Nonequilibrium Steady-State Transport in Quantum Impurity Models: A Thermofield and Quantum Quench Approach Using Matrix Product States

The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors $\lesssim 3\%$) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.

Figure 1

(a) The discretization combines a log-sector for high energy excitations with a lin-sector for the TW. (b) The log-sector is treated using NRG. Here, “holes” and “particles” are recombined. The effective low-energy basis of NRG is used as the local state space of one MPS chain element. For the lin-sector, holes (empty at $t=0$) and particles (filled at $t=0$) are treated separately. On the chain including the RI, we do a TDMRG calculation based on a Trotter decomposition in “odd” and “even” bonds [37].

Figure 2

Universal scaling of current vs voltage for the IRLM at the self-dual point for $T=0$, in units of the energy scale $T_B(t^{\prime })$, with negative differential conductance at large voltages, in excellent agreement with analytical results (solid curve, [15]). The inset shows the scaling of $T_B$ with $(t^{\prime }{)}^{3/4}$.

Figure 3

Numerical results for the SIAM with $\mathrm{\Gamma }={10}^{-3}$. For $U=12\mathrm{\Gamma }$, used in (b)–(d), we find $T_K=2.61\times {10}^{-5}$. [This implies $T_K=1.04T_K^{(\chi )}$, where $T_K^{(\chi )}=(1/4{\chi }_s)=(U\mathrm{\Gamma }/2{)}^{\frac{1}{2}}{e}^{\pi [(\mathrm{\Gamma }/2U)-(U/8\mathrm{\Gamma })]}$ is an alternative definition of the Kondo temperature based on the Bethe-ansatz result [50] for the static spin susceptibility ${\chi }_s$, at $B=T=0$]. (a) Conductance vs $V$ and $T$: squares show quench results in linear response as function of $T$, $g(T,0)$, in good agreement with the NRG results (solid line). Dots and triangles show quench results for the nonlinear conductance vs $V$ at $T=0$ for two different values of $U$. Inset: current vs $V$ for $U=0$ on a log-log scale, for two different temperatures, showing excellent agreement with analytical results. (b) Disappearance of the Kondo resonance in $g(T,V)$ with increasing $T$ at $B=0$, with $g(T,-V)=g(T,V)$, by symmetry. (c) Splitting of the resonance in $g(0,V)$ for finite $B$. Two subpeaks emerge at $V\approx \pm B$, as marked by the dashed lines. (d) Similar data as in (c) but plotted vs $V/B$ and on a linear scale. For $B=2T_K$, the peak position in the conductance $g(0,V)$ is still slightly below $B$, but for a higher magnetic field, the peak clearly moves towards $V/B\approx 1$. In (b)–(d), the squares indicate the NRG result for $V=0$.