Long-lived nonequilibrium states in the Hubbard model with an electric field

We study the single-band Hubbard model in the presence of a large spatially uniform electric field out of equilibrium. Using the Keldysh nonequilibrium formalism, we solve the problem using perturbation theory in the Coulomb interaction $U$. We present numerical results for the charge current, the total energy of the system, and the double occupancy on an infinite-dimensional hypercubic lattice with nearest-neighbor hopping. The system is isolated from an external bath and is in the paramagnetic state. We show that an electric field pulse can drive the system to a steady nonequilibrium state, which does not evolve into a thermal state. We compare results obtained within second-order perturbation theory (SOPT), self-consistent SOPT, and iterated perturbation theory (IPT). We also discuss the importance of initial conditions for a system which is not coupled to an external bath.

Figure 1

Second-order contribution to the self energy. The solid dot vertices correspond to the interaction $U$ and time changes along the horizontal direction. The solid lines represent either the Hartree-Fock (SOPT), or the dressed (self-consistent SOPT) Green's function, or the effective medium for the equivalent impurity problem (IPT).

Figure 2

Hartree-Fock self-energy diagram for the Hubbard model.

Figure 3

Single time self-energy diagrams for the Hubbard model.

Figure 4

The equilibrium density of states (DOS) versus frequency $\omega$, calculated using SOPT (solid), SCSOPT (dotted), and IPT (dashed) at $T=0.01$ for (a) $U=2$ and (b) $U=3$.

Figure 5

Nonequilibrium total energy for (a) $U=2, t_{on}=19$, (b) $U=2, t_{on}=0$, (c) $U=3, t_{on}=19$, (d) $U=3, t_{on}=0$, versus time calculated using SOPT (solid), SCSOPT (dotted), and IPT (dashed) for $T_0=0.01$.

Figure 6

Double occupancy for the interaction quench with $U=2$ in (red solid) SOPT, (green dotted) SCSOPT, (blue dashed) IPT.

Figure 7

Nonequilibrium (a) current density, (b) total energy, and (c) double occupancy versus time for different values of $T_0$ with $E=1, U=0.25$. Solid black lines correspond to the SOPT with $T_0=0.1$, solid bold red lines correspond to the SOPT with $T_0=1$, and dotted lines to the IPT calculation with $T_0=0.1$.

Figure 8

Nonequilibrium (a) double occupancy and (b) current density versus time in the self-consistent SOPT for different values of $U$ for a pulsed field with $E=1, T_0=0.1, t_f=2$.

Figure 9

Particle distribution function in the self-consistent SOPT at $t=100$ for a pulsed field with $E=1, U=0.25, T_0=0.1, t_f=1.625$.

Figure 10

$\mathrm{Im}{G}_{\varepsilon \overline{\varepsilon }}^K(t_1,t_2)$ as a function of two times $t_1,t_2$ for the pulsed field with $E=1, U=0.25, T_0=0.1, t_f=3$, and with fixed $\varepsilon =-2.625$ and $\overline{\varepsilon }=1.125$.

Figure 11

Double occupancy versus the length of the $E$ pulse $t_f$ in the self-consistent SOPT measured at $t=50$ for $T_0=0.1, E=1, U=0.26$ (dashed black), and $U=1$ (bold red).