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arXiv:cond-mat/0504771 · Journal of the Physical Society of Japan

Determination of the Phase Shifts for Interacting Electrons Connected to Reservoirs

Akira Oguri, Yunori Nisikawa, Alex C. Hewson

Published Sep 15, 2005• 30 citations
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Abstract

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.

DOI
10.1143/jpsj.74.2554

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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.
The ground-state properties of quartet quantum-dot arrays are studied using the numerical renormalization group (NRG) method with a four-site Hubbard model connected to two noninteracting leads. Specifically, we calculate the conductance and local charge in the dots from the many-body phase shifts, which can be deduced from the fixed-point eigenvalues of NRG. As a function of the on-site energy ϵd{ϵ}_{d}ϵd​ which corresponds to the gate voltage, the conductance shows alternatively wide peaks and valleys. Simultaneously, the total number of electrons Nel{N}_{\mathrm{el}}Nel​ in the four dots shows a quantized staircase behavior due to a large Coulomb interaction UUU. The conductance plateaus of the unitary limit emerging for odd Nel{N}_{\mathrm{el}}Nel​ are caused by the Kondo effect. The valleys of the conductance emerge for even Nel{N}_{\mathrm{el}}Nel​, and their width becomes substantially large at half-filling. It can be regarded as a kind of the Mott-Hubbard insulating behavior manifesting in a small system. These structures of the plateaus and valleys become weak for large values of the hybridization strength $\ensuremath{\Gamma}$ between the chain and leads. We also discuss the parallel conductance for the array connected to four leads.
We study the ground‐state properties of a triple quantum dot based on a three‐site Hubbard model connected to two non‐interacting leads. Using the numerical renormalization group (NRG), the many‐body phase shifts and dc conductance are calculated away from half‐filling as a function of the onsite energy εd, which corresponds to the gate voltage. The results for conductance g show the typical Kondo plateaus of the Unitary limit g ≈ 2e2/h at some finite ranges of the gate‐voltage corresponding to odd Nel (≈ 1, 3 and 5), where Nel is the total number of electrons in the triple dot. The conductance shows broad minima for even Nel(≈ 2 and 4). The local charge Nel shows a staircase behavior as a function of εd.
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.
We study transport through a double quantum dot with interdot hopping ttt, intradot repulsion UUU and interdot repulsion U′U^{\prime}U′, 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 ΓΓΓ. For small values of t/Γt/Γt/Γ and U′/UU^{\prime}/UU′/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 4e2/h4e^{2}/h4e2/h of two conducting modes. For large values of t/Γt/Γt/Γ or U′/UU^{\prime}/UU′/U, both conductances are suppressed to almost zero. The conductance in the two-terminal series configuration almost reaches a unitary limit value 2e2/h2e^{2}/h2e2/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.
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable ``zero-temperature'' fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.
We study the low-temperature transport properties of systems of parallel quantum dots described by the NNN-impurity Anderson model. We calculate the quasiparticle scattering phase shifts, spectral functions, and correlations as a function of the gate voltage for NNN up to 5. For any NNN, the conductance at the particle-hole symmetric point is unitary. For $N\ensuremath{\ge}2$, a transition from ferromagnetic to antiferromagnetic impurity spin correlations occurs at some gate voltage. For $N\ensuremath{\ge}3$, there is an additional transition due to an abrupt change in average impurity occupancy. For odd NNN, the conductance is discontinuous through both quantum phase transitions, while for even NNN only the magnetic transition affects the conductance. Similar effects should be experimentally observable in systems of quantum dots with ferromagnetic conduction-band-mediated interdot exchange interactions.