Many-body localization in a finite-range Sachdev-Ye-Kitaev model and holography

We study level statistics of a generalized Sachdev-Ye-Kitaev (SYK) model with two-body and one-body random interactions of finite range by exact diagonalization. Tuning the range of the one-body term, while keeping the two-body interaction sufficiently long ranged, does not alter substantially the spectral correlations, which are still given by the random matrix prediction typical of a quantum chaotic system. However, a transition to an insulating state, characterized by Poisson statistics, is observed by reducing the range of the two-body interaction. Close to the many-body metal-insulator transition, we show that spectral correlations share all features previously found in systems at the Anderson transition and in the proximity of the many-body localization transition. Our results suggest the potential relevance of SYK models in the context of many-body localization. It also offers a starting point for the exploration of a gravity dual of this phenomenon which we speculate to be related to the Hawking-Page transition.

Figure 1

Top: Level spacing distribution $P\left(s\right)$ for $N=30$, $\kappa =1$, $d=2$ and different cutoffs $D$. A 10th-order polynomial fitting has been employed to unfold the entire spectrum. We observe a transition from the GUE prediction [64] to Poisson statistics $P_P\left(s\right)=e^{-s}$ as $D$ decreases at $D_c\approx 5.6$. Inset: Qualitatively similar results are observed for $d\to \infty$. Therefore reducing the range of the one-body random term does not affect qualitatively level statistics. Bottom: Average adjacent gap ratio $\langle r\rangle$ as a function of $D$ for different $N$'s and $d=2$ [65]. The observed crossing point is a signature of a metal-insulator transition. Inset: Similar results are observed for $d=\infty$.

Figure 2

Top: $P\left(s\right)$ for $d=2$ and different $D\sim D_c\approx 5.6$. $P\left(s\right)$ does not depend on $N$, which is a feature of level correlations at the metal-insulator transition [55]. We observe level repulsion, typical of quantum chaos, for $s\ll 1$ while the decay is exponential for $s\gg 1$ as for Poisson statistics though with a different slope. Middle: Number variance ${\mathrm{\Sigma }}^2\left(L\right)$ Eq. (6) for $N=30$. It is asymptotically linear with a slope $\chi \sim 0.8$. Bottom: Spectral form factor $g\left(t\right)$ [Eq. (7)] for $N=30$, $\beta =0.001$, and $D\sim D_c$. No ramp or dip is observed for $D=5.1$, which is typical of an insulator, while for $D=5.8$ both a dip and small ramp start to form. This is consistent with a transition between these two values and with a critical $g\left(t\right)$ similar to that for an insulator. The latter is directly related to the linear, instead of logarithmic, growth of the number variance. All of these features have been previously found in systems metal-insulator transitions induced by disorder [19, 55, 56, 57]. In all cases, we have taken only the central $20\%$ part of spectrum and the unfolding has been carried out by the splines method that fits locally consecutive subsets of many ($>10$) eigenvalues with low-order polynomials.