Anomalous Thouless energy and critical statistics on the metallic side of the many-body localization transition

We study a one-dimensional XXZ spin chain in a random field on the metallic side of the many-body localization transition by level statistics. For a fixed interaction, and intermediate disorder below the many-body localization transition, we find that, asymptotically, the number variance grows faster than linear with a disorder-dependent exponent. This is consistent with the existence of an anomalous Thouless energy in the spectrum. In noninteracting disordered metals, this is an energy scale related to the typical time for a particle to diffuse across the sample. In the interacting case, it seems related to a more intricate anomalous diffusion process. This interpretation is not fully consistent with recent claims that for intermediate disorder, level statistics are described by a plasma model with power-law decaying interactions whose number variance grows slower than linear. As disorder is further increased, still on the metallic side, the Thouless energy is gradually washed out. In the range of sizes we can explore, level statistics are scale invariant and approach Poisson statistics at the many-body localization transition. Slightly below the many-body localization transition, spectral correlations, well described by critical statistics, are quantitatively similar to those of a high-dimensional, noninteracting, disordered conductor at the Anderson transition.

Figure 1

Average adjacent gap ratio $\langle r\rangle$ for the centered half of the spectrum for $L=18$ and different values of disorder where $\epsilon =\frac{E-E_{\min }}{E_{\max }-E_{\min }}$, where $E_{\min }$ and $E_{\max }$ are the minimum and maximum eigenvalues in the spectrum. The center of the spectrum is close to WD statistics for $h\le 2.5$, while for $h\ge 3.5$ it is closer to the Poisson statistics prediction. The MBL transition $h=h_c$ at the center of the spectrum occurs between these two values.

Figure 2

Finite-size scaling analysis of the adjacent gap ratio $\langle r\rangle$ after ensemble average. Left: Disorder dependence of the averaged adjacent gap ratio $\langle r\rangle$ (2) for different sizes. The crossing point $h=h_c$ corresponds to the mobility edge of the MBL transition. Right panel: Rescaled adjacent gap ratio so that all curves for different sizes collapse in a single curve. The best fit corresponds to $h_c=3.35\pm 0.05$ and $\nu =0.86\pm 0.13$, where $\nu$ is the critical exponent that controls deviations from spectral scale invariance and also the divergence of the localization length (see Appendix pp2 for more details).

Figure 3

Number variance ${\mathrm{\Sigma }}^2\left(N\right)$ (3) for a fixed size $L=18$ and different disorder strength $h$ across the MBL transition. For small $N$ and weak disorder, we clearly observe a logarithmic growth typical of a disordered metal. For larger $N$, the observed faster-than-linear power-law growth is a signature of the Thouless energy in the system. The saturation observed for even larger $N$ is a consequence of the finite number of levels used to compute ${\mathrm{\Sigma }}^2\left(N\right)$. For stronger disorder $\left(h>2.5\right)$, the number variance seem to approach Poisson statistics for all $N$, as in an Anderson insulator, even on the metallic side of the transition $h<h_c\approx 3.4$. In Sec. 4, we compare deviations from Poisson with the predictions of critical statistics.

Figure 4

Power-law exponent $\alpha$ that controls the asymptotic growth of the number variance ${\mathrm{\Sigma }}^2\left(N\right)\sim N^{\alpha }$ across the transition. We used a fitting function $a+b{N}^{\alpha }$, with $a,b,\alpha$ fitting parameters. For intermediate disorder $\left(h\le 2.5\right)$, the faster-than-linear $\left(\alpha >1\right)$ growth signals the existence of the Thouless energy. This does not agree with the results of [54] (see Fig. 11 for more details). As the MBL is approached, $h_c\approx 3.35$, the growth becomes linear with small deviations likely due to finite-size effects. These results, especially the faster-than-linear growth for $\left(h\le 2.5\right)$, are robust to changes in the fitting interval $[N_i,N_f]$ provided that roughly $N_f-N_i\ge 100$ and $N_i\gg 1,N_f\le 300$.

Figure 5

Size dependence of ${\mathrm{\Sigma }}^2\left(N\right)$ (3) for different disorder strength $h$. Deviations from logarithmic growth start to occur for larger $N$ as size increases. This is another indication that the power-law growth of ${\mathrm{\Sigma }}^2\left(N\right)$ for larger $N$ is a signature of the existence of the Thouless energy in this system. For $h>2.5$, the number variance is almost linear, ${\mathrm{\Sigma }}^2\left(N\right)\approx \chi N$, and scale invariant. It seems to be close to Poisson statistics as $h\to h_c$. No Thouless energy is observed in this region.

Figure 6

Level-spacing distribution $P\left(s\right)$ (5) for $L=18$ as a function of the disorder. A crossover from WD to Poisson statistics is clearly observed as disorder increases.

Figure 7

Size dependence of $P\left(s\right)$ for different disorder strength $h$ in the metallic phase. Well in the metallic phase, the scaling is consistent with that of a disordered metal with a Thouless energy in the spectrum. As the system approaches the MBL transition $h_c\approx 3.4,P\left(s\right)$ becomes scale invariant. $P\left(s\right)$ is closer to Poisson statistics, but the decay is slightly faster.

Figure 8

Power-law exponent $\beta$, which controls the strength of level repulsion, $P\left(s\right)\sim s^{\beta }$ for $s\ll 1$, around the MBL transition as a function of the system size $L$. The exponent $\beta$ decreases as disorder $h$ and $L$ increases. For $L=18$ and $h\sim H\sim h_c$, it is already much smaller than one. This is a strong indication that at the MBL transition, level statistics are close to Poisson statistics $\beta \approx 0$. Larger volumes are needed to fully confirm this prediction.

Figure 9

Level-spacing distribution $P\left(s\right)$ (5) close to the MBL transition $h_c\approx 3.35$. Level statistics are not close to semi-Poisson statistics for any value of $h$. The best fit for $h=3.25$ (red line), very similar to Poisson statistics, agrees with that of the disordered system at the Anderson transition in the $d\gg 1$ limit.

Figure 10

Left: Close to $h_c$, the number variance is well described by critical statistics. It is linear with a slope $1/2<\chi \le 1$, different from semi-Poisson statistics $\left(\chi =1/2,1/3,...\right)$. Right: ${\mathrm{\Delta }}_3\left(N\right)$ (11) for different disorder strength. Close to $h_c$, level statistics is well described by critical statistics ${\mathrm{\Delta }}_3\left(N\right)\approx \chi N/15$. It is linear with a slope $1/2<\chi \le 1$, different from semi-Poisson statistics $\left(\chi =1/2,1/3,...\right)$.

Figure 11

${\mathrm{\Sigma }}^2\left(N\right)$ (3) for different intermediate disorder strength well in the metallic phase. The growth of the number variance is faster than linear, in agreement with the existence of a Thouless energy in the system. In all cases, the fitting interval is $[20,200]$. The best fitting is not very sensitive on the fitting interval provided that the lower limit is $N\gg 1$ and the upper limit $100<N<300$. If the interval is too small, $[3,20]$ (red circles), the best fitting is very sensitive to the fitting interval. Moreover, the predicted slower-than-linear growth (red dots) seems to be an artifact of the fitting procedure. The numerical results (purple line) and the modified fitting (orange squares) clearly point to a much faster growth. We believe that this is the reason for the disagreement with the results of Ref. [54]. For comparison, we also show results for the Gaussian orthogonal ensemble (GOE) where spectral rigidity (4) is observed at all scales.

Figure 12

Comparison between the impact of local (crosses) and global (squares) unfoldings (definitions in Appendix pp1) on the level statistics for $L=18$. The density of states $\rho$ shows local methods are more accurate, but they destroy correlations in the number variance ${\mathrm{\Sigma }}^2\left(N\right)$. Local and global methods agree on the spacing distribution $P\left(s\right)$ and on short-range correlations.