Nonlinear Current Response of an Isolated System of Interacting Fermions

Nonlinear real-time response of interacting particles is studied on the example of a one-dimensional tight-binding model of spinless fermions driven by electric field. Using equations of motion and numerical methods we show that for a nonintegrable case at finite temperatures the major effect of nonlinearity can be taken into account within the linear response formalism extended by a renormalization of the kinetic energy due to the Joule heating. On the other hand, integrable systems show on constant driving a different universality with a damped oscillating current whereby the frequency is related but not equal to the Bloch oscillations.

Figure 1

Real-time response: (a) the kinetic energy $E_k(t)$, and (b) the renormalized current $I(t)/\mathbf{(}{E}_k(t)F\mathbf{)}$ obtained for $L=26$, $V=1.4$, and $W=1$. The electric field $F$ is switched on at $t=0$ and kept constant for $t>0$.

Figure 2

$I(t)$ for $L=26$, $V=1.4$, and $W=1$. Arrows indicate the instances of time, when the electric field is switched on (all panels) and off (upper left panel). Full lines represent results obtained via unrestricted numerical solution, whereas dotted and dashed lines show approximate results of the LR theory, $I_{\mathrm{LR}}(t)$, and the extended LR approach, $I_{\mathrm{ER}}(t)$, respectively.

Figure 3

Current-oscillation frequency $\omega$ (normalized to ${\omega }_B$ of NI fermions) vs field $F$ evaluated for $L=26$, $V=1$, $W=0$ (crosses). Inset: $I$ vs flux $|\varphi |$ (continuous lines). Dashed lines (inset) and squares (main panel) show results from the full diagonalization of the $L=14$ system.

Figure 4

Relation between current $I(t)$ and kinetic energy density $E_k(t)/L$ for parameters as in Fig. 3, for different fields $F/\pi =0.02–0.4$. Shown also are the results for NI fermions.