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

The Kondo effect is a fascinating many-body effect[1, 2], taking place in dilute magnetic alloys (MA), quantum dots (QD), and other novel quantum systems such as ultracold atomic gases [3] and quark matter [4]. It was shown in the 1970s that the low-energy behavior of the Kondo systems can be described by a quantum impurity version of the Fermi liquid (FL) theory [5, 6, 7, 8, 9]. In particular, using the numerical renormalization group (NRG) approach [10, 11, 12], Wilson et al. demonstrated that the low-lying excited states of the Kondo and the Anderson models exhibit a one-to-one correspondence with the excitations of the renormalized quasiparticles.

The quasiparticles are asymptotically free in the equilibrium ground state, where the perturbations from the environment or reservoirs, which may depend on external parameters such as frequency $\omega$, temperature $T$, bias voltages $eV$, etc., are absent. As these perturbations are adiabatically switched on, the quasiparticles capture the damping rate of order $\omega^{2}$, $T^{2}$, and $(eV)^{2}$ through the residual interactions and this significantly affects the transport properties [5, 6, 7, 8, 9, 13, 14]. When the electron-hole or time-reversal symmetry is broken by a potential or external fields, the quasiparticles also capture the energy shift of order $\omega^{2}$, $T^{2}$, and $(eV)^{2}$, i.e., the corrections in the same order as the ones due to the damping rate. The contributions of these higher-order energy shifts, however, had not been fully understood until very recently.

It has recently been clarified that these higher-order energy shifts of the quasiparticles can be described exactly in terms of the three-body correlations between impurity electrons. The proof was given in two different ways, which complement each other. One is given by Mora et al. and Filippone et al. [15, 16, 17, 18], extending Nozières’ description [5] that is based on an invariance against the “floating of Kondo resonance on the Fermi sea”. The other is based on the higher-order Fermi liquid relations [19, 20, 21], which can be derived from the Ward identities for the second derivatives of the self-energy, extending Yamada-Yosida’s field-theoretical approach [6, 7, 8, 9]. These proofs enabled one to express the next-leading order terms of the transport coefficients in terms of three-body correlation functions, and these formulations have been applied to the nonlinear currents, current noise, and thermocurrent through quantum impurities without electron-hole or time-reversal symmetry [22, 23, 24, 25, 26, 27, 28, 29].

The purpose of this paper is twofold. The first one is to extend the latest version of the FL theory for treating the thermoelectric transport coefficients of multilevel quantum dots and magnetic alloys described by the Anderson model with $N$ arbitrary impurity levels. The second one is to demonstrate how the next-leading order terms of various transport coefficients vary with the impurity level $\epsilon_{d}$ and Coulomb interaction $U$. Specifically, for the second purpose, we consider quantum dots and magnetic alloys having an SU($N$) symmetry, and calculate the three-body correlation functions for $N=4$ and $6$ using the NRG approach. There have been a number of intensive works which theoretically studied low-energy transport: the nonlinear current [30, 31, 13, 32, 33, 34, 35, 14, 36, 37, 18], nonlinear current noise [38, 39, 40, 41, 16, 42, 23], and thermoelectric transport[43, 15, 44, 21]. However, the parameter space for quantum impurities is so huge that many parts are still left unexplored. In this paper, we explore the whole region of the electron fillings, $0\leq N_{d}\leq N$, in which the occupation number $N_{d}$ of electrons in the impurity levels varies continuously across the various SU($N$) Kondo and intermediate valence regimes.

One of the advantages of the Kondo systems realized in quantum dots is that the information about the many-body quantum states can be probed in such a highly tunable way [45, 46, 47, 48]. For instance, recent experiments have succeeded in directly probing the Kondo screening cloud [49]. Furthermore, low-energy Fermi liquid behaviors have experimentally been confirmed for nonequilibrium currents [50, 51], current noise [52, 53, 54, 55, 56], and themocurrent [57, 58]. Internal degrees of freedoms of quantum impurities also bring an interesting variety to the Kondo effect. The systems having the SU($4$) symmetry have been realized, for instance, in the multiorbital semiconductor quantum dots and carbon nanotube (CNT) quantum dots [59], and have intensively been investigated theoretically [34, 60, 61, 62, 63, 64, 65, 15, 16, 66, 67, 26, 68, 22] and experimentally [69, 70, 71, 72, 73, 55]. Realization of the SU($N$) Kondo effects for various $N>2$ has also been proposed by several authors, using triple quantum dots [74, 75] and CNT [76]. In this paper, we provide a comprehensive FL view of the low-energy transport for SU(4) and SU(6) symmetric QD and MA.

Our results reveal the fact that the three-body correlations exhibit plateau structures, caused by the SU($N$) Kondo effects occurring at integer fillings $N_{d}=1$, $2$, $\ldots,N-1$. We also calculate the Lorenz number $L=\kappa/(T\sigma)$ for QD and MA, defined as the ratio of the thermal conductivity $\kappa$ to the electrical conductivity $\sigma$, and show how it deviates from the universal Wiedemann-Franz value $\pi^{2}/(3e^{2})$ as the temperature increases from $T=0$. Furthermore, we demonstrate how the three-body correlations affect the order $T^{2}$ electrical conductivity, the order $T^{3}$ thermal conductivity, and the order $(eV)^{3}$ nonlinear current and current noise. At quarter $N_{d}/N=1/4$ and three-quarters $N_{d}/N=3/4$ fillings, the SU($N$) Kondo effects occur for $N\equiv 0$ (mod $4$) while the intermediate valence fluctuations occur for $N\equiv 2$ (mod $4$). We show that this dependence on (mod $4$) causes a pronounced difference appearing in the peak structures of linear noise for $N=4$ and $6$.

This paper is organized as follows. In Sec. II, we describe the multilevel Anderson impurity model for quantum dots and magnetic alloys. Sections III and IV are devoted to the FL descriptions for the next-leading order terms of electrical and thermoelectric transport coefficients for quantum dots, applicable to arbitrary $N$ impurity-level structures. In Sec. V, the low-energy transport formulas for the SU($N$) quantum dots are described in terms of the five FL parameters. We present the NRG results for the three-body correlation functions in Sec. VI. The results for nonlinear current, current noise, and thermoelectric transport through quantum dots are discussed in Secs. VII and VIII. Section IX is devoted to the FL description of the three-body correlations in thermoelectric transport of dilute magnetic alloys. In Sec. X, we discuss the results for the electrical and thermal resistivities of the SU(4) and SU(6) magnetic alloys. Summary is given in Sec. XI. In Appendix, we provide details of the derivations for the transport formulas and additional NRG results for the FL parameters in the SU($N$) cases for $N=2$, $4$, and $6$, for comparison.

We consider a multi-orbital Anderson impurity coupled to two noninteracting leads on the left ($L$) and right ($R$): $\mathcal{H}=\mathcal{H}_{d}+\mathcal{H}_{c}+\mathcal{H}_{\mathrm{T}}$,

Here, the level index runs over $\sigma=1,2,\ldots,N$. The inter-electron interaction $U_{\sigma\sigma^{\prime}}$ generally depends on $\sigma$ and $\sigma^{\prime}$, with the requirements $U_{\sigma^{\prime}\sigma}=U_{\sigma\sigma^{\prime}}$ for $\sigma^{\prime}\neq\sigma$. For $N=2$, it describes the usual single-orbital Anderson model for spin $1/2$ fermions. The operator $d^{{\dagger}}_{\sigma}$ creates an impurity electron with spin $\sigma$ in the impurity level of energy $\epsilon_{d\sigma}$, and $n_{d\sigma}\equiv d^{{\dagger}}_{\sigma}d_{\sigma}$. Conduction electrons in the two lead at $j=L$ and $R$ obey the anti-commutation relation $\{c^{\phantom{\dagger}}_{\epsilon j\sigma},c^{\dagger}_{\epsilon^{\prime}j^{\prime}\sigma^{\prime}}\}=\delta_{jj^{\prime}}\,\delta_{\sigma\sigma^{\prime}}\delta(\epsilon-\epsilon^{\prime})$. The linear combination of the conduction electrons, $\psi_{j,\sigma}\equiv\int_{-D}^{D}d\epsilon\sqrt{\rho_{c}}\,c^{\phantom{\dagger}}_{\epsilon j\sigma}$ with $\rho_{c}=1/(2D)$, couples to the impurity level. The bare level width due to the tunnel couplings is given by $\Delta\equiv\Gamma_{L}+\Gamma_{R}$ with $\Gamma_{j}=\pi\rho_{c}v_{j}^{2}$. We consider the parameter region where the half band-width $D$ is much grater than the other energy scales, $D\gg\max(U_{\sigma\sigma^{\prime}},\Delta,|\epsilon_{d\sigma}|,T,|eV|)$. We use a unit $k_{B}=1$ throughout this paper.

Low-energy properties of the Anderson model can be described in terms of a set of Fermi liquid parameters defined with respect to the equilibrium ground state, i.e., the phase shift $\delta_{\sigma}$, the linear susceptibilities $\chi_{\sigma\sigma^{\prime}}$ and the three-body correlation functions $\chi_{\sigma\sigma^{\prime}\sigma^{\prime\prime}}^{[3]}$ between impurity electrons; see Appendix A for details. The phase shift is a primary parameter that reflects the charge distribution of impurity levels, through the Friedel sum rule: $\langle n_{d\sigma}\rangle\xrightarrow{\,T\to 0\,}\delta_{\sigma}/\pi$. In this paper, we explore the low-energy transport of quantum dots and magnetic alloys in the whole region of the electron fillings,

The current conservation, which follows from the Heisenberg equation of motion for the occupation number $n_{d\sigma}$,

also plays an essential role in the Fermi liquid description, through the Ward identities. Here, $\widehat{J}_{L,\sigma}$ represents the current flowing from the left lead to the dot, and $\widehat{J}_{R,\sigma}$ the current from the dot to the right lead:

In the next section, we will give a brief overview of the low-energy expansion formulas for nonlinear current and current noise, obtained previously in Refs.  27, 29, to show how the formulas can be expressed in terms of the FL parameters, including the three-body correlations. This is for comparison with the formulas for thermal electric transport coefficients, which we extend to multilevel quantum dots and magnetic alloys in this paper. Specifically, we will describe the derivation of the formulas for multilevel quantum dots and magnetic alloys in Sec. IV and Appendix G, respectively.

We consider the nonequilibrium steady state under a finite bias voltage $eV\equiv\mu_{L}-\mu_{R}$, applied between the two leads by setting the chemical potentials of the left and right leads to be $\mu_{L}$ and $\mu_{R}$, respectively. The retarded Green’s function plays a central role in the microscopic description of the Fermi liquid transport:

Here, $\langle\cdots\rangle_{V}$ represents a nonequilibrium steady-state average taken with the statistical density matrix, which is constructed at finite bias voltages $eV$ and temperatures $T$, using the Keldysh formalism [33, 13, 77].

In this section, we discuss thermoelectric transport through multilevel quantum dots in the linear-response regime, the low-energy behaviors of which can be deduced from the asymptotic form of the spectral function $A_{\sigma}(\omega)$ given in Appendix B.

The linear conductance $g$, thermopower $\mathcal{S}_{\mathrm{QD}}$ and thermal conductance $\kappa_{\mathrm{QD}}$ of a quantum dot can be expressed in the form [79, 80, 81, 22, 82]:

Here, $\mathcal{L}_{n,\sigma}^{\mathrm{QD}}$ for $n=0,1$, and $2$ is defined at $eV=0$ with respect to the thermal equilibrium, as

with $\mathcal{T}_{\sigma}(\omega)$, the transmission probability defined in Eq. (11). Note that thermal conductance $\kappa_{\mathrm{QD}}$ is the linear-response coefficient of the heat current $J_{Q}=\kappa_{\mathrm{QD}}\,\delta T$, flowing from the high-temperature side toward the low-temperature side, with $\delta T$ the temperature difference between the two sides.

The thermoelectric coefficients $\mathcal{L}_{n,\sigma}^{\mathrm{QD}}$ can be calculated, by substituting the low-energy asymptotic form of $A_{\sigma}(\omega)$, given in Eq. (111) into $\mathcal{T}_{\sigma}(\omega)$. At low temperatures, the component $\mathcal{L}_{0,\sigma}^{\mathrm{QD}}$ for $n=0$, which determines $g$, is given by

Here, $c_{T,\sigma}$ is the coefficient that we have already described in Eq. (14). The next component, $\mathcal{L}_{1,\sigma}^{\mathrm{QD}}$ for $n=1$, takes the form

Here, $\rho_{d\sigma}^{\prime}$ is the derivative of the density of states with respect to the frequency $\omega$, which can also be written in terms of the phase shift $\delta_{\sigma}$ and diagonal linear susceptibility $\chi_{\sigma\sigma}$, as shown in Eq. (101). Thus, the leading-order term of thermopower for quantum dots is given by

The thermal conductance $\kappa_{\mathrm{QD}}$ depends on the other component $\mathcal{L}_{2,\sigma}^{\mathrm{QD}}$ for $n=2$, the low-energy asymptotic form of which is given by

Therefore, the thermal conductance can be calculated up to terms of order $T^{3}$, by substituting these asymptotic forms into Eq. (22):

Here, the arithmetic mean (AM) of the phase shifts is defined by

Furthermore, the Lorenz number $L_{\mathrm{QD}}\equiv\kappa_{\mathrm{QD}}/(g\,T)$ can be calculated up to terms of order $T^{2}$:

The Wiedemann-Franz law holds between the leading-order terms of the linear conductance $g$ and the thermal conductance $\kappa_{\mathrm{QD}}$, such that the Lorenz number approaches the universal value $L_{\mathrm{QD}}\xrightarrow{T\to 0}\pi^{2}/(3e^{2})$ in the low-temperature limit. The Lorenz number deviates from this universal value as temperature increases, exhibiting the $T^{2}$ dependence.

This Hamiltonian $\mathcal{H}$, defined in Sec. II, has an SU($N$) symmetry in the case at which the impurity states become degenerate $\epsilon_{d\sigma}\equiv\epsilon_{d}$ for all $\sigma$ and the Coulomb interaction is isotropic $U_{\sigma\sigma^{\prime}}\equiv U$ for all $\sigma$ and $\sigma^{\prime}$.

In the atomic limit $v_{j}\to 0$ of the SU($N$) symmetric case, the total number of impurity electrons $N_{d}$ takes an integer value $M$ and exhibits the Coulomb-stair case behavior as a function of $\epsilon_{d}$. It consists of a series of plateaus of the width $U$ and the height $N_{d}=M$ for $M=1$, $2$, $\ldots$, $N-1$, emerging at $-MU<\epsilon_{d}<-(M-1)U$ around the midpoint $\epsilon_{d,M}^{\mathrm{mid}}\equiv-(M-\frac{1}{2})U$. We will use the following notation for the shifted impurity level $\xi_{d}$, which includes the Hartree-Fock energy shift defined with respect to the half filling in such a way that

The system has an electron-hole symmetry at $\xi_{d}=0$. When the tunneling couplings are switched on, the stair-case structure emerges for strong interactions $U\gg\Delta$.

In the SU($N$) symmetric case, the linear susceptibilities have two linearly independent components, i.e., the diagonal one $\chi_{\sigma\sigma}$ and the off-diagonal one $\chi_{\sigma\sigma^{\prime}}$ for $\sigma\neq\sigma^{\prime}.$ These two components determine the essential properties of quasiparticles:

Here, $T^{\ast}$ is a characteristic energy scale of the SU($N$) Fermi liquid, by which the $T$-linear specific heat of impurity electrons can be expressed in the form $\mathcal{C}_{\mathrm{imp}}=\frac{N\pi^{2}}{12}T/T^{*}$ [6, 8, 9]. The Wilson ratio $R$ corresponds to a dimensionless residual interaction between quasiparticles [83]; we will also use the rescaled ratio,

which is bounded in the range $0\leq\widetilde{K}\leq 1$.