Many-body wavefunctions for quantum impurities out of equilibrium. I. The nonequilibrium Kondo model

We present here the details of a method [A. B. Culver and N. Andrei, Many-body wavefunctions for quantum impurities out of equilibrium, Phys. Rev. B 103, L201103 (2021)] for calculating the time-dependent many-body wavefunction that follows a local quench. We apply the method to the voltage-driven nonequilibrium Kondo model to find the exact time-evolving wavefunction following a quench where the dot is suddenly attached to the leads at $t=0$. The method, which does not use Bethe ansatz, also works in other quantum impurity models and may be of wider applicability. We show that the long-time limit (with the system size taken to infinity first) of the time-evolving wavefunction of the Kondo model is a current-carrying nonequilibrium steady state that satisfies the Lippmann-Schwinger equation. We show that the electric current in the time-evolving wavefunction is given by a series expression that can be expanded either in weak coupling or in strong coupling, converging to all orders in the steady-state limit in either case. The series agrees to leading order with known results in the well-studied regime of weak antiferromagnetic coupling and also reveals a universal regime of strong ferromagnetic coupling with Kondo temperature $T_K^{\left(F\right)}=D{e}^{-\frac{3{\pi }^2}{8}\rho \left|J\right|}$ ($J<0, \rho \left|J\right|\to \infty$). In this regime, the differential conductance $dI/dV$ reaches the unitarity limit $2e^2/h$ asymptotically at large voltage or temperature.

Figure 1

Schematic of the quench process. Prior to $t=0$, the leads are filled with free electrons, with no tunneling to the dot allowed. From $t=0$ onward, the system evolves with the many-body Hamiltonian $H_{\text{Kondo}}$, with tunneling to and from the leads resulting in an electric current.

Figure 2

The wavefunction $|\mathrm{\Psi }\left(t\right)\rangle =|{\mathrm{\Psi }}^0\left(t\right)\rangle +\cdots +|{\mathrm{\Psi }}^N\left(t\right)\rangle$ for $N=1,2,$ and 3. Each line represents a quantum number ${\alpha }_j$ of the initial state ($j=1,\cdots ,N)$. Ordinary lines represent $c_{\alpha }^{†}\left(t\right)$ operators, while each line that ends on a circle represents a quantum number assigned to a “crossing state” (see main text). Sign factors, antisymmetrizations, and dependence on the time $t$ (and on the impurity quantum number $\beta$) are all implicit.

Figure 3

The $N$-body wavefunction of the two-lead Kondo model, either at arbitrary time [Eq. (2.39)] or the NESS [Eq. (2.61)] that is reached at long time with the system size taken to infinity first. Lines represent the momenta and spin quantum numbers of electrons in each lead. Any number of electrons, from lead 1 or lead 2, can be put into a crossing state (indicated by connecting lines), which is built from even sector operators only. For a fixed number $N$ of electrons, the wavefunction is a finite sum.

Figure 4

Schematic of the NESS obtained by taking the steady-state limit of $e^{-iHt}{e}^{i{H}^{\left(0\right)}t}|\mathrm{\Psi }\rangle$. The initial condition at $t=0$ becomes a boundary condition of two incoming Fermi seas, with a complicated result following the scattering off the dot.

Figure 5

The universal conductance $G\equiv \partial I_{\text{steady}\text{state}}/\partial V$ in the strong ferromagnetic regime at leading-log approximation. In contrast to the antiferromagnetic case in which $G$ is known to reach the unitarity limit $G_0\equiv 2e^2/h$ at $T=V=0$ [44], here the unitarity limit is reached asymptotically at large voltage or temperature. As the external scale is lowered to $T_K^{\left(\text{F}\right)}$ and below, the series in $1/g$ breaks down and another method is needed. Inset: the first correction beyond leading log in the quantity $\mathrm{\Delta }G\equiv G(T,V)-G(T=0,V)$ for $V\gg T_K^{\left(\text{F}\right)}$, with various values of $T$.

Figure 6

Kondo scaling picture. The two universal regimes are weak antiferromagnetic bare coupling ($0<g\ll 1, T_K=D{e}^{-1/\left(2g\right)}$) and strong ferromagnetic bare coupling ($g<0, \left|g\right|\gg 1, T_K^{\left(\text{F}\right)}=D{e}^{-3{\pi }^2\left|g\right|/8}$). The former has been much studied, and the latter is predicted by our calculations. In either case, the running coupling $g_R$ is close to the bare coupling if the system is probed at a high-energy scale (high relative to $T_K$ but always small compared to the bandwidth), but moves away from the bare coupling as the energy scale is reduced.

Figure 7

Sample numerical checks of our asymptotic result for $R^{(3,1,4,2)}[\{f,h\},\lambda ]$. Case 1 is $f\left(v\right)=h\left(v\right)=v/sin\mathrm{h}v$, which is used in the calculation of $G\left(T\right)$; case 2 is $f\left(v\right)=cosv$ and $h\left(v\right)=\mathrm{sinc}v$, which is used in the calculation of $I(T_1=0,T_2=0,V)$ [and hence, $G\left(V\right)$]. Only the real part of $R^{(3,1,4,2)}[\{f,h\},\lambda ]$ appears in the answer to the order we consider ($J^5$ or $1/J^5$), but the agreement for the imaginary part is similar.