Many-body localization transition in Rokhsar-Kivelson-type wave functions

We construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the random energy model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with the subsystem size: scaling with the entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a subextensive scaling between these limits. Near the phase transition point, the fluctuations of the Rényi entropies are non-Gaussian. We find that Rényi entropies with different Rényi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior.

Figure 1

(a) Phase diagram for the classical REM model as a function of $\beta$. The spin glass phase transition occurs at $\beta =\sqrt{2ln2}$. The shaded region between $\sqrt{ln2/2}$ and $\sqrt{2ln2}$ has non-Gaussian fluctuations. (b) Schematic phase diagram for the REM wave function in terms of the entanglement entropy showing different scaling behaviors in four different regimes as a function of $\beta$. In the regime (i), $\langle S_n\rangle$ is equal to $N_{A}ln2$ for all $N_A$. In the intermediate regime (ii), $\langle S_2\rangle$ is subextensive but does not saturate to a constant. Regime (iii) is the MBL phase. In this regime, the wave function is not localized in the Hilbert space. (iv) is also the MBL phase with the wave function localized in Hilbert space.

Figure 2

(a) The lower and upper bound for $\langle S_2\rangle$. The black dashed curve is for the lower bound and the red solid curve is for the upper bound. The system size $N=300$ and the ratio $t=1/3$. (b) The upper bound for $\langle S_n\rangle$. The setup is the same as (a).

Figure 3

A summary of our knowledge of the phase diagram of ${\mathrm{\Psi }}_{\mathrm{REM}+\mathrm{sign}}$, based on $\langle S_n\rangle$. The dotted black line indicates a bound to the left of which $\langle S_2\rangle$ analytically follows the $T=\infty$ volume law (ITV). The dotted red line indicates a bound to the right of which $\langle S_2\rangle$ analytically is strictly below the $T=\infty$ volume law. ${\beta }_{sg}$ indicates analytically the transition to a localized Hilbert space for any $N_A/N$ and a guaranteed constant for all $\langle S_n\rangle$. The solid purple dots indicate the numerically computed transition points for $\langle S_2\rangle$ (the purple line is a fit for the eye). In comparison, the blue dashed line indicates a bound to the right of which $\langle S_{\infty }\rangle$ analytically is strictly less then ITV, and suggests multifractality.

Figure 4

The numerical results for $\langle S_n\rangle$ with $n$ from 1 to $\infty$ at different values of $\beta$. The total system size is $N=30$. Each point in the plot is averaged over 2000 disorder realizations. At $\beta =0.3, S_n$ shows ITV behavior for all $n$. At $\beta =0.66,1.1, \langle S_n\rangle$ deviates from the ITV with increasing $\beta$, but the speed at which it moves away depends on its Rényi index. At $\beta =1.5, \langle S_n\rangle$ is a constant for all $n$.

Figure 5

$\frac{d\langle S_2\rangle }{d{N}_A}$ vs $\beta$ graph at $t=1/3$, where the slopes are obtained from finite differences. As $\beta$ increases, the slopes drop from $ln2$ towards 0 for all five different system sizes.

Figure 6

(a) Scaling collapse of $\langle S_2\rangle /S_T$ at $t=1/3$, where $S_T=N_{A}ln2$. It can be noticed that $b$ is very close to 0. (b) Scaling collapse of $\delta S_2/S_T$. The left insets show the original curves. Two scaling collapses give very close ${\beta }_c$ values. Error analysis is only performed on the scaling collapse of $\langle S_2\rangle /S_T$.

Figure 7

The distribution of $S_2/S_T$ at $\beta =0.72$. The total number of samples is 2000. The number of bins used to make the histogram is 100. The inset shows a log-log plot with a linear fit, which indicates a power law distribution. The number of bins in the log-log scale is 20 for $S_2/S_T$ from 0 to 1. However, as many of the bins have 0 sample, some points are not included in the log-log plot.

Figure 8

This graph uses the scaling parameters obtained from Fig. 6 to scale the $\langle S_2\rangle /S_T$ vs $N$ curves for $\beta$ values ranging from 0 to 2. The scaled graph clearly shows two branches: (1) curves with large $\beta$ flow to low entanglement entropy and (2) curves with small $\beta$ flow to $T=\infty$ entanglement entropy, i.e., $N_{A}ln2$. The right inset shows the original curves, where each curve corresponds to a different $\beta$.

Figure 9

(a) Scaling collapse of $\langle S_{\infty }\rangle /S_T$ at $t=1/3$, where $S_T=N_{A}ln2$. It can be noticed that $b$ is very close to 0. (b) Scaling collapse of $\delta S_{\infty }/S_T$. The left insets show the original curves. Two scaling collapses give very close ${\beta }_c$ values. Error analysis is only performed on the scaling collapse of $\langle S_{\infty }\rangle /S_T$.

Figure 10

This graph uses the scaling parameters obtained from Fig. 9 to scale the $\langle S_{\infty }\rangle /S_T$ vs $N$ curves for $\beta$ values ranging from 0 to 2. The scaled graph clearly shows two branches: (1) curves with large $\beta$ flow to low entanglement entropy and (2) curves with small $\beta$ flow to $T=\infty$ entanglement entropy, i.e., $N_{A}ln2$. The right inset shows the original curves, where each curve corresponds to a different $\beta$.

Figure 11

${\beta }_c$ vs $\frac{n-1}{n}$ curve at $t=1/3$. The linear fit line's slope is $-0.3213\pm 0.0034$, and its $y$ intercept is $0.9709\pm 0.0023$.

Figure 12

Entanglement spectrum for one randomly chosen disorder configuration at $\beta =0.3,0.66,1.1$, and $1.5$, with $N=30$ and $N_A=10$. The gaps in the entanglement spectrum become evident when $\beta$ gets larger.

Figure 13

$\langle S_2\rangle$ of REM+sign and REM, and their difference. The 1000 disorder configurations are the same for REM+sign and REM. Error bars here represent standard deviations. The difference of the $\langle S_2\rangle$ has well bounded variance for all $\beta$.

Figure 14

Difference of $\langle S_n\rangle$ between REM+sign and REM for $n=0.1,...,0.9$ in $0.1$ steps (triangles), and for $n=1.5, n=2,...,20$, and $\infty$ (full circles), at $N=30$ and $N_A=10$. The number of disorder configurations is 2000. The random configurations are not the same for REM+sign and REM.

Figure 15

Different behaviors of $\mathrm{\Delta }{S}_2=\langle S_2(\mathrm{REM}+\text{sign})\rangle -\langle S_2\left(\mathrm{REM}\right)\rangle$ on the $t$ vs $\beta$ graph. Region A represents the area where $\langle S_2(\mathrm{REM}+\text{sign})\rangle$ follows ITV and $\langle S_2\left(\mathrm{REM}\right)\rangle$ is a constant, so $\mathrm{\Delta }{S}_2$ obeys ITV. Region B represents the area where $\langle S_2(\mathrm{REM}+\text{sign})\rangle$ and $\langle S_2\left(\mathrm{REM}\right)\rangle$ are both ITV, and $\mathrm{\Delta }{S}_2$ is zero. As a result, $\mathrm{\Delta }{S}_2$ is also zero in region C based on the monotonicity argument. In region D, both $\langle S_2(\mathrm{REM}+\text{sign})\rangle$ and $\langle S_2\left(\mathrm{REM}\right)\rangle$ are localized in Hilbert space and $\mathrm{\Delta }{S}_2$ is zero. We do not have enough information to determine the behavior $\mathrm{\Delta }{S}_2$ in region E.

Figure 16

The numerical results for $\langle S_n^{\prime }\rangle$ without a random sign with $n$ from $1.5$ to $\infty$ at $\beta =0.4$. The total system size is $N=30$. Each point in the plot is the average of 2000 disorder realizations.