arXiv:1509.05948 · Physical Review B

Scaling of conductance through quantum dots with magnetic field

Ignacio J. Hamad, C. J. Gazza, J. A. Andrade, Armando A. Aligia, Pablo S. Cornaglia, P. Roura-Bas

Published Nov 9, 2015• 10 citations

Abstract

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.

DOI: 10.1103/physrevb.92.195113

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arXiv:1509.05948 · Physical Review B

Scaling of conductance through quantum dots with magnetic field

Ignacio J. Hamad, C. J. Gazza, J. A. Andrade, Armando A. Aligia, Pablo S. Cornaglia, P. Roura-Bas

Published Nov 9, 2015• 10 citations

arXiv HTML PDF

Abstract

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

{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.

DOI: 10.1103/physrevb.92.195113

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Delia Maria Fugger, Antonius Dorda, Frauke Schwarz +2 more

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arXiv:1708.05750New Journal of PhysicsDec 7, 201726 citations

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 +2 more

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.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.

arXiv:2404.05947Physical review. B./Physical review. BJul 23, 20243 citations

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}

\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

{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.

T

and magnetic field

B

of the conductance through a quantum dot described by the impurity Anderson model, for small

T

and

B

. We show that the equation used to extract the dependence on

B

from NRG results is incorrect out of the particle-hole symmetric case. As a consequence, in the Kondo regime, the correct NRG results have a weaker dependence on

B

and the disagreement between both approaches increases.

{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.

{c}_{A}

, describing its leading $\ensuremath{\sim}{\ensuremath{\varepsilon}}^{2}$ term, and the transport coefficients

{c}_{V}

and

{c}_{T}

, describing the leading $\ensuremath{\sim}{V}^{2}$ and $\ensuremath{\sim}{T}^{2}$ terms in the nonlinear differential conductance. A sign change in

{c}_{A}

{c}_{V}

is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. We find that the fields

{B}_{A}

{B}_{T}

, and

{B}_{V}

at which

{c}_{A}

{c}_{T}

, and

{c}_{V}

change sign, respectively, are all of order

{T}_{K}

, as expected, with

{B}_{A}={B}_{T}={B}_{V}=0.75073{T}_{K}

in the Kondo limit.

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.