Sparse Sachdev-Ye-Kitaev model, quantum chaos, and gravity duals

We study a sparse Sachdev-Ye-Kitaev (SYK) model with $N$ Majoranas where only $\sim kN$ independent matrix elements are nonzero. We identify a minimum $k\gtrsim 1$ for quantum chaos to occur by a level statistics analysis. The spectral density in this region, and for a larger $k$, is still given by the Schwarzian prediction of the dense SYK model, though with renormalized parameters. Similar results are obtained for a beyond linear scaling with $N$ of the number of nonzero matrix elements. This is a strong indication that this is the minimum connectivity for the sparse SYK model to still have a quantum gravity dual. We also find an intriguing exact relation between the leading correction to moments of the spectral density due to sparsity and the leading $1/d$ correction of Parisi’s U(1) lattice gauge theory in a $d$-dimensional hypercube. In the $k\to 1$ limit, different disorder realizations of the sparse SYK model show emergent random matrix statistics that for fixed $N$ can be in any universality class of the tenfold way. The agreement with random matrix statistics is restricted to short-range correlations, no more than a few level spacings, in particular in the tail of the spectrum. In addition, emergent discrete global symmetries in most of the disorder realizations for $k$ slightly below one give rise to $2^m$-fold degenerate spectra, with $m$ being a positive integer. For $k=3/4$, we observe a large number of such emergent global symmetries with a maximum $2^8$-fold degenerate spectra for $N=26$.

Figure 1

Two examples of intersections graphs $G$ for the dense SYK model. The left graph represents ${(\genfrac{}{}{0ex}{}{N}{q})}^{-2}{\sum }_{{\alpha }_1{\alpha }_2}(-1{)}^{c_{{\alpha }_1{\alpha }_2}}$ which is the $\eta$ defined in Eq. (16); the right graph represents ${(\genfrac{}{}{0ex}{}{N}{q})}^{-3}{\sum }_{{\alpha }_1{\alpha }_2{\alpha }_3}(-1{)}^{c_{{\alpha }_1{\alpha }_2}+c_{{\alpha }_1{\alpha }_3}+c_{{\alpha }_2{\alpha }_3}}$ which is the $T_6$ defined in Eq. (30).

Figure 2

An intersection graph example of Eq. (37). Note the left-hand side and the first term on the right-hand side have identical graphs but different labeling of the vertices, and hence the left represents $⟨{\eta }_G{⟩}_x$ whereas the first term on the right represents ${\eta }_G$.

Figure 3

The “merge and delete” procedures applied in the example of Eq. (38). We show how to obtain the subleading-moment intersection graphs from the leading-moment intersection graph in Fig. 2. There are three subleading graphs corresponding to the merging of ${\alpha }_1{\alpha }_2$, ${\alpha }_1{\alpha }_3$ and ${\alpha }_2{\alpha }_3$; we have drawn two of them because merging ${\alpha }_1{\alpha }_2$ and merging ${\alpha }_2{\alpha }_3$ result in identical graphs.

Figure 4

The accuracy of subleading Q-Hermite approximation. The sixth and eighth moments for $q=4$ are plotted with varying $N$. The solid lines represent the exact moments; the dashed lines represent the sum of leading and subleading moments, both with Q-Hermite approximation applied.

Figure 5

The accuracy of renormalized Q-Hermite approximation. The sixth and eighth moments for $q=4$ are plotted as a function of $N$. The solid lines represent the exact moments; the dashed lines represent the Q-Hermite approximation applied to the leading moments, but with a renormalized $\eta (k)=\eta +3/(kN)$.

Figure 6

Spectral density $\rho (E)$ obtained from the exact diagonalization of the Hamiltonian Eq. (1) for $N=24$ and 5000 disorder realizations. Left: For $\alpha >3$, a depletion of the eigenvalue density occurs for $E\sim 0$ that increases with $\alpha$. Right: For the critical scaling $\alpha =3$, we observe similar features for $k<1$. For $k>1$, the density is qualitatively similar to the dense SYK model.

Figure 7

Spectral density $\rho (E)$ obtained from the exact diagonalization of the Hamiltonian Eq. (1) for $k=4$ and comparison with the renormalized Q-Hermite prediction Eq. (44) for $N=32$ (left) and $N=26$ (right).

Figure 8

Left: spectral density $\rho (E)$ of the Hamiltonian Eq. (1) for $N=32$ and $k=\frac{3}{4}$ both with (blue points) and without the regularity condition (red points). Right: distribution of the smallest eigenvalue of theses ensembles of 1000 configurations compared to a Gaussian with average given by Eq. (45) and width given by Eq. (50).

Figure 9

Distribution of the ten smallest eigenvalues for an ensemble of 5000 configurations for $N=26$ and $k=1$ imposing the regularity condition (red curves, left) and without imposing the regularity condition (red curves, right). The analytical result given by the black curve has the right width but its average position disagrees with the numerical results.

Figure 10

Spectral density $\rho (E)$ for $N=26$ and an ensemble of 1000 disorder realizations without imposing the regularity condition. The red curve line is the renormalized Q-Hermite result Eq. (44) with $\eta$ a fitting parameter. The agreement is excellent even in the tail of the spectrum (right) where fluctuations are stronger. This is an indication that the sparse SYK model may have a gravity dual even for this large degree of sparseness.

Figure 11

Left: the root mean ensemble fluctuations of eigenvalues versus the eigenvalues for $N=32$, $q=4$ and various values of $k$ as shown in the legend of the figure. Right: The slope of these curves versus $1/k$ is shown in the right figure and is compared to the analytical result $1/\sqrt{kN}$. It is not clear why the point at $k=3/4$ deviates so much.

Figure 12

Left: the nearest neighbor spacing distribution $P(s)$ for $N=26$ and different scalings $p\propto 1/N^{\alpha }$ in the bulk of the spectrum. No regularity condition has been imposed. In agreement with the theoretical prediction, the critical scaling is at $\alpha =3$. For $\alpha >3$, corresponding to a more sparse Hamiltonian, we observe spectral quasi-degeneracy leading to an anomalous peak for small spacings and a gradual approach to Poisson statistics. Right: $P(s)$ for $\alpha =3$, $k=4$ and different values of $N$. We observe an excellent agreement with the predictions of RMT for the different values of $N$ corresponding to different universality classes.

Figure 13

Left: distribution function of the adjacent gap ratio $\rho (r)$ for $k=2$ with and without imposing the regularity condition. In the latter, we observe a peak in the $r\approx 0$ region while in the regular case the agreement with RMT prediction is excellent for all values of $r$. Right: $P(s)$ for $k=4$ with and without the regularity condition. We do not observe any difference even in the tail of the distribution.

Figure 14

The nearest neighbor spacing distribution $P(s)$ for $N=32$ and different $k$ with $p(\genfrac{}{}{0ex}{}{N}{4})=kN$ in linear (left) and logarithmic scale (right). In agreement with the theoretical prediction, we find good agreement with RMT for sufficiently large $k$. As $k$ decreases, we observe a bump for small spacings which suggests that the spectrum starts to develop a twofold degeneracy. For $k=1$, not shown, the degeneracy is exact for some realizations. We shall see that this is due to additional global symmetries induced by the increased sparseness. The regularity condition is imposed.

Figure 15

Distribution function of the adjacent gap ratio $\rho (r)$ in linear (left) and log (right) scale. For $k>1.25$, the agreement with the RMT prediction is excellent with no visible deviations. However, for $k=1.25$, it has a large peak (right plot) at small $r$ which suggests that, even for $k>1$, some disorder realization may have spectral degeneracies. The regularity condition is imposed.

Figure 16

Left: the nearest neighbor spacing distribution $P(s)$, $p(\genfrac{}{}{0ex}{}{N}{4})=kN$, $N=26$ and different values of $k$. Right: the same for $N=34$. Even for $k\gg 1$ spectral correlations deviate strongly from the RMT prediction (GUE). Results for different values of $N$ are qualitatively similar which reinforces the idea that the quantum chaos transition occurs at $k\gtrsim 1$. For $k=1$, we have noticed spectral degeneracies in some of the disorder realizations which we have removed for the calculation of $P(s)$. We do not fully understand the reason why $P(s)$ for $N=34$ and $k=4$ deviates from the RMT prediction more strongly than for smaller $k$.

Figure 17

Left: the nearest neighbor spacing distribution $P(s)$, $p(\genfrac{}{}{0ex}{}{N}{4})=kN$, $k=3/2$ and different values of $N$. Right: log scale. In this critical region, spectral correlations show deviations from the RMT prediction though level repulsion is still clearly observed. The $N$ dependence is relatively weak. These features are qualitatively similar to those of a critical system [76, 77] approaching a quantum chaos transition.

Figure 18

Adjacent gap ratio $⟨r⟩$, Eq. (56) for $p(\genfrac{}{}{0ex}{}{N}{4})=kN$ and different $k$’s corresponding to the lowest $2N$ eigenvalues of the Hamiltonian Eq. (1). As $k\to 1$, the adjacent gap ratio decreases and approaches the Poisson limit. We note that, for $k=1$, the spectrum of some disorder realizations, especially for larger values of $N$, shows a twofold degeneracy which was removed before the calculation of $⟨r⟩$. Emergent global symmetries are the origin of the spectral degeneracy. See text for more detail.

Figure 19

$P(s)$ for $N=34$ and $k=1$. The 90% of the eigenvalues around the center of the spectrum are considered. Each curve represents results for different disorder realizations, namely, without any ensemble average. For comparison, we also include the random matrix prediction for different universality classes and Poisson statistics.

Figure 20

Distribution of the adjacent gap ratios for an ensemble of 5000 disorder realizations of the sparse SYK Hamiltonian with $N=26$, $q=4$, $k=1$ and no regularity condition. The red line shows the analytical value of the adjacent gap ratio of the corresponding ensemble. We note that for convenience we set $\chi =1$ (0) for realizations with (without) chiral symmetry.

Figure 21

Connected spectral form factor $K_c(t)$ in units of the Heisenberg time. Unfolding was carried out by the Q-Hermite method [22]. For $k=4$, we find a large ramp and saturation in excellent agreement with the random matrix prediction. The peak for short time is a well-known finite size effect [22].

Figure 22

Connected form factor for the unfolded spectrum where we have grouped disorder realizations with the same global symmetries. No regularity condition is imposed. Surprisingly, despite the large degree of sparseness, $k=1$, the numerical results follows rather closely the predictions of random matrix theory. For each of the figures the value of $⟨r⟩+\chi$ is within 0.01 equal to the corresponding random matrix theory.

Figure 23

The spectral form factor of the sparse SYK model with $N=26$, $q=4$ and $k=1$ for realizations with adjacent ratio in the interval [0.41, 0.43] with no chiral symmetry in the left panel and with chiral symmetry in the right panel. Also shown are the analytical results for the superposition of $\mathrm{GOE}+\mathrm{GOE}$, $\mathrm{GOE}+\mathrm{GUE}$ and $\mathrm{GUE}+\mathrm{GUE}$; see Eq. (61) with $\alpha =1/2$. No regularity condition is imposed.

Figure 24

Level spacing distribution $P(s)$ for the same spectrum as Fig. 22. We observe both a good agreement with the RMT prediction and, depending on the disorder realization, results corresponding to different universality classes including the chiral random matrix ensembles ($\chi =1$).

Figure 25

Histogram of the 2-logarithm of the degeneracy of the spectrum for $N=26$ and $k=0.75$ both with the regularity condition (left) and without the regularity condition (right). The total number of disorder configurations is 1000.

Figure 26

Left: adjacent gap ratio distribution $\rho (r)$ for $N=26$ and $k=1$, 1000 disorder realizations, for both the bulk and the tail ($2N$ lowest eigenvalues) of the spectrum. The regularity condition is imposed. Right: the same for $P(s)$. We have fixed the nonzero $x_{ijkl}$ so that the system has a global symmetry that leads to a double degeneracy of the spectrum for all disorder realizations. This degeneracy is removed in the calculation of $P(s)$ and $\rho (r)$.