Entanglement transition through Hilbert-space localization

We study Hilbert-space localization of the many-body dynamics due to ergodicity breaking and analyze this effect in terms of the entanglement entropy and the entanglement spectrum. We find a transition from a regime driven by quantum tunneling to a regime that is dominated by boson-boson interaction, where the latter exhibits ergodicity breaking. Properties of this transition are captured by observation time averaging, which effectively suppresses the large dynamical entanglement fluctuations near the critical point. We employ this approach to the experimentally available bosonic Josephson junction. In this example, the transition from a tunneling regime to Hilbert-space localization reveals clear signatures in the entanglement entropy and entanglement spectrum. Interestingly, the transition point is reduced by quantum effects in comparison to the well-known result of the mean-field approximation in the form of self-trapping. This indicates that quantum fluctuations reduce the classical self-trapping. Different scaling with respect to the number of bosons, $N$, is found in the tunneling and the localization regime: While the entanglement entropy grows logarithmically with $N$ in the tunneling regime, it increases linearly in the localized regime. Our results indicate that entanglement provides a concept for a sensitive diagnosis for the transition from a quantum tunneling regime to Hilbert-space localization.

Figure 1

The entanglement entropy $\mathcal{S}$ for 100 bosons as a function of time $t$ for (a) $U=0.01$ and (b) $U=0.1$. We set $J=1$. As shown in the figure, the entanglement entropy fluctuates strongly in time, which is later suppressed with the observation time average.

Figure 2

The entanglement entropy phase transition with system parameters. (a) Schematic plot of the HSL phase and tunneling phase on the Bloch sphere. (b) Plot of $\mathcal{S}\left(J\right)$ for different $N\mathrm{s}$ with $U=1$. When $J$ is small, corresponding to the HSL phase, $\mathcal{S}$ increases with $J$. The system undergoes the phase transition to the tunneling regime at $J_c$, after which $\mathcal{S}$ is a constant. (c) The competition of $U$ and $J$, where we choose $N=20$ and the colors represent the magnitude of $\mathcal{S}$. (d),(e) The effects of the number of bosons on the phase transition. Specifically, (d) $\mathcal{S}(J,N)$ with $U=1$ and (e) $\mathcal{S}(U,N)$ with $J=1$. The four plots (b)–(e) indicate a clear entanglement phase transition when the system undergoes the transition from the HSL phase to the tunneling phase controlled by the $U,J$, and $N$.

Figure 3

The characterization of the critical transition point $u_c$. (a) Plot of $\mathcal{S}$ vs $J$ rescaled by $N$ with $U=1$. (b) Similarly, $\mathcal{S}\left(UN\right)$ with $J=1$. The phase transition point $u_c\sim 3.7$ for all the realizations. We plot the dependence of the entanglement entropy with $N$, (c) rescaled by $J^{-1}$ with $U=0.4$ and (d) rescaled by $U$ with $J=3$. Again, the entanglement entropy undergoes a phase transition at $u_c\sim 3.7$ and reaches the maximum at this critical value. Hence, $u_c$ is a generic value that characterizes the entanglement phase transition.

Figure 4

Scaling laws for the tunneling and HSL phases. (a) Plot of normalized entanglement entropy vs the number of bosons, $N$, with $u=1$. Here, $u<u_c\sim 3.7$, corresponding to the tunneling phase; $\tilde{\mathcal{S}}$ grows logarithmically with $N$. (c) Different scaling behaviors are found when the system is in the HSL phase, where $\tilde{\mathcal{S}}$ increases linearly with $N$. (b) Plot of $\tilde{\mathcal{S}}(u,N)$ with color representing its magnitude. The scaling law keeps $\tilde{\mathcal{S}}\sim lnN$ when $u<u_c$ and gradually becomes $\tilde{\mathcal{S}}\sim N$ after crossing the transition point $u_c$.

Figure 5

Entanglement spectrum as a function of (a) $U$ and (b) $J$. The transition to the HSL is indicated by the repulsion of the levels, which keeps constant in the tunneling phase and spreads in the HSL phase. Compared with the entanglement entropy, the critical transition of the entanglement spectrum is again characterized by $u_c$.

Figure 6

Evolution of the EE for $N=100$ bosons with $J=1$ and different values of the interaction strength $U$ on different timescales.

Figure 7

Evolution of the EE for different values of $U$, $J=1$, and $N=100$ on the same timescale.

Figure 8

The elements ${\rho }_n^L\left(t\right)$ of the reduced density matrix for different numbers of bosons $N$ (a) Deep in the tunneling regime ($u=1$) at fixed time $t=1000$ and (b) deep in the localized regime ($u=40$), where the maximum of ${\rho }_n^L\left(t\right)$ on the time interval $[0,2000]$ is plotted.