Bimodal entanglement entropy distribution in the many-body localization transition

We introduce the cut-averaged entanglement entropy in disordered periodic spin chains and prove it to be a concave function of subsystem size for individual eigenstates. This allows us to identify the entanglement scaling as a function of subsystem size for individual states in inhomogeneous systems. Using this quantity, we probe the critical region between the many-body localized (MBL) and ergodic phases in finite systems. In the middle of the spectrum, we show evidence for bimodality of the entanglement distribution in the MBL critical region, finding both volume law and area law eigenstates over disorder realizations as well as within single disorder realizations. The disorder-averaged entanglement entropy in this region then scales as a volume law with a coefficient below its thermal value. We discover in the critical region, as we approach the thermodynamic limit, that the cut-averaged entanglement entropy density falls on a one-parameter family of curves. Finally, we also show that without averaging over cuts the slope of the entanglement entropy vs subsystem size can be negative at intermediate and strong disorder, caused by rare localized regions in the system.

Figure 1

Top: Von Neumann entanglement entropies ($S$) for different fixed left cut positions as a function of subsystem size $\ell$ of a single eigenstate for one sample with system size $L=20$ and disorder strength $h=3.0$. Notice that the $S\left(\ell \right)$ curves are in general not differentiable. However, the cut-averaged entanglement entropy [CAEE, defined in Eq. (CAEE)] is a smooth and concave function of $\ell$. Bottom: Typical CAEE [denoted as $\overline{S}\left(\ell \right)$] sampled from different disorder strengths, for periodic Heisenberg chains of length $L=20$. Each curve is generated from a single eigenstate of one disorder realization. The slope of CAEE, abbreviated as SCAEE and defined in Eq. (SCAEE), can directly probe the volume law or area law scaling behavior of an eigenstate.

Figure 2

A typical arrangement of the subsystems to apply the strong subadditivity (SSA) inequality $S\left(A\right)+S\left(B\right)\ge S(A\cup B)+S(A\cap B)$. A continuous subsystem can be characterized sufficiently by its length $\ell$ and the position of its middle point $x$ in one dimension.

Figure 3

A typical arrangement of the subsystems to apply the strong subadditivity inequality $S(A\cup B)+S(B\cup C)\ge S\left(A\right)+S\left(C\right)$. A and C are nonoverlapping, which means $l<L/2$, where $L$ is the length of the entire periodic system.

Figure 4

Disorder-averaged mean entanglement entropy slope (top) and standard deviation of the mean entanglement entropy slope (bottom). The black dashed lines mark the rough intersecting positions of the curves of different system sizes. The red horizontal line for the bottom panel indicates the standard deviation of a uniform (box) distribution on the interval $[0,ln2]$, given by ${\sigma }_{\square }=\frac{ln2}{2\sqrt{3}}$. Note that the maximal variance of a bounded distribution on $[0,ln2]$ is given by a bimodal $\delta$ distribution, yielding ${\sigma }_{=}\frac{ln2}{2}\approx 0.34657$. The maximum can at most saturate at this value.

Figure 5

Variance peak locations at different subsystem ratio $\ell /L$ for various system sizes. The variance peak locations are extracted from standard deviation plots as in Fig. 4, where the discrete $\overline{S}\left(\ell \right)$ are interpolated by cubic splines. The solid lines are linear fit curves to data points. The variance peak location (h) rises with increasing system sizes. It is important to notice that for a particular $L$, the differences between the peak locations at different ratios become smaller as $L$ increases, which is reflected by the decreasing slope of the lines connecting different subsystem sizes.

Figure 6

Probability distribution of SCAEE evaluated at $\ell =L/4$ for systems of size $L=20$ at various disorder strengths. The black vertical line indicates the maximal slope of $ln2$.

Figure 7

System size dependence of the probability distributions of SCAEE evaluated at $\ell /L=1/4$. At weak disorder ($h=1$ and $h=2$), the distribution approaches a Gaussian distribution for large system sizes. For our finite systems, the zone showing critical behavior is roughly at $h=3.0$ and drifts to larger disorder strength for larger systems. We observe a clear signature of an emerging bimodal distribution. In the MBL phase ($h=4$ and $h=8$), the distribution is again unimodal and sharply peaked at zero slope. The black vertical line indicates the maximal slope of $ln2$. Note that the panels in the critical region are shown on a linear scale for clarity, while the other panels are on a logarithmic scale to exhibit the tails.

Figure 8

Discrete entanglement entropy “slope” ${\mathrm{\Delta }}^{-}S(L/4)=S(L/4)-S(L/4-1)$ of the raw entanglement entropy without averaging over all subsystem cuts. The appearance of negative slopes is associated with localized regions [21].

Figure 9

Probability density of the standard deviation of the SCAEE $\partial \overline{S}/\partial \ell$ of eigenstates from the middle of the spectrum ($\epsilon =0.5$) of one disorder realization with $\ell =L/4$. For each value of the disorder strength $h$, the distribution is sampled from $\sim 1000$ disorder realizations. Black curves indicate the mean of the distribution. Red dashed lines mark the largest standard deviations of the mean curves. A comparison of the results for system sizes $L=20$ and $L=12$ shows that for larger systems the states close to the critical point of one disorder realization become more diverse.

Figure 10

Histograms of horizontal slices ($h=2.6$ for $L=12$ and $h=2.8$ for $L=20$) through the mean curves' largest standard deviation points of Fig. 9. In comparison, the one disorder realization distribution for a larger system size has more weight shifted towards higher standard deviation, which indicates a higher likelihood of finding bimodality in a single sample.

Figure 11

SCAEE distribution of a single disorder realization that has the largest standard deviation among the 500 samples at $L=16$. Each sample contains 6000 eigenstates, with disorder strength $h$ corresponding to the variance peak location of SCAEE evaluated at $\ell /L=1/4$ as in Fig. 4. The inset shows SCAEE vs energy density of each eigenstate of this disorder sample. Bimodality is clearly visible, and not likely due to the curvature of mobility edge.

Figure 12

Two-dimensional histogram of CAEE vs SCAEE for systems of size $L=20$ and $\ell =L/4$, at disorder strength $h=2.0,3.0,3.2,4.0$. The mean curves for $L=12,16$ are also shown in the figure. The color bar indicates the bin counts on a logarithmic scale. The red straight lines indicate the upper ($\overline{S}/L\le ln2/4$) and lower ($\overline{S}/L\ge \frac{1}{4}\partial \overline{S}/\partial \ell$) bounds of the entanglement entropy. The mean curves are nearly converged, indicating possible universal behavior.

Figure 13

Two-dimensional histogram of second-order derivative of CAEE vs SCAEE for systems of size $L=20$ and $\ell =L/4$, at $h=2.0,3.0,3.2,4.0$. The mean curves for $L=12,16$ are also shown on the graphs. The color bar indicates the bin counts at a log scale. The mean curves are nearly converged, indicating possible universal behavior.

Figure 14

Standard deviation of vertical slices of CAEE vs SCAEE histograms as in Fig. 12, for systems of size $L=20$ and $\ell =L/4$, at $h=2.0,3.0,3.2,4.0$. The standard deviations decreases with system size. The third and the fourth cumulants (not included) of the vertical slices of the histograms shows that the vertical distributions become more Gaussian with increasing system size.

Figure 15

Standard deviation of vertical slices of second-order derivative of CAEE vs SCAEE histograms as in Fig. 13, for systems of size $L=20$ and $\ell =L/4$, at $h=2.0,3.0,3.2,4.0$. The standard deviations decrease with system size. The third and the fourth cumulants (not included) of the vertical slices of the histograms show that the vertical distributions become more Gaussian with increasing system size.