula for the current noise generated in the steady flow up to terms of order $(eV)^3$ at zero temperature $T=0$. To this end, we calculate the Keldysh vertex functions $\Gamma_{\sigma\sigma';\sigma'\sigma}^{ \nu_1\nu_2;\nu_3\nu_4} (\omega,\omega'; \omega',\omega)$, which depend on branches $\nu_1, \nu_2, \nu_3$ and $\nu_4$ of the time-loop contour and on spin degrees of freedom $\sigma$ and $\sigma'$, up to linear-order terms with respect to $eV$, $T$, and frequencies $\omega$ and $\omega'$. The coefficients of these linear-order terms are determined by a set of the parameters, defined with respect to the equilibrium ground state: the phase shift, static susceptibilities, and nonlinear three-body susceptibilities of the impurity electrons. The low-energy expressions of the vertex components are shown to satisfy the Ward identities with the Keldysh Green's functions expanded up to terms of order $\omega^2$, $(eV)^2$, and $T^2$. We also find that the imaginary part of the Ward identities can be described in terms of the $eV$-dependent collision integrals for a single-quasiparticle excitation and that for a single quasiparticle-quasihole pair excitation. These collision integrals ensure the current conservation of the next-to-leading order Fermi-liquid transport due to the quasiparticles with a finite damping rate., Comment: 43 pages, 10 figures, typos have been corrected

We study the nonlinear conductance through a quantum dot, specifically its dependence on the asymmetries in the tunnel couplings and bias voltages $V$, at low energies. Extending the microscopic Fermi-liquid theory for the Anderson impurity model, we obtain an exact formula for the steady current $I$ up to terms of order $V^3$ in the presence of these asymmetries. The coefficients for the nonlinear terms are described in terms of a set of the Fermi-liquid parameters: the phase shift, static susceptibilities, and three-body correlation functions of electrons in the quantum dots, defined with respect to the equilibrium ground state. We calculate these correlation functions, using the numerical renormalization group approach (NRG), over a wide range of impurity-electron filling that can be controlled by a gate voltage in real systems. The NRG results show that the order $V^2$ nonlinear current is enhanced significantly in the valence fluctuation regime. It is caused by the order $V$ energy shift of the impurity level, induced in the presence of the tunneling or bias asymmetry. Furthermore, in the valence fluctuation regime, we also find that the order $V^3$ nonlinear current exhibits a shoulder structure, for which the three-body correlations that evolve for large asymmetries play an essential role., 18 pages, 27 figures. Typos have been corrected

Behavior of quantum liquids is a fascinating topic in physics. Even in a strongly correlated case, the linear response of a given system to an external field is described by the fluctuation-dissipation relations based on the two-body correlations in the equilibrium. However, to explore nonlinear non-equilibrium behaviors of the system beyond this well-established regime, the role of higher order correlations starting from the three-body correlations must be revealed. In this work, we experimentally investigate a controllable quantum liquid realized in a Kondo-correlated quantum dot and prove the relevance of the three-body correlations in the nonlinear conductance at finite magnetic field, which validates the recent Fermi liquid theory extended to the non-equilibrium regime., Recent theory has shown that the non-equilibrium response of a Kondo model can be described by the Fermi liquid theory with three-body correlations. Here, the authors experimentally measure such correlations in the nonlinear conductance of a Kondo-correlated quantum dot.

We present a microscopic Fermi-liquid view on the low-energy transport through an Anderson impurity with $N$ discrete levels, at arbitrary electron filling $N_d$. It is applied to nonequilibrium current fluctuations, for which the two-quasiparticle collision integral and the three-body correlations that determine the quasiparticle energy shift play important roles. Using the numerical renormalization group up to $N=6$, we find that for strong interactions the three-body fluctuations are determined by a single parameter other than the Kondo energy scale in a wide filling range $1 \lesssim N_d \lesssim N-1$. It significantly affects the current noise for $N>2$ and the behavior of noise in magnetic fields., 5 pages, 8 figures, supplemental material containing 8 figures

We study finite-temperature properties of the Kondo effect in a carbon nanotube (CNT) quantum dot using the Wilson numerical renormalization group (NRG). In the absence of magnetic fields, four degenerate energy levels of the CNT consisting of spin and orbital degrees of freedom give rise to the SU(4) Kondo effect. We revisit the universal scaling behavior of the SU(4) conductance for quarter- and half-filling in a wide temperature range. We find that the filling dependence of the universal scaling behavior at low temperatures $T$ can be explained clearly with an extended Fermi-liquid theory. This theory clarifies that a $T^{2}$ coefficient of conductance becomes zero at quarter-filling whereas the coefficient at half-filling is finite. We also study a field-induced crossover from the SU(4) to SU(2) Kondo state observed at the half-filled CNT dot. The crossover is caused by the matching of the spin and orbital Zeeman splittings, which lock two levels among the four at the Fermi level even in magnetic fields $B$. We find that the conductance shows the SU($4$) scaling behavior at $\mu_{B}B, Comment: 27 pages, 14 figures

We study spin-entanglement of the quasiparticles of the local Fermi liquid excited in nonlinear current through a quantum dot described by the Anderson impurity model with two degenerate orbitals coupled to each other via an exchange interaction. Applying the renormalized perturbation theory, we obtain the precise form of the cumulant generating function and cross-correlations for the currents with spin angled to arbitrary directions, up to third order in the applied bias voltage. It is found that the exchange interaction gives rise to spin-angle dependency in the cross-correlation between the currents through the two different orbitals, and also brings an intrinsic cross-correlation of currents with three different angular momenta.

Universal properties of entangled many-body states are controlled by their symmetry and quantum fluctuations. By magnetic-field tuning of the spin-orbital degeneracy in a Kondo-correlated quantum dot, we have modified quantum fluctuations to directly measure their influence on the many-body properties along the crossover from $SU(4)$ to $SU(2)$ symmetry of the ground state. High-sensitive current noise measurements combined with the non-equilibrium Fermi liquid theory clarify that the Kondo resonance and electron correlations are enhanced as the fluctuations, measured by the Wilson ratio, increase along the symmetry crossover. Our achievement demonstrates that non-linear noise constitutes a measure of quantum fluctuations that can be used to tackle quantum phase transitions., Comment: 5 pages, 4 figures

We study the higher-order Fermi-liquid relations of Kondo systems for arbitrary impurity-electron fillings, extending the many-body quantum theoretical approach of Yamada-Yosida. It includes partly a microscopic clarification of the related achievements based on Nozi\`{e}res' phenomenological description: Filippone, Moca, von Delft, and Mora [Phys.\ Rev.\ B {\bf 95}, 165404 (2017)]. In our formulation, the Fermi-liquid parameters such as the quasi-particle energy, damping, and transport coefficients are related to each other through the total vertex $\Gamma_{\sigma\sigma';\sigma'\sigma} (\omega, \omega'; \omega', \omega)$, which may be regarded as a generalized Landau quasi-particle interaction. We obtain exactly this function up to linear order with respect to the frequencies $\omega$ and $\omega'$ using the anti-symmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half-filling through the non-linear susceptibilities. We also apply the formulation to non-equilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field., Comment: 5 pages, 2 figures, Introduction has been revised

We study the low-energy behavior of the vertex function of a single Anderson impurity away from half-filling for finite magnetic fields, using the Ward identities with careful consideration of the anti-symmetry and analytic properties. The asymptotic form of the vertex function $\Gamma_{\sigma\sigma';\sigma'\sigma}^{}(i\omega,i\omega';i\omega',i\omega)$ is determined up to terms of linear order with respect to the two frequencies $\omega$ and $\omega'$, as well as the $\omega^2$ contribution for anti-parallel spins $\sigma'\neq \sigma$ at $\omega'=0$. From these results, we also obtain a series of the Fermi-liquid relations beyond those of Yamada-Yosida. The $\omega^2$ real part of the self-energy $\Sigma_{\sigma}^{}(i\omega)$ is shown to be expressed in terms of the double derivative $\partial^2\Sigma_{\sigma}^{}(0)/\partial \epsilon_{d\sigma}^{2}$ with respect to the impurity energy level $\epsilon_{d\sigma}^{}$, and agrees with the formula obtained recently by Filippone, Moca, von Delft, and Mora in the Nozi\`{e}res phenomenological Fermi-liquid theory [Phys.\ Rev.\ B {\bf 95}, 165404 (2017)]. We also calculate the $T^2$ correction of the self-energy, and find that the real part can be expressed in terms of the three-body correlation function $\chi_{\uparrow\downarrow,-\sigma}^{[3]} = \partial \chi_{\uparrow\downarrow}/\partial \epsilon_{d,-\sigma}^{}$. We also provide an alternative derivation of the asymptotic form of the vertex function. Specifically, we calculate the skeleton diagrams for the vertex function $\Gamma_{\sigma\sigma;\sigma\sigma}^{}(i\omega,0;0,i\omega)$ for parallel spins up to order $U^4$ in the Coulomb repulsion $U$. It directly clarifies the fact that the analytic components of order $\omega$ vanish as a result of the cancellation of four related Feynman diagrams which are related to each other through the anti-symmetry operation., Comment: 15 figures, 19pages, and supplemental material (15 pages), typo found in (6.15) has been corrected

We theoretically study an emergent SU(2) symmetry which is suggested by recent magneto-transport measurements, carried out near two electrons filling of a carbon nanotube quantum dot. It emerges in the case where the spin and orbital Zeeman splittings cancel each other out for two of the one-particle dot levels among four. Using the Wilson numerical renormalization group, we show that a crossover from the SU(4) to SU(2) Fermi-liquid behavior occurs at two impurity-electrons filling as magnetic field increases. We also find that the quasiparticles are significantly renormalized as the remaining two one-particle levels move away from the Fermi level and are frozen at high magnetic fields. In order to clarify how the ground state evolves during such a crossover, we also reexamine the SU(N) Kondo singlet state for M impurity-electrons filling in the limit of strong exchange interactions. We show that the nondegenerate Fermi-liquid fixed point of Nozi\`{e}es and Blandin can be described as a bosonic Perron-Frobenius vector for M hard-core bosons, each of which consists of one impurity-electron and one conduction hole. This interpretation can also be applied to the Fermi-liquid fixed-point without the SU(N) symmetry., Comment: 20 pages, 10 figures, Sec.III B. has been revised. J.Phys.Soc.Jpn.in press

We describe a formulation to deduce the phase shifts, which determine the ground-state properties of interacting quantum-dot systems with the inversion symmetry, from the fixed-point eigenvalues of the numerical renormalization group (NRG). Our approach does not assume the specific form of the Hamiltonian nor the electron-hole symmetry, and it is applicable to a wide class of quantum impurities connected to noninteracting leads. We apply the method to a triple dot which is described by a three-site Hubbard chain connected to two noninteracting leads, and calculate the dc conductance away from half-filling. The conductance shows the typical Kondo plateaus of Unitary limit in some regions of the gate voltages, at which the total number of electrons N_el in the three dots is odd, i.e., N_el =1, 3 and 5. In contrast, the conductance shows a wide minimum in the gate voltages corresponding to even number of electrons, N_el = 2 and 4. We also discuss the parallel conductance of the triple dot connected transversely to four leads, and show that it can be deduced from the two phase shifts defined in the two-lead case., Comment: 9 pages, 12 figures: Fig. 12 has been added to discuss T_K

We derive an exact expression for the differential conductance for a quantum dot in an arbitrary magnetic field for small bias voltage. The derivation is based on the symmetric Anderson model using renormalized perturbation theory and is valid for all values of the on-site interaction $U$ including the Kondo regime. We calculate the critical magnetic field for the splitting of the Kondo resonance to be seen in the differential conductivity as function of bias voltage. Our calculations for small field show that the peak position of the component resonances in the differential conductance are reduced substantially from estimates using the equilibrium Green's function. We conclude that it is important to take the voltage dependence of the local retarded Green's function into account in interpreting experimental results, Comment: 8 pages, 4 figures; Replaced by a fully revised version with minor corrections in the text

We study the low-energy properties of a Hubbard chain of finite size N_C connected to two noninteracting leads using the numerical renormalization group (NRG) method. The results obtained for N_C = 3 and 4 show that the low-lying eigenstates have one-to-one correspondence with the free quasi-particle excitations of a local Fermi liquid. It enables us to determine the transport coefficients from the fixed-point Hamiltonian. At half-filling, the conductance for even N_C decreases exponentially with increasing U showing a tendency towards the development of a Mott-Hubbard gap. In contrast, for odd N_C, the Fermi-liquid nature of the low-energy states assures perfect transmission through the Kondo resonance. Our formulation to deduce the conductance from the fixed-point energy levels can be applied to various types of interacting systems., Comment: One typo found in Eq.(3) in previous version has been corrected

Interacting quantum many-body systems constitute a fascinating playground for researchers since they form quantum liquids with correlated ground states and low-lying excitations, which exhibit universal behaviour. In fermionic systems, such quantum liquids are realized in helium-3 liquid, heavy fermion systems, neutron stars and cold gases. Their properties in the linear-response regime have been successfully described by the theory of Fermi liquids. However, non-equilibrium properties beyond this regime have still to be established and remain a key issue of many-body physics. Here, we show a precise experimental demonstration of Landau Fermi-liquid theory extended to the non-equilibrium regime in a 0-D system. Combining transport and ultra-sensitive current noise measurements, we have unambiguously identified the SU(2) and SU(4) symmetries of quantum liquid in a carbon nanotube tuned in the universal Kondo regime. We find that, while the electronic transport is well described by the free quasi-particle picture around equilibrium, a two-particle scattering process due to residual interaction shows up in the non-equilibrium regime. By using the extended Fermi-liquid theory, we obtain the interaction parameter "Wilson ratio" $R=1.9 \pm 0.1$ for SU(2) and $R=1.35 \pm 0.1$ for SU(4) as well as the corresponding effective charges, characterizing the quantum liquid behaviour. This result, in perfect agreement with theory, provides a strong quantitative experimental background for further developments of the many-body physics. Moreover, we discovered a new scaling law for the effective charge, signalling as-yet-unknown universality in the non-equilibrium regime. Our method to address quantum liquids through their non-equilibrium noise paves a new road to tackle the exotic nature of quantum liquids out-of-equilibrium in various physical systems., Comment: accepted for publication in Nature Physics

We study the nonequilibrium Keldysh Green's function for an N-orbital Anderson model at high bias voltages, extending a previous work, which for the case only with the spin degrees of freedom N=2, to arbitrary N. Our approach uses an effective non-Hermitian Hamiltonian that is defined with respect to a Liouville-Fock space in the context of a thermal field theory. The result correctly captures the relaxation processes at high energies, and is asymptotically exact not only in the high-bias limit but also in the high-temperature limit at thermal equilibrium. We also present an explicit continued-fraction representation of the Green's function. It clearly shows that the imaginary part is recursively determined by the decay rate of intermediate states with at most N-1 particle-hole-pair excitations. These high-bias properties follow from the conservations of a generalized charge and current in the Liouville-Fock space. We also examine temperature dependence of the spectral function in equilibrium, comparing the exact results with both the finite-T and infinite-T results of the non-crossing approximation (NCA)., 16 pages, 6 figures, typos have been corrected and DOI has been provided

Transport through small interacting systems connected to noninteracting leads is studied based on the Kubo formalism using a Eliashberg theory of the analytic properties of the vertex part. The transmission probability, by which the conductance is expressed as $g = (2e^2/h) \int d\epsilon (- \partial f / \partial \epsilon) {\cal T}(\epsilon)$, is introduced for interacting electrons. Here $f(\epsilon)$ is the Fermi function, and the transmission probability ${\cal T}(\epsilon)$ is defined in terms of a current vertex or a three-point correlation function. We apply this formulation to a series of Anderson impurities of size N (=1,2,3,4), and calculate ${\cal T}(\epsilon)$ using the order $U^2$ self-energy and current vertex which satisfy a generalized Ward identity. The results show that ${\cal T}(\epsilon)$ has much information about the excitation spectrum: ${\cal T}(\epsilon)$ has two broad peaks of the upper and lower Hubbard bands in addition to N resonant peaks which have direct correspondence with the noninteracting spectrum. The peak structures disappear at high temperatures., Comment: 18 pages, 16 figures

We describe some exact high-energy properties of a single Anderson impurity connected to two noninteracting leads in a nonequilibrium steady state. In the limit of high bias voltages, and also in the high-temperature limit at thermal equilibrium, the model can be mapped on to an effective non-Hermitian Hamiltonian consisting of two sites, which correspond to the original impurity and its image that is defined in a doubled Hilbert space referred to as Liouville-Fock space. For this, we provide a heuristic derivation using a path-integral representation of the Keldysh contour and the thermal field theory, in which the time evolution along the backward contour is replicated by extra degrees of freedom corresponding to the image. We find that the effective Hamiltonian can also be expressed in terms of charges and currents. From this, it can be deduced that the dynamic susceptibilities for the charges and the current fluctuations become independent of the Coulomb repulsion U in the high bias limit. Furthermore, the equation of motions for the Green's function and two other higher-order correlation functions constitute a closed system. The exact solution obtained from the three coupled equations extends the atomic-limit solution such that the self-energy correctly captures the imaginary part caused by the relaxation processes at high energies. The spectral weights of the upper and lower Hubbard levels depend sensitively on the asymmetry in the tunneling couplings to the left and right leads., Comment: 12 pages, 5 figures, typos have been corrected

We study non-equilibrium current fluctuations through a quantum dot, which includes a ferromagnetic Hund's rule coupling $J$, in the low-energy Fermi liquid regime using the renormalized perturbation theory. The resulting cumulant for the current distribution in the particle-hole symmetric case, shows that spin-triplet and spin-singlet pairs of quasiparticles are formed in the current due to the Hund's rule coupling and these pairs enhance the current fluctuations. In the fully screened higher-spin Kondo limit, the Fano factor takes a value $F_b = (9M+6)/ (5M+4)$ determined by the orbital degeneracy $M$. We also investigate crossover between the small and large $J$ limits in the two-orbital case M=2, using the numerical renormalization group approach., Comment: 5 pages, 3 figures

We study the Kondo effect in side-coupled double quantum dots with particular focus on the crossover between two distinct singlet ground states, using the numerical renormalization group. The crossover occurs as the quantized energy level of the embedded dot, which is connected directly to the leads, is varied. In the parameter region where the embedded dot becomes almost empty or doubly occupied, the local moment emerging in the other dot at the side of the path for the current is screened via a superexchange process by the conduction electrons tunneling through the embedded dot. In contrast, in the other region where the embedded dot is occupied by a single electron, the local moment emerges also in the embedded dot, and forms a singlet bond with the moment in the side dot. Furthermore, we derive two different Kondo Hamiltonians for these limits carrying out the Schrieffer-Wolff transformation, and show that they describe the essential feature of the screening for each case., 9 pages, 9 figures

We study low-energy transport through a quantum dot coupled to one normal and two superconducting (SC) leads in a junction of Y-shape. In this geometry a crossover between Kondo dominated and Cooper-pairing dominated states occurs by tuning the parameters such as the quantized energy level of the dot and the Josephson phase, which induces a supercurrent flowing between the two SC leads through the dot. Furthermore, Andreev scattering takes place at the interface between the dot and normal lead. The low-lying energy states of this system can be described by a local Fermi-liquid theory for interacting Bogoliubov particles. In a description based on an Anderson impurity model we calculate transport coefficients, renormalized parameters and spectral function, using Wilson's numerical renormalization group (NRG) approach, in the limit of large SC gap. Our results demonstrate how the Andreev resonance level approaches the Fermi level in the crossover region between Cooper-pairing singlet state and strong coupling situation as the impurity level or Josephson phase are varied. The strong coupling situation shows a Kondo effect with a significantly renormalized resonance width. The crossover is smeared when the coupling between the dot and normal lead is large. Furthermore, asymmetry in the Josephson junction suppresses the cancellations of the SC proximity for finite Josephson phase, and it favors the SC singlet state rather than the Kondo singlet., Comment: 12 pages, 11 figures, some brief explanations are added

We apply a recently developed 1/(N-1) expansion to the full counting statistics for the N-fold degenerate Anderson impurity model in the Kondo regime. This approach is based on the perturbation theory in the Coulomb interaction U and is different from the conventional large N theories, such as the usual 1/N expansion and non-crossing approximation. We have confirmed that the calculations carried out up to order 1/(N-1)^2 agree closely with those of the numerical renormalization group at N=4, where the degeneracy is still not so large. This ensures the applicability of our approach for N>4. We present the results of the cumulants of the probability distribution function for a nonequilibrium current through a quantum dot in the particle-hole symmetric case., 4 pages, 6 figures, typos have been corrected

We study the interplay between the Kondo and Andreev-Josephson effects in a quantum dot coupled to one normal and two superconducting (SC) leads. In the large gap limit, the low-energy states of this system can be described exactly by a local Fermi liquid for the interacting Bogoliubov particles. The phase shift and the renormalized parameters for the Bogoliubov particles vary depending on the Josephson phase between the two SC leads. We explore the precise features of a crossover that occurs between the Kondo singlet and local Cooper-pairing states as the Josephson phase varies, using the numerical renormalization group approach., 4 pages, 4 figures, contribution to SCES 2011

We investigate out of equilibrium transport through an orbital Kondo system realized in a single quantum dot, described by the multiorbital impurity Anderson model. Shot noise and current are calculated up to the third order in bias voltage in the particle-hole symmetric case, using the renormalized perturbation theory. The derived expressions are asymptotically exact at low energies. The resulting Fano factor of the backscattering current $F_b$ is expressed in terms of the Wilson ratio $R$ and the orbital degeneracy $N$ as $F_b =\frac{1 + 9(N-1)(R-1)^2}{1 + 5(N-1)(R-1)^2}$ at zero temperature. Then, for small Coulomb repulsions $U$, we calculate the Fano factor exactly up to terms of order $U^5$, and also carry out the numerical renormalization group calculation for intermediate $U$ in the case of two- and four-fold degeneracy ($N=2,\,4$). As $U$ increases, the charge fluctuation in the dot is suppressed, and the Fano factor varies rapidly from the noninteracting value $F_b=1$ to the value in the Kondo limit $F_b=\frac{N+8}{N+4}$, near the crossover region $U\sim \pi \Gamma$, with the energy scale of the hybridization $\Gamma$., Comment: 10 pages, 4 figures

The triangular triple quantum dot is an interesting system which can demonstrate various types of the Kondo effects, such as the one due to the local spin S=1 moment caused by the Nagaoka ferromagnetic mechanism and the SU(4) Kondo effect. We theoretically study the low-temperature properties and the Kondo energy scale of the triangular triple quantum dot, using the Wilson numerical renormalization group. We have explored a wide parameter region of the electron-filling and distortions which break the symmetry of an equilateral structure. Our results give a comprehensive overview of how the Kondo behavior varies in the different the regions in the wide parameter space of the triangular triple quantum dot., 18 pages; 21 figures (Figs.17 and 20 are added)

We study the transport through side-coupled double quantum dots, connected to normal and superconducting (SC) leads with a T-shape configuration. We find, using the numerical renormalization group, that the Coulomb interaction suppresses SC interference in the side dot, and enhances the conductance substantially in the Kondo regime. This behavior stands in total contrast to a wide Kondo valley seen in the normal transport. The SC proximity penetrating into the interfacial dot pushes the Kondo clouds, which screens the local moment in the side dot, towards the normal lead to make the singlet bond long. The conductance shows a peak of unitary limit as the cloud expands. Furthermore, two separate Fano structures appear in the gate-voltage dependence of the Andreev transport, where a single reduced plateau appears in the normal transport., 6 pages, 5 figures

We study the conductance through a triangular triple quantum dot, which are connected to two noninteracting leads, using the numerical renormalization group (NRG). It is found that the system shows a variety of Kondo effects depending on the filling of the triangle. The SU(4) Kondo effect occurs at half-filling, and a sharp conductance dip due to a phase lapse appears in the gate-voltage dependence. Furthermore, when four electrons occupy the three sites on average, a local S=1 moment, which is caused by the Nagaoka mechanism, is induced along the triangle. The temperature dependence of the entropy and spin susceptibility of the triangle shows that this moment is screened by the conduction electrons via two separate stages at different temperatures. The two-terminal and four-terminal conductances show a clear difference at the gate voltages, where the SU(4) or the S=1 Kondo effects occurring., Comment: 4 pages, 4 figs: typos just below (4) are corrected, results are not affected

We study transport through a quantum dot coupled to normal and superconducting leads using the numerical renormalization group method. We show that the low-energy properties of the system are described by the local Fermi liquid theory despite of the superconducting correlations penetrated into the dot due to a proximity effect. We calculate the linear conductance due to the Andreev reflection in the presence of the Coulomb interaction. It is demonstrated that the maximum structure appearing in the conductance clearly characterizes a crossover between two distinct spin-singlet ground states, i.e. the superconducting singlet state and the Kondo singlet state. It is further elucidated that the gate-voltage dependence of the conductance shows different behavior in the superconducting singlet region from that in the Kondo singlet region., 10 pages, 6 figures; a typo in eq. (B.5) corrected, which does not affect any other results of the paper

We study ground-state properties of a triangular triple quantum dot connected to two superconducting (SC) leads. In this system orbital motion along the triangular configuration causes various types of quantum phases, such as a Kondo effect with a four-fold degenerate state and the Nagaoka ferromagnetic mechanism, depending on the electron filling. The ground state also evolves as the Cooper pairs penetrate from the SC leads. We describe the phase diagram in a wide range of the parameter space, varying the gate voltage, the couplings between the dots and leads, and also the Josephson phase between the SC gaps. The results are obtained in the limit of large SC gap, carrying out exact diagonalization of an effective Hamiltonian. We also discuss a classification of the quantum states according to the fixed point of the Wilson numerical renormalization group (NRG). Furthermore, we show that the Bogoliubov zero-energy excitation determines the ground state of a $\pi$ Josephson junction at small electron fillings., Comment: 6 pages, 7 figures. Typos have been corrected

We reexamine basic aspects of a nonequilibrium steady state in the Kondo problem for a quantum dot under a bias voltage using a reduced density matrix, which is obtained in the Fock space by integrating out one of the two conduction channels. The integration has been carried out by discretizing the conduction channels preserving the two-fold degeneracy due to the left-going and right-going scattering states. The remaining subspace is described by a single-channel Anderson model, and the statistical weight is determined by the reduced density matrix. In the noninteracting case, it can be constructed as the mixed states that show a close similarity to the high-temperature distribution in equilibrium. Specifically, if the system has an inversion symmetry, the one-particle states in an energy window between the two chemical potentials \mu_R and \mu_L are occupied, or unoccupied, completely at random with an equal weight. The Coulomb interaction preserves these aspects, and the correlation functions can be expressed in a Lehmann-representation form using the mixed-state statistical weight., Comment: 8 pages, 3 figures

Asymmetry in the Josephson couplings between two superconductors through a quantum dot is studied based on a single impurity Anderson model using the numerical renormalization group (NRG). Specifically, we examine how the difference between the couplings \Gamma_L and \Gamma_R affects the ground state, which is known to show a quantum phase transition between a nonmagnetic singlet and a magnetic doublet depending on the various parameters; the Coulomb interaction U, onsite potential \epsilon_d, level width \Gamma_L, \Gamma_R, caused by the hybridization, and superconducting gaps \Delta_L and \Delta_R for the leads on the left and right. Our results show that whether the local moment is fully screened or not depends substantially on the asymmetry in the couplings \Gamma_L \neq \Gamma_R. It tends to make the singlet ground state stable, while the size and phase difference of the two superconducting gaps tend to suppress the screening. We also discuss some general symmetry properties of the system and their relation to the current conservation., Comment: 11 pages,11 figs: Equation (A9) has been corrected. It does not affect any other results of the paper

We study transport through a triangle triple quantum dot connected to two noninteracting leads using the numerical renormalization group (NRG). The triangle has a high-spin ground state of S=1 caused by a Nagaoka ferromagnetism, when it is isolated and has one extra electron introduced into a half-filling. The results show that the conduction electrons screen the local moment via two separate stages with different energy scales. The half of the S=1 is screened first by one of the channel degrees, and then at very low temperature the remaining half is fully screened to form a Kondo singlet. The transport is determined by two phase shifts for quasi-particles with even and odd parities, and then a two-terminal conductance in the series configuration is suppressed $g_{\rm series} \simeq 0$, while plateau of a four-terminal parallel conductance reaches a Unitary limit value $g_{\rm parallel} \simeq 4e^2/h$ of two conducting modes., Comment: 2pages, 2figures: fig1 is revised to show a narrow dip found in the series conductance

In this report, we describe a recent development in a Fermi liquid theory for the Kondo effect in quantum dots under a finite bias voltage $V$. Applying the microscopic theory of Yamada and Yosida to a nonequilibrium steady state, we derive the Ward identities for the Keldysh Green's function, and determine the low-energy behavior of the differential conductance $dI/dV$ exactly up to terms of order $(eV)^2$ for the symmetric Anderson model. These results are deduced from the fact that the Green's function at the impurity site is a functional of a nonequilibrium distribution $f_{\text{eff}}(\omega)$, which at $eV=0$ coincides with the Fermi function. Furthermore, we provide an alternative description of the low-energy properties using a renormalized perturbation theory (RPT). In the nonequilibrium state the unperturbed part of the RPT is determined by the renormalized free quasiparticles, the distribution function of which is given by $f_{\text{eff}}(\omega)$. The residual interaction between the quasiparticles $\widetilde{U}$, which is defined by the full vertex part at zero frequencies, is taken into account by an expansion in the power series of $\widetilde{U}$. We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime., Comment: 8 pages, to appear in a special edition of JPSJ "Kondo Effect -- 40 Years after the Discovery", typos are corrected

We propose a minimal model for the Josephson current through a quantum dot in a Kondo regime. We start with the model that consists of an Anderson impurity connected to two superconducting (SC) leads with the gaps $\Delta_{\alpha}=|\Delta_{\alpha}| e^{i \theta_{\alpha}}$, where $\alpha = L, R$ for the lead at left and right. We show that, when one of the SC gaps is much larger than the others $|\Delta_L| \gg |\Delta_R|$, the starting model can be mapped exactly onto the single-channel model, which consists of the right lead of $\Delta_R$ and the Anderson impurity with an extra onsite SC gap of $\Delta_d \equiv \Gamma_L e^{i \theta_L}$. Here $\theta_L$ and $\Gamma_L$ are defined with respect to the starting model, and $\Gamma_L$ is the level width due to the coupling with the left lead. Based on this simplified model, we study the ground-state properties for the asymmetric gap, $|\Delta_L| \gg |\Delta_R|$, using the numerical renormalization group (NRG) method. The results show that the phase difference of the SC gaps $\phi \equiv \theta_R -\theta_L$, which induces the Josephson current, disturbs the screening of the local moment to destabilize the singlet ground state typical of the Kondo system. It can also drive the quantum phase transition to a magnetic doublet ground state, and at the critical point the Josephson current shows a discontinuous change. The asymmetry of the two SC gaps causes a re-entrant magnetic phase, in which the in-gap bound state lies close to the Fermi level., Comment: 23 pages, 13 figures, typos are corrected

We study correlation effects on the transport through a quantum dot superlattice using a two-dimensional Hubbard model connected to two noninteracting leads. To calculate the zero-temperature conductance away from half-filling, we have used the non-magnetic solution of the Hartree-Fock approximation for the unperturbed Green's function, and then take into account the electron correlation beyond the mean field theory through the second-order self-energy corrections with respect to the residual interaction. When the value of the onsite potential $\epsilon_d$ or the repulsion $U$ is changed, the conductance shows a number of peaks corresponding to the resonance states passing through the Fermi energy. The behavior of the resonance states for correlated electrons can be investigated using the eigenvalue of the effective Hamiltonian for a quasi-particle of a local Fermi liquid. We calculate the eigenvalue using the second-order self-energy, and show explicitly that it can be compared with the conductance peak., Comment: 26 pages included 14 figures, to appear in J. Phys. Soc. Jpn. vol. 73 No.1

We show that the transmission probability through a small interacting region connected to noninteracting leads, can be written in terms of a retarded product of a three-point correlation function defined in the real time. Our proof is based on the Kubo formalism, and uses an Eliashberg theory for analytic properties of vertex functions. The aim of this short report is to add a new viewpoint to the transport theory described in the previous paper: A.O., J. Phys. Soc. Jpn. 70 (2001) 2666., 2 pages, 2 figures, submitted to J. Phys. Soc. Jpn

We study the high- and low-voltage properties of the out-of-equilibrium Anderson model for quantum dots, using a functional method in the Keldysh formalism. The Green's function at the impurity site can be regarded as a functional of a nonequilibrium distribution function. The dependence of the Green's function on the bias voltage V and temperature T arises through the nonequilibrium distribution function. From this behavior as a functional, it is shown that the nonequilibrium Green's function at high-voltage limit is identical to the equilibrium Green's function at high-temperature limit. This correspondence holds when the couplings of the dot and two leads, at the left and right, are equal. In the opposite limit, for small eV, the low-energy behavior of the Green's function can be described by the local Fermi-liquid theory up to terms of order $(eV)^2$. These results imply that the correlation effects due to the Coulomb interaction U can be treated adiabatically in the two limits, at high and low bias voltages., Comment: 6 pages, 4 figures: to appear in J. Phys. Soc. Jpn. 71, No.12 (2002)

We study the effects of disorders on the transport through small interacting systems based on a two-dimensional Hubbard cluster of finite size connected to two noninteracting leads. This system can be regarded as a model for the superlattice of quantum dots or atomic network of the nanometer size. We calculate the conductance at T=0 using the order $U^2$ self-energy in an electron-hole symmetric case. The results show that the conductance is ensitive to the randomness when the resonance states are situated near the Fermi energy., Comment: 2 pages, 3 figures, to be published in Physica E, proceedings Low Temperature Physics 23 (Hirosima, Japan)

We apply a theory for the transmission probability of small interacting systems, which was formulated based on the Kubo formalism in our previous study, to a series of quantum dots described by the N-impurity Anderson model. In this report, we present the transmission pobability for the system of N=2 calculated using the order $U^2$ self-energy and vertex corrections. Particularly, we examine the features in the two typical parameter regions, $t\Gamma$, where the Kondo effect or the inter-dot correlation dominates. Here, $t$ is the inter-dot transfer and $\Gamma$ is the level broadening caused by the coupling with the noninteracting leads., Comment: 2 pages, 2 figures: proccedings of LT23 (Hiroshima, August, 2002)

We study the effects of electron correlation on transport through an interacting region connected to multi-mode leads based on the perturbation expansion with respect to the inter-electron interaction. At zero temperature the conductance defined in the Kubo formalism can be written in terms of a single-particle Green's function at the Fermi energy, and it can be mapped onto a transmission coefficient of the free quasiparticles described by an effective Hamiltonian. We apply this formulation to a two-dimensional Hubbard model of finite size connected to two noninteracting leads. We calculate the conductance in the electron-hole symmetric case using the order $U^2$ self-energy. The conductance shows several maximums in the $U$ dependence in some parameter regions of $t_y/t_x$, where $t_x$ ($t_y$) is the hopping matrix element in the $x$- ($y$-) directions. This is caused by the resonance occurring in some of the subbands, and is related with the $U$ dependence of the eigenvalues of the effective Hamiltonian., 17 pages, 12 figures, to be published in J.Phys.Soc.Jpn. 71(2002)No.1

We study low-energy properties of the Anderson impurity under a finite bias voltage $V$ using the perturbation theory in $U$ of Yamada and Yosida in the nonequilibrium Keldysh diagrammatic formalism, and obtain the Ward identities for the derivative of the self-energy with respect to $V$. The self-energy is calculated exactly up to terms of order $\omega^2$, $T^2$ and $V^2$, and the coefficients are defined with respect to the equilibrium ground state. From these results, the nonlinear response of the current through the impurity has been deduced up to order $V^3$., Comment: 8 pages, 1 figure

We study effects of electron correlation on the transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasi-particles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at $\omega=0$. Specifically, we apply the method to a Hubbard chain of finite size $N$ ($=1, 2, 3, ...$), and calculate the self-energy within the second order in $U$ in the electron-hole symmetric case. When the couplings between the chain and the reservoirs on the left and right are small, the conductance for even $N$ decreases with increasing $N$ showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in the previous work. We also study the effects of the asymmetry in the two couplings. While the perfect transmission due to the Kondo resonance occurs for any odd $N$ in the symmetric coupling, the conductance for odd $N$ decreases with increasing $N$ in the case of the asymmetric coupling., Comment: 27 pages, RevTeX, 14 figures, to be published in Phys. Rev. B

Transport through a Hubbard chain of size N (=1,2,3,...) connected to reservoirs is studied at T = 0 in an electron-hole symmetric case based on the second-order perturbation theory in U. The result shows a typical even-odd property corresponding to a Kondo or Mott-Hubbard physics. In this report, specifically, we study the dependence of the conductance on the coupling between the chain and reservoirs, which was not examined in detail in our previous report [Phy. Rev. B 59, 12240 (1999)]., 2 pages, LaTeX, submitted to LT22 (Physica B)

The dc conductance through a finite Hubbard chain of size N coupled to two noninteracting leads is studied at T = 0 in an electron-hole symmetric case. Assuming that the perturbation expansion in U is valid for small N (=1,2,3,...) owing to the presence of the noninteracting leads, we obtain the self-energy at \omega = 0 analytically in the real space within the second order in U. Then, we calculate the inter-site Green's function which connects the two boundaries of the chain, G_{N1}, solving the Dyson equation. The conductance can be obtained through G_{N1}, and the result shows an oscillatory behavior as a function of N. For odd N, a perfect transmission occurs independent of U. This is due to the inversion and electron-hole symmetries, and is attributed to a Kondo resonance appearing at the Fermi level. On the other hand, for even N, the conductance is a decreasing function of N and U., Comment: 11 pages, RevTeX, 6 figures, to be published in Phys. Rev. B 59 (1999)

We study transport through a double quantum dot with interdot hopping $t$, intradot repulsion $U$ and interdot repulsion $U^{\prime}$, using the numerical renormalization group (NRG) method. At half-filling, the conductances in two-terminal series and four-terminal parallel configuration are calculated via two phase shifts for quasi-particles of double quantum dot connected to two noninteracting leads with hybridization strength $\Gamma$. For small values of $t/\Gamma$ and $U^{\prime}/U$, conductance in the two-terminal series configuration is suppressed to almost zero. In this region, plateau of conductance in the four-terminal parallel configuration appears and almost reaches a unitary limit value $4e^{2}/h$ of two conducting modes. For large values of $t/\Gamma$ or $U^{\prime}/U$, both conductances are suppressed to almost zero. The conductance in the two-terminal series configuration almost reaches a unitary limit value $2e^{2}/h$ only around cross-over regions of electron-configuration in double quantum dot. Through the behavior of the local charge and some thermodynamic quantities, we discuss the relation between transport and electron-configuration., Comment: To be published in "Journal of Physics: Condensed Matter"