Figure 1

Kadanoff-Baym-Keldysh contour for the two-time Green’s functions in the nonequilibrium case. We take the contour to run from $t_{\mathit{min}}=-5$ to $t_{\mathit{max}}$ and back, and then extend downward parallel to the imaginary axis for a distance of $\beta$. The field is turned on at $t=0$; i.e., the vector potential is nonzero only for positive times.Reuse & Permissions

Figure 2

(Color online) Scaled nonequilibrium current for different values of $U$ with $E=0.125$. All of these cases are metals in equilibrium. We used a quartic extrapolation formula with five $\Delta t$ values for $U=0.5$ ($\Delta t=0.1$, 0.067, 0.05, 0.04, and 0.033), while we used a quadratic extrapolation formula with three $\Delta t$ values for $U=1$ ($\Delta t=0.025$, 0.02, and 0.0167).Reuse & Permissions

Figure 3

(Color online) Scaled nonequilibrium current for different values of $U$ (0, 0.5, 1, and 1.5) with $E=0.25$. All of these cases are metals in equilibrium except for $U=1.5$ which is a near-critical Mott insulator. We used a quadratic extrapolation formula with three $\Delta t$ values for $U=0.5$ ($\Delta t=0.1$, 0.067 and 0.05, 0.04, and 0.033), $U=1$ ($\Delta t=0.05$, 0.04, and 0.033), and $U=1.5$ ($\Delta t=0.033$, 0.025, and 0.02).Reuse & Permissions

Figure 4

(Color online) Scaled nonequilibrium current for different values of $U$ with $E=0.5$. In panel (a), we show cases that are metals in equilibrium. In panel (b), we show one anomalous metal $(U=1)$ and two Mott insulators ($U=1.5$ and $U=2$); note the reduced scale for the vertical axis vs that in panel (a). We used a quadratic extrapolation formula for $U⩽1$ ($\Delta t=0.1$, 0.067, and 0.05 for $U=0.125$, $U=0.25$, and $U=0.5$; $\Delta t=0.067$, 0.05, and 0.04 for $U=1$), a cubic extrapolation formula for $U=1.5$ ($\Delta t=0.067$, 0.05, 0.04, and 0.033), and a quartic extrapolation formula for $U=2$ ($\Delta t=0.05$, 0.04, 0.033, 0.025, and 0.02).Reuse & Permissions

Figure 5

(Color online) Scaled nonequilibrium current for different values of $U$ with $E=1$. In panel (a), we show cases that are metals in equilibrium ($U=0$, 0.125, 0.25, and 0.5). In panel (b), we show one anomalous metal $(U=1)$ and three Mott insulators ($U=1.5$, 2, and 3); note the reduced scale for the vertical axis vs that in panel (a). We used a quadratic extrapolation formula for $U⩽0.5$ ($\Delta t=0.1$, 0.067, and 0.05), a cubic extrapolation formula for $U=1$ ($\Delta t=0.1$, 0.067, 0.05, and 0.04), a quadratic extrapolation for $U=1.5$ ($\Delta t=0.067$, 0.05, and 0.04), a quartic extrapolation formula for $U=2$ ($\Delta t=0.067$, 0.05, 0.04, 0.033, and 0.025), and a quadratic extrapolation formula for $U=3$ ($\Delta t=0.02$, 0.0167, and 0.0143).Reuse & Permissions

Figure 6

(Color online) Scaled nonequilibrium current for different values of $U$ with $E=2$. In panel (a), we show cases that are metals in equilibrium ($U=0$, 0.125, 0.25, 0.5, and 1). In panel (b), we show Mott insulators ($U=1.5$, 2, 3, and 4); note the reduced scale for the vertical axis versus that in panel (a). We could not fit the $U$ labels onto panel (a). The $U=0$ case (black, amplitude unchanged with time) has no damping to the oscillations. The $U=0.125$ case (red, amplitude steadily decreases for the entire $t$ range in the figure) looks like a damped oscillator, because the beat period is about 50. The $U=0.25$ case (green, amplitude decreases out to $t=25$ then increases) shows one beat period (it has the second largest amplitude near $t=30$). The $U=0.5$ case (blue, third largest amplitude at $t=30$) shows a few beats. The $U=1$ case (magenta, smallest amplitude throughout the figure) is the most irregular looking of the curves. The Mott insulators all show the same irregular oscillations, with decreasing initial amplitude as the interaction strength is increased. We used a quadratic extrapolation formula $U⩽0.5$ ($\Delta t=0.1$, 0.067, and 0.05), a cubic extrapolation formula for $U=1$ ($\Delta t=0.1$, 0.067, 0.05, and 0.04), and for $U=1.5$ ($\Delta t=0.067$, 0.05, 0.04, and 0.033), a quadratic extrapolation formula for $U=2$ ($\Delta t=0.04$, 0.033, and 0.025), a cubic extrapolation formula for $U=3$ ($\Delta t=0.04$, 0.033, 0.025, and 0.0167), and a linear extrapolation for $U=4$ ($\Delta t=0.0167$ and 0.0143).Reuse & Permissions

Figure 7

Unscaled nonequilibrium current $(\Delta t=0.1)$ for $E=1$ and (a) $U=0.125$, (b) $U=0.25$, and (c) $U=0.5$. The longer time cutoff allows us to see the initial development of beat like behavior in the current response.Reuse & Permissions

Figure 8

Unscaled nonequilibrium current $(\Delta t=0.1)$ for $E=2$ and (a) $U=0.125$, (b) $U=0.25$, and (c) $U=0.5$. Here the beats are clearly visible, but one can also see a dephasing at long times, because the beat amplitude does not go to zero.Reuse & Permissions

Figure 9

Local many-body density of states with $U=0.5$ for time $T=95\text{units}$ after the electric field was turned on and (a) $E=0.5$, (b) $E=1$, and (c) $E=2$; the data are from unscaled results with $\Delta t=0.1$. The Wannier-Stark ladder would have delta functions at the Bloch frequencies $nE$. Here the many-body density of states is broadened into bands about the Bloch frequencies with a width of approximately $U$. Note how the central band becomes sharply peaked at $\omega =\pm U∕2$ as $E$ increases. This is the source of the two frequencies separated by $U$ that creates the beats in the current as a function of time.Reuse & Permissions