Nonlinear response to electric field in extended Hubbard models

The electric-field response of a one-dimensional ring of interacting fermions, where the interactions are described by the extended Hubbard model, is investigated. By using an accurate real-time propagation scheme based on the Chebyshev expansion of the evolution operator, we uncover various nonlinear regimes for a range of interaction parameters that allows modeling of metallic and insulating (either charge density wave or spin density wave insulators) rings. The metallic regime appears at the phase boundary between the two insulating phases and provides the opportunity to describe either weakly or strongly correlated metals. We find that the fidelity susceptibility of the ground state as a function of magnetic flux piercing the ring provides a very good measure of the short-time response. Even completely different interacting regimes behave in a similar manner at short time scales as long as the ground-state fidelity susceptibility is the same. Depending on the strength of the electric field we find various types of responses: persistent currents in the insulating phase, a dissipative regime, or damped Bloch-like oscillations with varying frequencies or even irregular in nature. Furthermore, we also consider the dimerization of the ring and describe the response of a correlated band insulator. In this case the distribution of the energy levels is more clustered and the Bloch-like oscillations become even more irregular.

Figure 1

Ground-state fidelity susceptibility for a ring with $L=10$ and $N_{\uparrow }=N_{\downarrow }=5$ at ${\varphi }_{\mathrm{anti}}=0.1\pi$ as a function of interactions. The inset shows the ground-state fidelity susceptibility as a function of $\varphi /2\pi$ for different sets of parameters. The points represent specifically chosen pairs of parameters $U,V$ in order to model a weakly interacting metal (circle), a SDW insulator (triangle), a CDW insulator (upside down triangle), and a strongly interacting metal (diamond).

Figure 2

Current as a function of time for a ring with $U=1.5$, $V=0.82$, $L=10$, $N_{\uparrow }=N_{\downarrow }=5$, and for different electric field strengths. Inset: The frequency of the BO for different electric fields and the same parameters of the main graph with ${\omega }_b=F$.

Figure 3

(a) Eigenvalues of the instantaneous Hamiltonian as function of time for a system with $L=6$ at half filling, $U=1.5$, $V=0.82$, $F=0.4$. The colors and the size of the points are given by the overlap of the time-dependent wave function with the instantaneous eigenstates, $|\langle n(\varphi \left(t\right)\left)\right|\psi \left(t\right)\rangle |$. (b) The same as (a) but with a dimerization parameter $\eta =0.4$ and $F=4.0$.

Figure 4

(a) ${\left|\langle n\left(\varphi \left(t\right)\right)|\psi \left(t\right)\rangle \right|}^2$ for $L_{\mathrm{sites}}=8$ and $N_{\uparrow }=N_{\downarrow }=4,U=1.5,V=0.82$, and $F=0.025$ at $tF=2.505$ as a function of the current of each state; $\left|n\right(\varphi \left(t\right))\rangle$ are the eigenstates of the instantaneous Hamiltonian. (b) ${\gamma }_n=\arg \left[\langle n\left(\varphi \left(t\right)\right)|\psi \left(t\right)\rangle \right]$ for the same parameters.

Figure 5

Current as function of time for a dimerized ring with $L_{\mathrm{sites}}=10$ and $N_{\uparrow }=N_{\downarrow }=5,U=1.5,V=0.82,\eta =0.4$ (see the definition of the hopping parameter following Eq. (1)) and for different electric field strengths. The inset shows the frequency of the Bloch oscillations for different electric fields and the same parameters of the main graph with ${\omega }_b=F$.

Figure 6

(a) Current as function of time for very small field $F=0.002$ for different interactions. (b) The energy dispersion of the first three excited states of the instantaneous Hamiltonian together with the overlap of these states with $\left|\psi \right(t)\rangle$ as function of time for $U=4.0$, $V=2.16$. The inset of panel (b) shows a zoom-in of the anticrossing region. Colors represent the overlap of the time-dependent wave function with the instantaneous eigenstates.

Figure 7

Current as a function of time for different interactions and different field strengths. The inset shows the square of the overlap of $\left|\psi \right(t)\rangle$ with the instantaneous ground state of $\widehat{H}\left(\varphi \left(t\right)\right)$ for different interactions and $F=0.2$.

Figure 8

Current as a function of time for different interactions and different field strengths and different sizes of the system.

Figure 9

(a) CDW order parameter as a function of time for a system with $L=12$ at half filling derived with $F=0.2$. (b) SDW order parameter as function of time and the same parameters as in plot (a).