Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2404.05947 · Physical review. B./Physical review. B

Thermoelectric transport and current noise through a multilevel Anderson impurity: Three-body Fermi liquid corrections in quantum dots and magnetic alloys

Yoshimichi Teratani, Kazuhiko Tsutsumi, Kaiji Motoyama, Rui Sakano, Akira Oguri

Published Jul 23, 2024• 3 citations
arXivHTMLPDF

Abstract

We present a comprehensive Fermi liquid description for thermoelectric transport and current noise, applicable to multilevel quantum dots (QD) and magnetic alloys (MA) without electron-hole or time-reversal symmetry. Our formulation for the low-energy transport is based on an Anderson model with $N$ discrete impurity levels, and is asymptotically exact at low energies, up to the next-leading order terms in power expansions with respect to temperature $T$ and bias voltage $eV$. The expansion coefficients can be expressed in terms of the Fermi liquid parameters, which include the three-body correlation functions defined with respect to the equilibrium ground state in addition to the linear susceptibilities and the occupation number ${N}_{d}^{}$ of impurity electrons. We apply this formulation to the $\mathrm{SU}(N)$ symmetric QD and MA, and calculate the correlation functions for $N=4$ and 6, using numerical renormalization group approach. The three-body correlations are shown to be determined by a single parameter over a wide range of electron fillings $1\ensuremath{\lesssim}{N}_{d}^{}\ensuremath{\lesssim}N\ensuremath{-}1$ for strong Coulomb interactions $U$, and they also exhibit the plateau structures due to the $\mathrm{SU}(N)$ Kondo effects at integer values of ${N}_{d}^{}$. We find that the Lorenz number $L=\ensuremath{\kappa}/(T\ensuremath{\sigma})$ for QD and MA, defined as the ratio of the thermal conductivity $\ensuremath{\kappa}$ to the electrical conductivity $\ensuremath{\sigma}$, deviates from the universal Wiedemann-Franz value ${\ensuremath{\pi}}^{2}/(3{e}^{2})$ as the temperature increases from $T=0$, showing the ${T}^{2}$ dependence, the coefficient for which depends on the three-body correlations away from half filling. Furthermore, we find that the current noise for the SU(4) quantum dots and that for SU(6) show a pronounced difference at the quarter ${N}_{d}^{}/N=1/4$ and $3/4$ fillings. In particular, the linear noise for $N=4$ exhibits a flat peak while the peak for $N=6$ shows a round shape, reflecting the fact that, at these filling points, the $\mathrm{SU}(N)$ Kondo effects occur for $N\ensuremath{\equiv}0$ (mod 4), whereas the intermediate-valence fluctuations occur for $N\ensuremath{\equiv}2$ (mod 4). We also demonstrate the role of three-body correlations on the nonlinear current noise and the other transport coefficients.

DOI
10.1103/physrevb.110.035308

Related papers

Three-body Fermi liquid corrections for an infinite- U SU( N ) Anderson impurity model

Kaiji Motoyama, Yoshimichi Teratani, Kazuhiko Tsutsumi +4 more

We study the three-body Fermi liquid effects in the SU($N$) Anderson impurity model within the strong interaction limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, where the occupation number ${N}_{d}^{}$ of the impurity levels varies over the range of $0<{N}_{d}^{}<1$. The three-body correlations arise among the impurity electrons when the electron-hole symmetry, the time-reversal symmetry, or both are broken by external fields or potentials. They play a key role in low-energy transport, particularly in the next-to-leading-order terms of power expansion with respect to temperature $T$ and bias voltage $eV$. Using the numerical renormalization group approach, we calculate the differential conductance and nonlinear current noise through quantum dots, as well as the thermal conductivity of both quantum dots and magnetic alloys. Specifically, we focus on the SU(2) and SU(4) cases, demonstrating how the three-body correlations evolve in the strong interaction limit across the $1/N$-filling Kondo regime at ${N}_{d}^{}\ensuremath{\simeq}1.0$ and the valence fluctuation regime where the occupation number ${N}_{d}^{}$ varies rapidly with the position ${\ensuremath{\epsilon}}_{d}^{}$ of the impurity levels. We also show that the three-body correlations strongly couple with asymmetries in the tunnel couplings between quantum dots and reservoirs. For $N=4$, this coupling significantly affects the order ${(eV)}^{3}$ nonlinear current through the quarter-filling Kondo state, where the electron-hole symmetry is broken. In particular, in the $U\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the expansion coefficients exhibit clear plateau structures associated with the Kondo effect at ${N}_{d}^{}\ensuremath{\simeq}1.0$. Our formulation and analysis pave the way for deducing the three-body correlations from the low-energy experiments on the next-to-leading-order transport.

arXiv:2411.10778•Physical review. B./Physical review. B•Jun 4, 2025•1 citation

Fermi Liquid Theory for Nonlinear Transport through a Multilevel Anderson Impurity

Yoshimichi Teratani, Rui Sakano, Akira Oguri

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≲N_{d}≲N-1. It significantly affects the current noise for N>2 and the behavior of noise in magnetic fields.

arXiv:2001.08348•Physical Review Letters•Nov 17, 2020•18 citations

Higher-Order Fermi-Liquid Corrections for an Anderson Impurity Away from Half Filling

Akira Oguri, Alex C. Hewson

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 and Yosida. It includes, partly, a microscopic clarification of the related achievements based on Nozi\`eres' phenomenological description: Filippone, Moca, von Delft, and Mora [Phys. Rev. B 95, 165404 (2017)]. In our formulation, the Fermi-liquid parameters such as the quasiparticle energy, damping, and transport coefficients are related to each other through the total vertex ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}};{\ensuremath{\sigma}}^{\ensuremath{'}}\ensuremath{\sigma}}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}};{\ensuremath{\omega}}^{\ensuremath{'}},\ensuremath{\omega})$, which may be regarded as a generalized Landau quasiparticle interaction. We obtain exactly this function up to linear order with respect to the frequencies $\ensuremath{\omega}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}$ using the antisymmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half filling through the nonlinear susceptibilities. We also apply the formulation to nonequilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field.

arXiv:1709.06385•Physical Review Letters•Mar 23, 2018•32 citations

Role of bias and tunneling asymmetries in nonlinear Fermi-liquid transport through an SU(N) quantum dot

Kazuhiko Tsutsumi, Yoshimichi Teratani, Kaiji Motoyama +2 more

We study how bias and tunneling asymmetries affect nonlinear current through a quantum dot with $N$ discrete levels in the Fermi liquid regime, using an exact low-energy expansion of the current derived up to terms of order ${V}^{3}$ with respect to the bias voltage. The expansion coefficients are described in terms of the phase shift, the linear susceptibilities, and the three-body correlation functions, defined with respect to the equilibrium ground state of the Anderson impurity model. In particular, the three-body correlations play an essential role in the order ${V}^{3}$ term, and their coupling to the nonlinear current depends crucially on the bias and tunnel asymmetries. The number of independent components of the three-body correlation functions increases with $N$ the internal degrees of the quantum dots, and it gives a variety in the low-energy transport. We calculate the correlation functions over a wide range of electron fillings of the Anderson impurity model with the $\mathrm{SU}(N)$ internal symmetry, using the numerical renormalization group. We find that the order ${V}^{3}$ nonlinear current through the $\mathrm{SU}(N)$ Kondo state, which occurs at electron fillings of 1 and $N\ensuremath{-}1$ for strong Coulomb interactions, significantly varies with the three-body contributions as tunnel asymmetries increase. Furthermore, in the valence fluctuation regime toward the empty or fully occupied impurity state, a sharp peak emerges in the coefficient of ${V}^{3}$ current in the case at which bias and tunneling asymmetries cooperatively enhance the charge transfer from one of the electrodes.

arXiv:2305.01162•Physical review. B./Physical review. B•Jul 6, 2023•4 citations

Fermi-liquid theory for the single-impurity Anderson model

Christophe Mora, Cătălin Paşcu Moca, Jan von Delft +1 more

We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of ${T}^{2}, {V}^{2}$, and ${B}^{2}$ are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.

arXiv:1409.3451•Physical Review B•Aug 10, 2015•54 citations

Scaling of conductance through quantum dots with magnetic field

Ignacio J. Hamad, C. J. Gazza, J. A. Andrade +3 more

Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where $U$ is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy ${E}_{d}$, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and $\mathrm{NRG}+\mathrm{RPT}$ are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once ${E}_{d}$ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in $U$ agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of $U$. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.

arXiv:1509.05948•Physical Review B•Nov 9, 2015•10 citations
We present a comprehensive Fermi liquid description for thermoelectric transport and current noise, applicable to multilevel quantum dots (QD) and magnetic alloys (MA) without electron-hole or time-reversal symmetry. Our formulation for the low-energy transport is based on an Anderson model with NNN discrete impurity levels, and is asymptotically exact at low energies, up to the next-leading order terms in power expansions with respect to temperature TTT and bias voltage eVeVeV. The expansion coefficients can be expressed in terms of the Fermi liquid parameters, which include the three-body correlation functions defined with respect to the equilibrium ground state in addition to the linear susceptibilities and the occupation number Nd{N}_{d}^{}Nd​ of impurity electrons. We apply this formulation to the SU(N)\mathrm{SU}(N)SU(N) symmetric QD and MA, and calculate the correlation functions for N=4N=4N=4 and 6, using numerical renormalization group approach. The three-body correlations are shown to be determined by a single parameter over a wide range of electron fillings 1\ensuremath≲Nd\ensuremath≲N\ensuremath−11\ensuremath{\lesssim}{N}_{d}^{}\ensuremath{\lesssim}N\ensuremath{-}11\ensuremath≲Nd​\ensuremath≲N\ensuremath−1 for strong Coulomb interactions UUU, and they also exhibit the plateau structures due to the SU(N)\mathrm{SU}(N)SU(N) Kondo effects at integer values of Nd{N}_{d}^{}Nd​. We find that the Lorenz number L=\ensuremathκ/(T\ensuremathσ)L=\ensuremath{\kappa}/(T\ensuremath{\sigma})L=\ensuremathκ/(T\ensuremathσ) for QD and MA, defined as the ratio of the thermal conductivity \ensuremathκ\ensuremath{\kappa}\ensuremathκ to the electrical conductivity \ensuremathσ\ensuremath{\sigma}\ensuremathσ, deviates from the universal Wiedemann-Franz value \ensuremathπ2/(3e2){\ensuremath{\pi}}^{2}/(3{e}^{2})\ensuremathπ2/(3e2) as the temperature increases from T=0T=0T=0, showing the T2{T}^{2}T2 dependence, the coefficient for which depends on the three-body correlations away from half filling. Furthermore, we find that the current noise for the SU(4) quantum dots and that for SU(6) show a pronounced difference at the quarter Nd/N=1/4{N}_{d}^{}/N=1/4Nd​/N=1/4 and 3/43/43/4 fillings. In particular, the linear noise for N=4N=4N=4 exhibits a flat peak while the peak for N=6N=6N=6 shows a round shape, reflecting the fact that, at these filling points, the SU(N)\mathrm{SU}(N)SU(N) Kondo effects occur for N\ensuremath≡0N\ensuremath{\equiv}0N\ensuremath≡0 (mod 4), whereas the intermediate-valence fluctuations occur for N\ensuremath≡2N\ensuremath{\equiv}2N\ensuremath≡2 (mod 4). We also demonstrate the role of three-body correlations on the nonlinear current noise and the other transport coefficients.
We study the three-body Fermi liquid effects in the SU(NNN) Anderson impurity model within the strong interaction limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, where the occupation number Nd{N}_{d}^{}Nd​ of the impurity levels varies over the range of 0<Nd<10<{N}_{d}^{}<10<Nd​<1. The three-body correlations arise among the impurity electrons when the electron-hole symmetry, the time-reversal symmetry, or both are broken by external fields or potentials. They play a key role in low-energy transport, particularly in the next-to-leading-order terms of power expansion with respect to temperature TTT and bias voltage eVeVeV. Using the numerical renormalization group approach, we calculate the differential conductance and nonlinear current noise through quantum dots, as well as the thermal conductivity of both quantum dots and magnetic alloys. Specifically, we focus on the SU(2) and SU(4) cases, demonstrating how the three-body correlations evolve in the strong interaction limit across the 1/N1/N1/N-filling Kondo regime at ${N}_{d}^{}\ensuremath{\simeq}1.0$ and the valence fluctuation regime where the occupation number Nd{N}_{d}^{}Nd​ varies rapidly with the position ${\ensuremath{\epsilon}}_{d}^{}$ of the impurity levels. We also show that the three-body correlations strongly couple with asymmetries in the tunnel couplings between quantum dots and reservoirs. For N=4N=4N=4, this coupling significantly affects the order (eV)3{(eV)}^{3}(eV)3 nonlinear current through the quarter-filling Kondo state, where the electron-hole symmetry is broken. In particular, in the $U\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the expansion coefficients exhibit clear plateau structures associated with the Kondo effect at ${N}_{d}^{}\ensuremath{\simeq}1.0$. Our formulation and analysis pave the way for deducing the three-body correlations from the low-energy experiments on the next-to-leading-order transport.
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≲N_{d}≲N-1. It significantly affects the current noise for N>2 and the behavior of noise in magnetic fields.
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 and Yosida. It includes, partly, a microscopic clarification of the related achievements based on Nozi\`eres' phenomenological description: Filippone, Moca, von Delft, and Mora [Phys. Rev. B 95, 165404 (2017)]. In our formulation, the Fermi-liquid parameters such as the quasiparticle energy, damping, and transport coefficients are related to each other through the total vertex ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}};{\ensuremath{\sigma}}^{\ensuremath{'}}\ensuremath{\sigma}}(\ensuremath{\omega},{\ensuremath{\omega}}^{\ensuremath{'}};{\ensuremath{\omega}}^{\ensuremath{'}},\ensuremath{\omega})$, which may be regarded as a generalized Landau quasiparticle interaction. We obtain exactly this function up to linear order with respect to the frequencies $\ensuremath{\omega}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}$ using the antisymmetry and analytic properties. The coefficients acquire additional contributions of three-body fluctuations away from half filling through the nonlinear susceptibilities. We also apply the formulation to nonequilibrium transport through a quantum dot, and clarify how the zero-bias peak evolves in a magnetic field.
We study how bias and tunneling asymmetries affect nonlinear current through a quantum dot with NNN discrete levels in the Fermi liquid regime, using an exact low-energy expansion of the current derived up to terms of order V3{V}^{3}V3 with respect to the bias voltage. The expansion coefficients are described in terms of the phase shift, the linear susceptibilities, and the three-body correlation functions, defined with respect to the equilibrium ground state of the Anderson impurity model. In particular, the three-body correlations play an essential role in the order V3{V}^{3}V3 term, and their coupling to the nonlinear current depends crucially on the bias and tunnel asymmetries. The number of independent components of the three-body correlation functions increases with NNN the internal degrees of the quantum dots, and it gives a variety in the low-energy transport. We calculate the correlation functions over a wide range of electron fillings of the Anderson impurity model with the SU(N)\mathrm{SU}(N)SU(N) internal symmetry, using the numerical renormalization group. We find that the order V3{V}^{3}V3 nonlinear current through the SU(N)\mathrm{SU}(N)SU(N) Kondo state, which occurs at electron fillings of 1 and $N\ensuremath{-}1$ for strong Coulomb interactions, significantly varies with the three-body contributions as tunnel asymmetries increase. Furthermore, in the valence fluctuation regime toward the empty or fully occupied impurity state, a sharp peak emerges in the coefficient of V3{V}^{3}V3 current in the case at which bias and tunneling asymmetries cooperatively enhance the charge transfer from one of the electrodes.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
Using different techniques, and Fermi-liquid relationships, we calculate the variation with the applied magnetic field (up to second order) of the zero-temperature equilibrium conductance through a quantum dot described by the impurity Anderson model. We focus on the strong-coupling limit $U\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$, where UUU is the Coulomb repulsion and $\mathrm{\ensuremath{\Delta}}$ is half the resonant-level width, and consider several values of the dot level energy Ed{E}_{d}Ed​, ranging from the Kondo regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\gg}\mathrm{\ensuremath{\Delta}}$ to the intermediate-valence regime ${\ensuremath{\epsilon}}_{F}\ensuremath{-}{E}_{d}\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. We have mainly used the density-matrix renormalization group (DMRG) and the numerical renormalization group (NRG) combined with renormalized perturbation theory (RPT). Results for the dot occupancy and magnetic susceptibility from the DMRG and NRG+RPT\mathrm{NRG}+\mathrm{RPT}NRG+RPT are compared with the corresponding Bethe ansatz results for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, showing an excellent agreement once Ed{E}_{d}Ed​ is renormalized by a constant Haldane shift. For $U<3\mathrm{\ensuremath{\Delta}}$ a simple perturbative approach in UUU agrees very well with the other methods. The conductance decreases with the applied magnetic field for dot occupancies ${n}_{d}\ensuremath{\sim}1$ and increases for ${n}_{d}\ensuremath{\sim}0.5$ or ${n}_{d}\ensuremath{\sim}1.5$ regardless of the value of UUU. We also relate the energy scale for the magnetic-field dependence of the conductance with the width of the low-energy peak in the spectral density of the dot.