Approach to a stationary state in a driven Hubbard model coupled to a thermostat

We investigate the dynamics of the Hubbard model in a static electric field in order to identify the conditions necessary to reach a nonequilibrium stationary state. We show that, for a generic electric field, the convergence to a stationary state requires coupling to a thermostatting bath that absorbs the work done by the external field. Following the real-time dynamics of the system, we show that a nonequilibrium stationary state is reached for essentially any value of the coupling to the bath. We characterize the properties of such nonequilibrium stationary states by studying suitable physical observables, pointing out the existence of an analog of the Pomeranchuk effect as a function of the electric field. We map out a phase diagram in terms of dissipation and electric field strengths and identify the dissipation values at which the steady current is largest for a given field.

Figure 1

Dynamics of the local current $J$ for $U=6.0$, increasing coupling to thermostat, $\Lambda$, and $E=4.7$ (a) and 1.26 (b).

Figure 2

(a) Dynamics of the local current from different realizations of the initial conditions: electric field quench on the noninteracting $U=0.0$ (black dashed line) and interacting $U=4.0$ (red solid line) states, smooth switching on of the electric field for $U=0.0$, and $\tau =2.0$ (blue; cf. text). The data are for $E=2.0$ and $\Lambda =0.05$. The inset shows the profile of the electric field. (b) Effective temperature $T_{\mathrm{eff}}\left(\Omega \right)$ as a function of time for $U=6$, $E=1.9$.

Figure 3

Nonequilibrium dynamics of the double occupations $d$ for $U=6$, $E=1.25$, and increasing $\Lambda$.

Figure 4

(a) Linear-nonlinear crossover of the stationary current as a function of the electric field $E$. Data for $\Lambda =0.025$ and increasing $U$. (b) Effective temperature $T_{\mathrm{eff}}\left(\Omega \right)$ of the NSS as a function of the applied external field and for increasing coupling to the thermostat. Data are for $U=6$ and $E=1.9$.

Figure 5

Double occupancy of the NSS $d_{\mathrm{nss}}$ as a function of the electric field for $U=6$ and increasing coupling to the thermostat $\Lambda$. The equilibrium solution for the same value of the interaction (dotted line) is reported for comparison.

Figure 6

Top panel: Phase diagram of the thermostatted nonequilibrium Hubbard model. The diagram is obtained from the normalized local current $J$ as a function of electric field strength $E$ and coupling to the thermal bath, $\Lambda$. The data are for $U=4$. Horizontal lines indicate the cuts shown in the bottom panel. Bottom panel: Behavior of local current $J$ and entropy $S$ as a function of $\Lambda$, for $E=1$ (circles) and 2 (triangles).