Nonequilibrium Statistics of Biased Kondo Resonance

TL;DR

The paper tackles nonequilibrium steady-state behavior of biased quantum impurity systems, focusing on the Kondo regime and the associated resonance. It develops a nonequilibrium extension of numerical renormalization group (NRG) by implementing Hershfield's scattering-state formalism with a density matrix based on the operator $\hat{H}-V\hat{Y}$, using the original bath basis for simplicity. The main findings show a bias-induced splitting of the Kondo resonance into two peaks at $\omega \approx \pm V/2$, a multiscale nonequilibrium distribution with population inversion and broad excitations up to $|\omega|\gtrsim V$, and a current saturation regime in the low-temperature limit for $V/2$ between $T_K$ and a few $T_K$. The results validate the method against noninteracting limits and reveal how the Kondo singlet breaks under bias, with potential to extend to nonequilibrium DMFT and more complex impurities.

Abstract

Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale $T_K$. The distribution shows distinct multi-scale spectral features at energy $ω$ below the Kondo scale ($ω\lesssim T_K$) and near the bias ($ω\gtrsim V$), leading to the nonequilibrium temperature $T_{\rm loc}$ local to the Kondo dot scaling as $k_BT_{\rm loc}\approx V$ for $V\gg T_K$. The current-voltage relation in the low-temperature limit ($T\ll T_K$) deviates from the unitary limit as the bias exceeds the Kondo scale ($V/2\gtrsim T_K$) and reaches the current saturation regime.

Figures (5)

• Figure 1: (a) Scattering state represented as a linear combination of bonding and anti-bonding states computed on finite chains. By choosing a certain linear superposition of the bonding and anti-bonding states, the wave $\psi_L$ can be made to have only the right-moving component in the $R$-reservoir. (b) The nonequilibrium NRG scheme for an Anderson impurity hewson1997kondo coupled to biased reservoirs. The $L/R$ reservoir is made up of the Wilson chain of length $N$ with the $N_{\rm max}-N$ environment orbitals at energy levels at zero. The chemical potentials apply up to $\pm V/2$ to $L/R$-reservoirs, respectively, including the environment orbitals. The levels from the Wilson chains follow the conventional scaling of the Wilson tight-binding parameters wilsonRMP1975.
• Figure 2: Current calculated in the non-interacting resonant level model. The current expectation value $I$ (per spin, see text for definition) scaled with the conductance quantum $G_0=e^2/(2\pi\hbar)$ follows the exact result (dashed line) in the low bias limit (blown up in the inset). The infinitesimal parameter $\eta$ of Eq. (\ref{['eq:y']}) is varied below the hybridization $\Gamma=2\pi t^2/(2D)=0.0079$ with the hopping $t=0.05$ and the (half) bandwidth $D=1$. (See main text for the definition of $\eta_0$. The level energy is set to the particle-hole symmetric limit $\epsilon_d=0$. The $z$-averaging oliveiraPRB1994 was used over 8 values of $z$ ($N_z=8$).
• Figure 3: NRG flow of eigenvalues of $(\hat{H}-V\hat{Y})/D_n$ per $(Q,S)$ at (odd integer) chain lengths $n$, $\bar{E}^Y_n(Q,S)$, with bias up to $V/2\sim T_K$. $Q$ and $S$ are charge and spin quantum numbers from the equilibrium ground state, and $D_n$ is the RG energy scale, Eq. (\ref{['eq:dn']}). The $(1,0)$ sector, Kondo-screened doubly-occupied impurity, approaches the Kondo fixed point at $V=0$. As the bias grows, the RG flow departs from the fixed point at the chain length (indicated by an arrow for each bias with corresponding color that gives the maximum density-matrix weight).
• Figure 4: Spectral evolution of Anderson model under bias $V$. (a) Particle-hole (p-h) symmetric limit with $\epsilon_d=0$ in the strong Kondo regime ($U/\Gamma=10$, $\Gamma=\Gamma_L+\Gamma_R$) at a temperature much lower than the Kondo temperature $T_K=0.0044$. The right inset shows the whole frequency range demonstrating the sharpness of the Kondo resonance. From top to bottom, $V/T_K=0, 0.2,0.4,0.6, 0.8, 1,2,3,4,5,6,7,8,9,10$ with selected bias values shown in the legend. The peak-splitting begins at $V/2\approx T_K$ with the peak positions at $\omega\approx \pm\frac{1}{2} V$. (b) p-h asymmetric limit $(\epsilon_d=U/4)$ with the same set of bias $V$.
• Figure 5: (a) The distribution function $f(\omega)$ in the p-h symmetric limit with $U/\Gamma=10$ at bias values $V=0,\cdots,10$ with the spacing $\Delta V=1$. $f(\omega)$ strongly deviates from the Fermi-Dirac function, displaying multiscale features of population inversion at $\omega\sim T_K$ and the broad excitation peaks at $\omega\gtrsim V$. The inset shows the nonequilibrium electron temperature $T_{\rm loc}$ local to the Kondo dot. In the high bias limit, $T_{\rm loc}\approx V$. (b) $IV$ relation at temperatures $T=0.1\,T_K$ and $T_K$. In the low temperature limit at $T=0.1\, T_K$, the unitary conductance limit in the small field limit starts to deviate and reach the current saturation regime for $V\gtrsim T_K$. At higher temperature $T=T_K$, no sign of current saturation is observed.