Nonequilibrium quantum dissipation in spin-fermion systems

Dissipative processes in nonequilibrium many-body systems are fundamentally different than their equilibrium counterparts. Such processes are of great importance for the understanding of relaxation in single-molecule devices. As a detailed case study, we investigate here a generic spin-fermion model, where a two-level system couples to two metallic leads with different chemical potentials. We present results for the spin relaxation rate in the nonadiabatic limit for an arbitrary coupling to the leads using both analytical and exact numerical methods. The nonequilibrium dynamics is reflected by an exponential relaxation at long times and via complex phase shifts, leading in some cases to an “antiorthogonality” effect. In the limit of strong system-lead coupling at zero temperature we demonstrate the onset of a Marcus-like Gaussian decay with voltage difference activation. This is analogous to the equilibrium spin-boson model, where at strong coupling and high temperatures, the spin excitation rate manifests temperature activated Gaussian behavior. We find that there is no simple linear relationship between the role of the temperature in the bosonic system and a voltage drop in a nonequilibrium electronic case. The two models also differ by the orthogonality-catastrophe factor existing in a fermionic system, which modifies the resulting line shapes. Implications for current characteristics are discussed. We demonstrate the violation of pairwise Coulomb gas behavior for strong coupling to the leads. The results presented in this paper form the basis of an exact, nonperturbative description of steady-state quantum dissipative systems.

Figure 1

Decay of $C_f\left(t\right)$ as computed by both the RG approach and standard linear discretization for $\nu =0.1$. Constant discretization with 200 states per band, $\Delta \mu =0$ (dotted), showing an artificial rise of the correlation function due to discretization errors; logarithmic discretization (RG) with 100 states per band, $\Lambda =1.1$, $\Delta \mu =0$ (full); logarithmic discretization (RG) with 100 states per band, $\Lambda =1.1$, $\Delta \mu =0.007$ (dashed). $\Lambda$ is a logarithmic scale parameter, where for each conduction energy there are states with energies $ϵ∕{\Lambda }^m$, $m=1,2,\dots$. Inset: The finite bias case exhibits an exponential decay at long times. The slope agrees with the theoretical value, Eq. (19). The Fermi sea bandwidth is $2D_0$ in all plots with $D_0=1$.

Figure 2

The correlation function $C_f\left(t\right)$ for $\Delta \mu =0.24$, $\nu =0.95$, manifesting (a) a power-law decay at short times $\Delta \mu t<1$ (notice the log-log scale), (b) a Gaussian decay at intermediate times $\Delta \mu t\sim 1–10$, and an exponential decay at long times $\Delta \mu t⪢1$.

Figure 3

Evidence for the Gaussian decay at intermediate times $\Delta \mu t\sim 1–10$ and strong coupling $\nu =0.95$, $\Delta \mu =0.16$ (full), $\Delta \mu =0.24$ (dotted), and $\Delta \mu =0.32$ (dashed-dotted). The inset, which includes the three lines one on top of the other, reveals that $C_f\left(t\right)t^{\beta }\propto e^{-\kappa {\left(\Delta \mu t\right)}^2}$, $\beta =0.13$, with $\kappa \sim 0.03$.

Figure 4

Time dependence of the correlation function $\mid C_f\left(t\right)\mid$ for $\Delta \mu =0.24$, $\nu =0.95$. The fitting function $y_G=e^{-\kappa {\left(\Delta \mu t\right)}^2}{\left(Dt\right)}^{-\beta }$ (dotted) with $\kappa =0.03$, $D=6$, and $\beta =0.13$ compared to exact numerical solution (full). The Gaussian (dashed-dotted) and the power-law (dashed) parts of $y_G$ are also displayed separately.

Figure 5

The correlation function $C_f\left(t\right)$ for different coupling strengths $\nu$, manifesting the increasing dominance of intermediate-time Gaussian behavior at strong coupling. $\Delta \mu =0.24$ in all plots.

Figure 6

Testing the validity of Eqs. (19, 27) for describing the long-time and intermediate-time behaviors, respectively. Top: The relaxation rate $\Gamma$. Numerical results calculated from the slope of $\ln \left[C_f\left(t\right)t^{\beta }\right]$ vs $\Delta \mu t$ at long times (squares); analytical results using Eq. (19) (dashed line). Bottom: The coefficient $\kappa$. The numerical slope of $\ln \left[C_f\left(t\right)t^{\beta }\right]$ vs $\Delta {\mu }^2{t}^2$ at intermediate times (squares); analytical results using Eq. (27) (dashed line). These data were computed with $\Delta \mu =0.24$, and $\beta$ was extracted from the short-time dynamics for each value of $\nu$.

Figure 7

The turnover from a Gaussian decay to an exponential relaxation. Top: Comparison between the numerical correlation function $C_f\left(t\right)$ (full) and the fitting functions $y_G$ (dotted) and $y_E$ (dashed) defined in the text, $\nu =0.1$, $\Delta \mu =0.24$. Inset: Exposing the turnover by taking away the short-time and intermediate-time terms $t^{-\beta }{e}^{-\kappa \Delta {\mu }^2{t}^2}$ with $\beta =0.002$, $\kappa =5\times {10}^{-4}$. Bottom: Same with $\nu =0.95$, leading to $\beta =0.13$, $\kappa =0.03$. While $y_G$ explicitly includes the Gaussian decay, correct to $\Delta \mu t\sim 10$, the function $y_E$ captures the correct slope at longer times.

Figure 8

Orthogonality and antiorthogonality effects in the nonequilibrium system $\Delta \mu =0.4$. [(a) and (b)] $\nu =0.3$, $\alpha =0.5$, manifesting the standard orthogonality effect $(\tilde{\gamma }<0)$. [(c) and (d)] $\nu =0.8$, $\alpha =0$, revealing the antiorthogonality effect $(\tilde{\gamma }>0)$.

Figure 9

Map of orthogonality $[({\delta }_L^2+{\delta }_R^2)>0]$ and antiorthogonality $[({\delta }_L^2+{\delta }_R^2)<0]$ behaviors using expression (7) with $\alpha ={\alpha }_1={\alpha }_2$.

Figure 10

Resolving the complex part of the power-law exponent $\tilde{\gamma }$. ${\alpha }_1=0$, ${\alpha }_2=0$ (full); ${\alpha }_1=0.1$, ${\alpha }_2=0$ (dashed); ${\alpha }_1=0.2$, ${\alpha }_2=0$ (dashed-dotted); ${\alpha }_1=0.2$, ${\alpha }_2=0.2$ (dotted); $\Delta \mu =0.24$ and $\nu =0.95$ in all plots.

Figure 11

Golden rule relaxation rate calculated for the weak coupling limit $\nu =0.5$ ($\beta =0.043$, $\Gamma =0.08$); $\Delta \mu =0$ (dashed line), $\Delta \mu =0.24$ (solid line), and $\Delta \mu =0.48$ (dotted line). The inset presents an expanded view of low frequency regime.

Figure 12

Fermi’s golden rule rate, Eq. (33), as a function of frequency. $\kappa =0.03$ and $\beta =0.13$ $(\nu =0.95)$. Inset: Voltage activated excitation rate $(\omega <0)$.

Figure 13

Spin polarization as a function of potential bias $\Delta \mu$ for different energy biases $\omega =0.1$ (full), $\omega =0.2$ (dashed), and $\omega =0.3$ (dashed-dotted) for $\kappa =0.03$, $\beta =0.13$ $(\nu =0.95)$. Inset: Mean-field calculation of the current $I\propto {⟨n_d(+)⟩}^2\Delta \mu$ for the same frequencies as in the main plot.

Figure 14

Schematic representation of spin-flip events on the Keldysh contour. Plotted are examples for particular two, four, and six spin-flip processes, respectively.

Figure 15

Testing the Coulomb gas picture for out-of-equilibrium situations. Comparison between the exact fourth order correlation function $C_4(t_1,t_2,t_3,t_4)$, Eq. (42) (full), and the pairwise approximation ${\tilde{C}}_4(t_1,t_2,t_3,t_4)$, Eq. (43) (dashed). $\tau =t_{i+1}-t_i$ is the distance between spin-flip events. All other parameters are the same as in Fig. 5.

Figure 16

Testing the Coulomb gas picture for out-of-equilibrium situations. Comparison between the exact expression sixth order correlation function $C_6(t_1,t_2,t_3,t_4,t_5,t_6)$, Eq. (42) (full), and the pairwise approximation ${\tilde{C}}_6(t_1,t_2,t_3,t_4,t_5,t_6)$, Eq. (43) (dashed). $\tau =t_{i+1}-t_i$ is the distance between spin-flip events. All other parameters are the same as in Fig. 5.