Fermi liquid theory for the Anderson model out of equilibrium

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 figur

Determination of the phase shifts for interacting electrons connected to reservoirs

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_

Perturbation Study of the Conductance through an Interacting Region Connected to Multi-Mode Leads

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.Comment: 17 pages, 12 figures, to be published in J.Phys.Soc.Jpn. 71(2002)No.

NRG approach to the transport through a finite Hubbard chain connected to reservoirs

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 correcte

Fermi liquid theory for the nonequilibrium Kondo effect at low bias voltages

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 correcte

Out-of-equilibrium Anderson model at high and low bias voltages

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

Quasi-particle description for the transport through a small interacting system

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.