Relaxation dynamics in a Hubbard dimer coupled to fermionic baths: Phenomenological description and its microscopic foundation

We study relaxation dynamics in a strongly interacting two-site Fermi-Hubbard model that is induced by coupling each site to a local fermionic bath. To derive the proper form of the Lindblad operators that enter an effective description of the system-bath coupling in different temperature regimes, we employ a diagrammatic real-time technique for the time evolution of the reduced density matrix. In spite of a local coupling to the baths, the found Lindblad operators are nonlocal in space. We compare with the local approximation, where those nonlocal effects are neglected. Furthermore, we propose an improvement on the commonly used secular approximation (rotating-wave approximation), referred to as coherent approximation, which turns out superior in all studied parameter regimes (and equivalent otherwise). We look at the relaxation dynamics for several important observables and compare the methods for early and late times in various temperature regimes.

Figure 1

Single site coupled to a hot (red), cold (blue), and finite-temperature (gray) bath. Electron-electron interaction $U$ gives rise to two excitation energies $\varepsilon$ and $\varepsilon +U$.

Figure 2

(a) Particle asymmetry $〈\mathrm{\Delta }N〉$ and (b) energy $E=〈H_{\text{s}}〉$ as functions of time starting from an initial state ${\rho }_0=|\uparrow ,0\rangle \langle \uparrow ,0|$. We compare the local secular approximation ${\mathcal{L}}_{\text{loc}}^{\text{S}}$ (blue line) and the nonlocal secular ${\mathcal{L}}^{\text{S}}$ approximation (green line) with the exact results given by the local coherent approximation ${\mathcal{L}}_{\text{loc}}^{\text{C}}={\mathcal{L}}^{\text{C}}$ (orange line). In (c) we resolve the differences $E^{\text{C}}-E^{\text{S}}$ between the nonlocal secular approximation ${\mathcal{L}}^{\text{S}}$ and the exact results ${\mathcal{L}}^{\text{C}}$. The parameters are $U=5\mathrm{\Gamma }, J=2\mathrm{\Gamma }$, and $T\to \infty$.

Figure 3

(a) Particle asymmetry $〈\mathrm{\Delta }N〉$ and (b) spin asymmetry $〈\mathrm{\Delta }{S}_z〉$ as functions of time starting from an initial state ${\rho }_0=|\uparrow ,0\rangle \langle \uparrow ,0$|. We compare the local methods ${\mathcal{L}}_{\text{loc}}^{\text{S}}={\mathcal{L}}_{\text{loc}}^{\text{C}}$ (blue line) with the nonlocal secular ${\mathcal{L}}^{\text{S}}$ (green line) and the nonlocal coherent approximation ${\mathcal{L}}^{\text{C}}$ (black line). Moreover, we show the full solution to the time-dependent problem from Eq. (16) (orange line). The parameters are $U=10\mathrm{\Gamma }, J=2\mathrm{\Gamma }$, and $T\to 0$.

Figure 4

(a) Spin squared ${\mathbf{S}}^2$ and (b) parity $P={(-1)}^N$ as functions of time starting from an initial state ${\rho }_0=|S\rangle \langle S$|. We compare the local methods ${\mathcal{L}}_{\text{loc}}^{\text{S}}={\mathcal{L}}_{\text{loc}}^{\text{C}}$ (blue line) with the nonlocal secular ${\mathcal{L}}^{\text{S}}$ (green line) and the nonlocal coherent approximation ${\mathcal{L}}^{\text{C}}$ (black line). The latter two methods (green and black line) agree for the shown observables. Moreover, we show the full solution to the time-dependent problem from Eq. (16) (orange line). The parameters are $U=10\mathrm{\Gamma }, J=2\mathrm{\Gamma }$, and $T\to 0$.

Figure 5

Spin asymmetry $\mathrm{\Delta }{S}_z$ as a function of time starting from an initial state ${\rho }_0=|\uparrow ,0\rangle \langle \uparrow ,0$|. We compare the results of the coherent approximation ${\mathcal{L}}^{\text{C}}$ for zero temperature (black line) with finite temperatures $k_{\text{B}}T=\mathrm{\Gamma }/2$ (orange line) and $k_{\text{B}}T=\mathrm{\Gamma }$ (green line). The spin oscillations decay on a rather long time scale. The remaining parameters are $U=10\mathrm{\Gamma }$ and $J=2\mathrm{\Gamma }$.