Characteristic length scales from entanglement dynamics in electric-field-driven tight-binding chains

We study entanglement dynamics in the nearest-neighbor fermionic chain that is subjected to both dc and ac electric fields. The dynamics gives the well-known Bloch oscillations in the dc field case provided that the system size is larger than the Bloch length, whereas in the ac field case the entropy is bounded and oscillates with the driving frequency at the points of dynamic localization, and has a logarithmic growth at other points. A combined ac+dc field yields super-Bloch oscillations for system sizes larger than the super-Bloch length, which puts a constraint on the device size in a typical nonequilibrium setup to observe super-Bloch oscillations where the device is connected to the leads. Entanglement entropy provides useful signatures for all of these phenomena, and an alternate way to capture the various length scales involved.

Figure 1

Mean square single-particle width of the wave packet at $t=5T_B$ (top) and $t=2T_{\text{SB}}$ (bottom) as a function of system size rescaled to Bloch length and super-Bloch length, respectively. The transition shows the requirement of a device size greater than Bloch length and super-Bloch length to observe the oscillations. (Parameters: $A=2.0$ and $\omega =1.0$.)

Figure 2

Entanglement entropy as a function of $L/L_B$ and $L/L_{\text{SB}}$ for a dc field (top) and combined ac+dc field with slight detuning (bottom) at $t=5T_B$ and $t=3T_{\text{SB}}$ ($A=2.0$ and $\omega =1.0$). The initial state is a half-filled state where all the particles are filled on the left half of the chain. The entropy becomes zero when the system size is larger than the Bloch length and super-Bloch length, respectively.

Figure 3

Mean square width of an initially localized wave packet for an ac driving. Mean square width is unbounded for arbitrary $A/\omega$ (left), whereas it is bounded and periodic when $A/\omega$ is a root of the Bessel function of zeroth order (center) (parameters are $L=100$ and $\omega =1.0$). Entanglement entropy (right) for a half-filled localized state. For $\omega =0.0$, the entropy shows Bloch oscillations, whereas for other values the entropy is bounded/unbounded depending on the ratio of $A/\omega$. (Parameters are $L=100$, $J=0.5$, and $A=0.5$ for all curves except the blue one.)

Figure 4

Destruction of Wannier-Stark localization (left) for resonantly tuned driving $F=n\omega$, $n=1,2,...$, and dynamic localization (center) under the action of both static electric field and ac driving from the dynamics of entanglement entropy. Super-Bloch oscillations captured by entanglement entropy for a detuning from the resonant driving (right). The parameters are $L=100$ for all plots $J=0.5$ for (a), (b) and $A=3.0$, $J=0.5$ for (c).