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

In this paper, we consider the SU($N$) symmetric case, at which the impurity level has the $N$-fold degeneracy: $\epsilon_{d\sigma}\equiv\epsilon_{d}$ for $\sigma=1,2,\ldots,N$. For convenience, we use a shifted impurity energy level, defined by $\xi_{d}\equiv\epsilon_{d}+(N-1)U/2$ in the following. Note that the system additionally has an electron-hole symmetry at $\xi_{d}=0$.

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

$\displaystyle T^{\ast}\,\equiv\,\frac{1}{4\chi_{\sigma\sigma}}\,,\qquad\quad R\,\equiv\,1-\frac{\chi_{\sigma\sigma^{\prime}}}{\chi_{\sigma\sigma}}\,.$

(17)

Here, $T^{\ast}$ is a characteristic energy scale of the SU($N$) Fermi liquid, for instance, the $T$-linear specific heat of impurity electrons is scaled in the form $\mathcal{C}_{\mathrm{imp}}=\frac{N\pi^{2}}{12}T/T^{*}$. The Wilson ratio $R$ corresponds to a dimensionless residual interaction between quasiparticles [61]: we will use the following rescaled Wilson ratio $\widetilde{K}$ which is bounded in the range $0\leq\widetilde{K}\leq 1$,

$\displaystyle\widetilde{K}\,\equiv\,(N-1)(R-1)\,.$

(18)

The differential conductance for SU($N$) symmetric quantum dots can be expressed in the following form, scaling the bias voltage $eV$ by $T^{\ast}$,

	$\displaystyle\frac{dI}{dV}\,$	$\displaystyle=\ Ng_{0}\left[\,\sin^{2}\delta\,+\,C_{V}^{(2)}\,\frac{eV}{T^{\ast}}\,-\,C_{V}^{(3)}\left(\frac{eV}{T^{\ast}}\right)^{2}\,+\,\cdots\right],$		(19)
	$\displaystyle g_{0}\,$	$\displaystyle\equiv\ \frac{e^{2}}{h}\,\frac{4\Gamma_{L}\Gamma_{R}}{(\Gamma_{L}+\Gamma_{R})^{2}}\ =\,\frac{e^{2}}{h}\bigl{(}1-\gamma_{\mathrm{dif}}^{2}\bigr{)}\,.$		(20)

The dimensionless coefficients $C_{V}^{(2)}$ and $C_{V}^{(3)}$, can be deduced from the general formulas given in Eqs. (55) and (56), taking into account the SU($N$) symmetry:

$\displaystyle C_{V}^{(2)}\,=$

$\displaystyle\ \frac{\pi}{4}\left[\,\alpha_{\mathrm{dif}}\left(1-\widetilde{K}\right)\,-\,\gamma_{\mathrm{dif}}\,\widetilde{K}\,\right]\,\sin 2\delta\,.$

(21)

For $N=2$, this reproduces the previous result obtained by Aligia for the spin-1/2 Anderson model [57, 58]. $C_{V}^{(2)}$ consists of a linear combination of $\alpha_{\mathrm{dif}}$ and $\gamma_{\mathrm{dif}}$, and it identically vanishes $C_{V}^{(2)}\to 0$ when both the tunnel couplings and the bias voltages are symmetrical $\alpha_{\mathrm{dif}}=\gamma_{\mathrm{dif}}=0$. The magnitude is determined also by $\widetilde{K}$ and $\sin 2\delta$ which can be related to the derivative of the spectral function $\rho_{d}^{\prime}$, given in Eq. (13): $\sin 2\delta/T^{\ast}=4\Delta\rho_{d}^{\prime}$. Furthermore, $C_{V}^{(2)}$ is an odd function of $\xi_{d}$. Note that $C_{V}^{(2)}$ depends on the Coulomb interaction only through the real part of the self-energy.

In contrast, $C_{V}^{(3)}$ given in the following depends on both the real part that determines the high-order energy shifts and imaginary part that destroy phase coherence [62, 63], specifically on the order $\omega^{2}$ and $(eV)^{2}$ terms. The exact formula of the coefficient $C_{V}^{(3)}$ for order $(eV)^{3}$ nonlinear current is composed of two parts: $W_{V}$ and $\Theta_{V}$ which represent the two-body contribution and three-body one, respectively,

	$\displaystyle C_{V}^{(3)}\,=$	$\displaystyle\ \frac{\pi^{2}}{64}\,\bigl{(}W_{V}\,+\,\Theta_{V}\bigr{)}\,,$		(22)
	$\displaystyle W_{V}\equiv$	$\displaystyle\ -\cos 2\delta\Biggl{[}1+3\alpha_{\mathrm{dif}}^{2}-6\Bigl{(}\alpha_{\mathrm{dif}}^{2}+\alpha_{\mathrm{dif}}\gamma_{\mathrm{dif}}\Bigr{)}\,\widetilde{K}$
		$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\left\{\frac{5}{N-1}+3\alpha_{\mathrm{dif}}^{2}+6\alpha_{\mathrm{dif}}\gamma_{\mathrm{dif}}+\frac{3(N-2)}{N-1}\,\gamma_{\mathrm{dif}}^{2}\right\}\,\widetilde{K}^{2}\Biggr{]},$		(23)
	$\displaystyle\Theta_{V}\equiv$	$\displaystyle\Bigl{[}1+3\alpha_{\mathrm{dif}}^{2}\Bigr{]}\Theta_{\mathrm{I}}+3\Bigl{[}1+3\alpha_{\mathrm{dif}}^{2}+4\alpha_{\mathrm{dif}}\gamma_{\mathrm{dif}}\Bigr{]}\,\widetilde{\Theta}_{\mathrm{II}}$
		$\displaystyle\ +6\Bigl{[}\alpha_{\mathrm{dif}}^{2}+2\alpha_{\mathrm{dif}}\gamma_{\mathrm{dif}}+\gamma_{\mathrm{dif}}^{2}\Bigr{]}\,\widetilde{\Theta}_{\mathrm{III}}\,.$		(24)

$C_{V}^{(3)}$ depends on the asymmetry parameters through the quadratic terms, i.e., $\,\alpha_{\mathrm{dif}}^{2}$, $\,\alpha_{\mathrm{dif}}\gamma_{\mathrm{dif}}$, and $\,\gamma_{\mathrm{dif}}^{2}$. Effects of the three-body corrections enter through the dimensionless parameters:

	$\displaystyle\Theta_{\mathrm{I}}\,\equiv$	$\displaystyle\ \frac{\sin 2\delta}{2\pi}\,\frac{\chi_{\sigma\sigma\sigma}^{[3]}}{\chi_{\sigma\sigma}^{2}}\,,$		(25)
	$\displaystyle\widetilde{\Theta}_{\mathrm{II}}\,\equiv$	$\displaystyle\ (N-1)\,\frac{\sin 2\delta}{2\pi}\,\frac{\chi_{\sigma\sigma^{\prime}\sigma^{\prime}}^{[3]}}{\chi_{\sigma\sigma}^{2}}\,,$		(26)
	$\displaystyle\widetilde{\Theta}_{\mathrm{III}}\,\equiv$	$\displaystyle\ \frac{(N-1)(N-2)}{2}\,\frac{\sin 2\delta}{2\pi}\,\frac{\chi_{\sigma\sigma^{\prime}\sigma^{\prime\prime}}^{[3]}}{\chi_{\sigma\sigma}^{2}}\,,$		(27)

for $\sigma\neq\sigma^{\prime}\neq\sigma^{\prime\prime}\neq\sigma$. In particular, $\widetilde{\Theta}_{\mathrm{III}}$ represents the correlation between three different levels which appears for $N\geq 3$. This component $\widetilde{\Theta}_{\mathrm{III}}$ will give no contribution to $C_{V}^{(3)}$ through Eq. (24) if there is no tunnel or bias asymmetry [44, 64]: $W_{V}$ and $\Theta_{V}$ take the following form at $\alpha_{\mathrm{dif}}=\gamma_{\mathrm{dif}}=0$,

	$\displaystyle W_{V}\,\xrightarrow{\,\alpha_{\mathrm{dif}}=\gamma_{\mathrm{dif}}=0\,}$	$\displaystyle\ -\cos 2\delta\,\left(1\,+\,\frac{5\widetilde{K}^{2}}{N-1}\right),$		(28)
	$\displaystyle\Theta_{V}\,\xrightarrow{\,\alpha_{\mathrm{dif}}=\gamma_{\mathrm{dif}}=0\,}$	$\displaystyle\ \,\Theta_{\mathrm{I}}\,+\,3\,\widetilde{\Theta}_{\mathrm{II}}\,.$		(29)

When both the chemical potentials and the tunnel couplings are inverted such that $(\alpha_{\mathrm{dif}}$, $\gamma_{\mathrm{dif}})\to(-\alpha_{\mathrm{dif}}$, $-\gamma_{\mathrm{dif}})$, the coefficients $C_{V}^{(2)}$ and $C_{V}^{(3)}$ exhibit odd and even properties, respectively: $C_{V}^{(2)}\to-C_{V}^{(2)}$ and $C_{V}^{(3)}\to C_{V}^{(3)}$ as shown in Appendix B. These formulas of $C_{V}^{(2)}$ and $C_{V}^{(3)}$ of the Anderson model for arbitrary level structures $\epsilon_{d\sigma}$ are consistent, in the limit of strong interaction $U\to\infty$, with the corresponding formulas for the SU($N$) Kondo model obtained by Mora et al. [40] at integer-filling points $\bigl{\langle}n_{d}\bigr{\rangle}=1,2,\ldots,N-1$, with $n_{d}\equiv\sum_{\sigma}n_{d\sigma}$.

The Fermi-liquid parameters of an SU(4) quantum dot, which can be deduced from the phase shift and the linear susceptibilities, are shown in Fig. 1 as a function of $\xi_{d}$ for several different values of $U/(\pi\Delta)=1/3$, $2/3$, $5/3$, $10/3$, $5$. For large $U/(\pi\Delta)\gtrsim 2.0$, the occupation number in Fig. 1(a) shows a Coulomb-staircase behavior with the plateaus of integer height $\langle n_{d}\rangle\simeq 1.0$, $2.0$, $3.0$ and the steps at $\xi_{d}\simeq-U$, $0$, $U$: the structure becomes clearer for stronger interactions. Figure 1(b) shows $\sin^{2}\delta$ corresponding to the linear term of the differential conductance in Eq. (1). For strong interactions $U/(\pi\Delta)\gtrsim 2.0$, the Kondo ridges which reflect the step structure of the occupation number of the values $\langle n_{d}\rangle\simeq 1,2,3$ evolve at $\xi_{d}\simeq U,0,-U$, respectively, as $U$ increases.

Figure 1(c) shows the rescaled Wilson ratio $\widetilde{K}\equiv(N-1)(R-1)$. It has a wide plateau that reaches the strong-coupling limit value $\widetilde{K}\simeq 1$ in the region $-1.5U\lesssim\xi_{d}\lesssim 1.5U$, for large interactions $U/(\pi\Delta)\gtrsim 3.0$. This is caused by the fact that charge fluctuations are suppressed in this region and it makes the charge susceptibility vanish: $\chi_{c}\propto\chi_{\sigma\sigma}+(N-1)\chi_{\sigma\sigma^{\prime}}\to 0$. The shallow dips of $\widetilde{K}$ at $\xi_{d}\simeq\pm 0.5U$ is caused by the charge fluctuations at the steps of the Coulomb staircase structure of $\langle n_{d}\rangle$.

Correspondingly, the renormalization factor $z$ in Fig. 1(d) exhibits a broad valley structure at $-1.5U\lesssim\xi_{d}\lesssim 1.5U$, which becomes deeper as $U$ increases. It has a clear local minimum for $U/(\pi\Delta)\gtrsim 2.0$ at $\xi_{d}\simeq$ $0$ and $\pm U$, where $\langle n_{d}\rangle$ takes integer values: it also has a local maximum at intermediate valence states in between the two adjacent minima. Figure 1(e) shows the gate voltage $\xi_{d}$ dependence of the inverse of the characteristic energy scale, $1/T^{\ast}$, scaled by $T_{K}$ that is defined as the value of $T^{\ast}$ at the elelctron-hole symmetric point $\xi_{d}=0$. It shows an oscillatory behavior for strong interactions $U/(\pi\Delta)\gtrsim 3.0$, reflecting the dependence of $z$ on $\xi_{d}$. In particular, $1/T^{\ast}$ reaches a local maximum at the integer-filling points where the SU($4$) Kondo effect occurs.

Some of the two-body correlation functions contribute to the low-energy transport through the derivative of the impurity density of state: $\rho_{d}^{\prime}=(\chi_{\sigma\sigma}/\Delta)\sin 2\delta$ given in Eq. (13). For instance, the coefficient $C_{V}^{(2)}$ is proportional to $\rho_{d}^{\prime}$. Figure 1(f) shows $\sin 2\delta$. It is an odd function of $\xi_{d}$ and vanishes at $\xi_{d}=0$ where the phase shift takes the value $\delta=\pi/2$. For strong interactions $U/(\pi\Delta)\gtrsim 3.0$, it has a wide maximum (minimum) at $\xi_{d}\simeq U$ ($-U$), where the $1/4$-filling ($3/4$-filling) SU($4$) Kondo occurs.

We next consider the three-body correlations between electrons passing through an SU(4) symmetric dot. Figure 2 shows $\Theta_{\mathrm{I}}$, $\widetilde{\Theta}_{\mathrm{II}}$, and $\widetilde{\Theta}_{\mathrm{III}}$ as a function of $\xi_{d}/U$ for several different values of $U$. These dimensionless three-body correlation functions vanish in the electron-hole symmetric case $\xi_{d}=0$. For strong interactions $U/(\pi\Delta)\gtrsim 3.0$, a plateau of the width $U$ emerges at $\xi_{d}\simeq 0$ and $\pm U$, i.e., at integer filling points $\langle n_{d}\rangle\simeq 1,$ $2$, $3$. The plateau structure evolves further as interaction $U$ increases. Specifically, the correlation function between three different levels $\widetilde{\Theta}_{\mathrm{III}}$ appears for SU($N$) quantum dots with $N\geq 3$, and contributes to the nonlinear conductance when there are some asymmetries in the tunnel couplings or the bias voltages.

A comparison of the three independent components $\Theta_{\mathrm{I}}$, $-\widetilde{\Theta}_{\mathrm{II}}$, and $\widetilde{\Theta}_{\mathrm{III}}$ is made in Fig. 2(d) for a large interaction $U/(\pi\Delta)=5.0$. It shows that these three components approach each other very closely over a wide range of gate voltages $-1.5U\lesssim\xi_{d}\lesssim 1.5U$,

$\displaystyle\Theta_{\mathrm{I}}\,\simeq\,-\widetilde{\Theta}_{\mathrm{II}}\,\simeq\,\widetilde{\Theta}_{\mathrm{III}}\,.$

(30)

This is caused by the fact that the derivatives of the two independent components of linear susceptibilities, $|\frac{\partial\chi_{\sigma\sigma}}{\partial\epsilon_{d}}|$ and $|\frac{\partial\chi_{\sigma\sigma^{\prime}}}{\partial\epsilon_{d}}|$, become much smaller than an inverse of the characteristic energy scale $(T^{\ast})^{-2}$ in a wide range of electron fillings $1\lesssim\langle n_{d}\rangle\lesssim N-1$, in addition to the suppression of the charge fluctuations $\chi_{c}\simeq 0$, mentioned above [see Appendix C] [44].

We next summarize the low-energy Fermi-liquid properties of SU($6$) quantum dots. The phase shift and the renormalized parameters that can be deduced from the two-body correlations are plotted as a function of the gate voltage $\xi_{d}/U$ in Fig. 3, for several different values of $U/(\pi\Delta)=2/5$, $1.0$, $2.0$, $5.0$, i.e., from weak to strong interactions. The behaviors of these parameters are similar to those of the SU($4$) quantum dots. However, the number of different SU($N$) Kondo states occurring at integer fillings increases with $N$, i.e., $\langle n_{d}\rangle=1$, $2$, $\ldots$, $N-1$. It takes place at $\xi_{d}\simeq 0.0$, $\pm U$, …, $\pm\frac{N-2}{2}U$, and gives an interesting variety in low-energy transport. As $N$ increases, quantum fluctuations caused by the Coulomb interaction $U$ is suppressed, and in particular, the mean-field theory becomes exact in the large $N$ limit of the finite-$U$ Anderson impurity model [69]. Hence for larger $N$, electron-correlation effects occur for stronger interactions $U$.

In Fig. 3(a), we can see that the Coulomb staircase behavior of the occupation number $\langle n_{d}\rangle$ emerges for a large interaction $U/(\pi\Delta)=5.0$, which has steps at $\xi_{d}\simeq 0.0$, $\pm U$, $\pm 2U$ for SU($6$) quantum dots. The transmission probability $\sin^{2}\delta$, shown in Fig. 3(b), reaches the unitary limit value $\sin^{2}\delta=1.0$ at half filling $\xi_{d}=0$. We can see that the plateau structure develops as $U$ increases, and it becomes visible at a strong interaction $U/(\pi\Delta)=5.0$.

Figure 3(c) shows that the rescaled Wilson ratio $\widetilde{K}$ for $N=6$ has a wide flat peak at $-(N-1)U/2\lesssim\xi_{d}\lesssim(N-1)U/2$, the height of which increases with $U$. In particular, it approaches saturation value $\widetilde{K}\simeq 1$ at $U/(\pi\Delta)=5.0$ due to the suppression of charge fluctuations, as mentioned.

The renormalization factor $z$ for SU($6$) quantum dots, shown in Fig.  3(d), exhibits a broad valley structure similar to the one for the SU(4) symmetric case. The valley becomes deeper as $U$ increases, and the local minima emerge at the integer-filling points, reflecting the electron correlations due to the SU(6) Kondo effects. Figure 3(e) shows that the inverse of the characteristic energy $1/T^{\ast}$, which is scaled by the value $T_{K}$ defined at half filling $\xi_{d}=0$. It has peaks situated at $\xi_{d}\simeq 0$, $\pm U$, $\pm 2U$ for a strong interaction $U/(\pi\Delta)=5.0$, although the ones at $\pm 2U$ are still developing. These peaks correspond to the SU(6) Kondo temperature at each of the integer-filling points.

Figure 3(f) shows $\sin 2\delta$ for $N=6$, which is proportional to the derivative of the density of states $\rho_{d}^{\prime}=(\chi_{\sigma\sigma}/\Delta)\sin 2\delta$, and determines the magnitude of the coefficient $C_{V}^{(2)}$ of the nonlinear current, as mentioned. This factor $\sin 2\delta$ is an odd function of $\xi_{d}$, and at a strong interaction $U/(\pi\Delta)=5.0$ it takes a broad peak (dip) at $\delta=\pi/4$ ($3\pi/4$) corresponding to the half-integer filling $\langle n_{d}\rangle=1.5$ ($4.5$) where charge fluctuations are not fully suppressed.

Figure 4 shows three-body correlations between electrons passing through an SU(6) quantum dot, calculated as a function of $\xi_{d}$ for several values of interactions $U/\pi\Delta=2/5$, $1.0$, $2.0$, $5.0$. These dimensionless correlation functions $\Theta_{\mathrm{I}}$, $\widetilde{\Theta}_{\mathrm{II}}$, and $\widetilde{\Theta}_{\mathrm{III}}$ vanish at half filling $\xi_{d}=0$, and away from half filling, they are enhanced as $U$ increases. We can see that for a strong interaction $U/(\pi\Delta)=5.0$ there emerges either a wide peak, a wide dip, or a plateau at $\xi_{d}\simeq\pm U$, $\pm 2U$, i.e., at integer filling points. The component between three different levels $\widetilde{\Theta}_{\mathrm{III}}$ also contributes to the order $(eV)^{3}$ term of nonlinear current through an SU($6$) quantum dot when there are some asymmetries in the tunnel couplings or the bias voltages.

In Fig. 4(d) the three independent components $\Theta_{\mathrm{I}}$, $-\widetilde{\Theta}_{\mathrm{II}}$, and $\widetilde{\Theta}_{\mathrm{III}}$ are compared in a strong interaction case $U/(\pi\Delta)=5.0$. It shows that these three components approach each other very closely $\Theta_{\mathrm{I}}\simeq-\widetilde{\Theta}_{\mathrm{II}}\simeq\widetilde{\Theta}_{\mathrm{III}}$ over a wide range of the gate voltage $-(N-1)U/2\lesssim\xi_{d}\lesssim(N-1)U/2$ with $N=6$. This is due to the fact that the diagonal component $\chi_{\sigma\sigma\sigma}^{[3]}$ dominates the three-body correlation, and the derivatives $|\frac{\partial\chi_{\sigma\sigma}}{\partial\epsilon_{d}}|$ and $|\frac{\partial\chi_{\sigma\sigma^{\prime}}}{\partial\epsilon_{d}}|$ become much smaller than $(T^{\ast})^{-2}$ in a wide range of electron fillings $1\lesssim\langle n_{d}\rangle\lesssim N-1$ [see Appendix C] [44].