Effect of long-range hopping and interactions on entanglement dynamics and many-body localization

We numerically investigate the dynamics of entanglement in a chain of spinless fermions with nonrandom but long-range hopping and interactions, and with random on-site energies. For moderate disorder in the absence of interactions, the chain hosts delocalized states at the top of the band which undergo a delocalization-localization transition with increasing disorder. We find an interesting regime in this noninteracting disordered chain where the long-time entanglement entropy scales as $S\left(t\right)\sim lnt$ and the saturated entanglement entropy scales with system size $L$ as $S(L,t\to \infty )\sim lnL$. We further study the interplay of long-range hopping and interactions on the growth of entanglement and the many-body localization (MBL) transition in this system. We develop an analogy to higher-dimensional short-range systems to compare and contrast such behavior with the physics of MBL in a higher dimension.

Figure 1

Time evolution of entanglement $S\left(t\right)$ in the long-range noninteracting model at different values of $\alpha$ for strong disorder $\eta \approx 20J$. The chain lengths $L$ are $50,100,200,...,1600$ in all except the middle row where $L=3200$ is also included. The red dotted straight line in the middle row is a guide to the eye for emphasizing the logarithmic growth at $\alpha =2$ in the center.

Figure 2

System-size $\left(L\right)$ dependence of asymptotic entanglement ($S_{\infty }$) in the long-range noninteracting model at $\eta \approx 20J$. $S_{\infty }$ shows an algebraic dependence on $L$ for smaller $\alpha$, as shown by a log-log plot in (a), while it grows as $lnL$ at $\alpha =2$, highlighted by a log-linear plot in (b). Different curves correspond to different values of $\alpha$. In the inset of (a) we show the exponent $\gamma$ of algebraic $L$ dependence ($S_{\infty }\sim L^{\gamma }$) as a function of $\alpha$. The two solid black lines in the inset represent the $\gamma$ values corresponding to an area law in 2D ($1/2$) and 3D ($2/3$) short-range models.

Figure 3

Eigenstate and dynamical entanglement properties in a long-range interacting model at different values of $\alpha (=1,2,3)$ (rows). The first column shows the eigenstate entanglement $S_e$ in the middle of the spectrum as a function of disorder ($\eta$) for $\beta =1$. Time evolution of entanglement $S\left(t\right)$ for $\beta (=1,\infty )$ (second and third columns respectively) for $\eta \approx 20J$. The last column shows the fluctuations in asymptotic dynamical entanglement $\delta S_{\infty }$ with disorder $\eta$ at three $\alpha$ values for $\beta =1$.

Figure 4

A cartoon phase diagram of the spinless fermion model in 1D with long-range hopping at strong disorder without (bottom panel) and with (top panel) interactions decaying with distance $r$ as $U/r^{\beta }$. The dashed blue line at $\alpha \approx 3$ divides the long-range noninteracting model in quasi-AL [with some growth of long-time entanglement, $S\left(t\right)]$ and AL [without any growth of long-time $S\left(t\right)]$ phases, and it divides the long-range interacting model in quasi-MBL (size-dependent eigenstate entanglement $S_e$ and particle transport during long-time dynamics) and MBL (size-independent $S_e$ and the dephasing mechanism) phases. The dashed red line at $\alpha \approx 2$ points to the region of logarithmic growth of long-time entanglement in the long-range noninteracting model. The maximum group velocity $v_g^{\max }$ of quasiparticles in the long-range (noninteracting and interacting) model is unbounded for $\alpha <2$ and bounded for $\alpha >2$. The growth of long-time entanglement in the MBL phase is algebraic for small $\beta$ and logarithmic for large $\beta$.

Figure 5

Single-particle level-spacing ratio ($\tilde{r}$) in the long-range noninteracting model for the highest energy eigenstate (top row), the second highest energy eigenstate (middle row), and the eigenstate in the middle of the spectrum (bottom row) for three values of $\alpha$ ($1.25,1.5,1.75$). The two solid lines in each subplot show the GOE ($\approx 0.536$) and the Poisson ($\approx 0.386$) values for $\tilde{r}$.

Figure 6

Single-particle eigenstate entanglement $S_e$ in the long-range noninteracting model in the highest energy eigenstate (top row), the second highest energy eigenstate (middle row), and the eigenstate in the middle (bottom row) for three values of $\alpha$ ($1.25,1.5,1.75$).

Figure 7

Comparison of entanglement dynamics between a long-range noninteracting model ($J\ne 0,U=0$) and a short-range interacting model ($\alpha ,\beta \to \infty ,J,U\ne 0$). Here (a) interacting delocalized phase ($\eta \approx J$) and (b) many-body localized phase ($\eta \approx 20J$) of the short-range interacting model, and (c) $\alpha =1,\eta \approx 20J$ and (d) $\alpha =2,\eta \approx 20J$ of the long-range noninteracting model. The chain lengths are $6,8,...,16$ in (a) and (b) while those are $50,100,200,...,1600$ in (c) and (d). In the inset of (c) we zoom in to show the length dependence at short times of the long-range model.

Figure 8

Left column: Localization-delocalization as evidenced by the level-spacing ratio $\tilde{r}$ as a function of disorder strength $\eta$. Different rows correspond to growing $\alpha =1,1.5,2,3$ from top to bottom, with different curves in each panel representing different system sizes, $L$. A localization-delocalization transition is clearly evident for all $\alpha >1$. The two solid lines in each panel show the GOE ($\approx 0.536$) and the Poisson ($\approx 0.386$) values for $\tilde{r}$. Right column: thermalization and its breakdown. Shown is the disorder-averaged standard deviation of eigenstate expectation values of a local observable as a function of system size $L$ for different values of disorder strength $\eta$ (different colors) and $\alpha =1,1.5,2,3$ from top to bottom. The local observable used is the number operator in the middle of the chain $n_{L/2}$ and $10\%$ eigenstates near the middle of the spectrum are used to calculate the standard deviation. For $\alpha >1$, a system-size-independent standard deviation at large $L$ indicates breakdown of eigenstate thermalization.

Figure 9

System-size dependence of asymptotic dynamical entanglement ($S_{\infty }$) and particle number fluctuations (${\mathcal{F}}_{\infty }$) in the long-range interacting model for $\eta \approx 20J$ at different values of exponents ($\alpha ,\beta$).