Nonequilibrium Dynamical Mean-Field Theory

The many-body formalism for dynamical mean-field theory is extended to treat nonequilibrium problems. We illustrate how the formalism works by examining the transient decay of the oscillating current that is driven by a large electric field turned on at time $t=0$. We show how the Bloch oscillations are quenched by the electron-electron interactions, and how their character changes dramatically for a Mott insulator.

Figure 1

Kadanoff-Baym-Keldysh contour for the two-time Green’s functions in the nonequilibrium case. We take the contour to run from $-5$ to $t_{\max }$ and back, and then extends 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.

Figure 2

Scaled nonequilibrium current for different values of $U$. Note how the Bloch oscillations are rapidly reduced in amplitude as the scattering increases.

Figure 3

Nonequilibrium current for (a) $U=0.5$ ($\Delta t=0.1$) and (b) $U=1$ (scaled from $\Delta t=0.1$ and 0.0667) and longer times. Note how the Bloch oscillations are still present but become more erratic at large times. The current is multiplied by either 6 or 10 to enhance it at large times.

Figure 4

Nonequilibrium current for $U=2$ (the tail is multiplied by 10 and shifted by 0.02). Note how the “Bloch oscillations” are now quite irregular.