Sachdev-Ye-Kitaev model: Non-self-averaging properties of the energy spectrum

The short time (large energy) behavior of the Sachdev-Ye-Kitaev model (SYK) is one of the main reasons for the growing interest garnered by this model. True chaotic behavior sets in at the Thouless time, which can be extracted from the energy spectrum. In order to do so, it is necessary to unfold the spectrum, i.e., to filter out global tendencies. Using a simple ensemble average for unfolding results in a parametically low estimation of the Thouless energy. By examining the behavior of the spectrum as the distribution of the matrix elements is changed into a log-normal distribution, it is shown that the sample-to-sample level spacing variance determines this estimation of the Thouless energy. Using the singular value decomposition method, which filters out these global sample-to-sample fluctuations, the Thouless energy becomes parametrically much larger, essentially of the order of the band width. It is shown that the SYK model is non-self-averaging even in the thermodynamic limit which must be taken into account in considering its short time properties.

Figure 1

The nearest-neighbor level spacing statistics as manifested in the behavior of the ratio statistics $r_s$ for different system sizes $L$ of the CSYK model with a box distribution are indicated by the black circles. The number of fermions is $N=L/2$ for even $L$ and $N=(L+1)/2$ for odd $L$. The $r_s$ values expected for GOE (dashed red), GUE (dashed green), and GSE (dashed blue) are marked. The expected fourfold symmetry is seen.

Figure 2

The level number variance, $\langle {\delta }^{2}n\left(E\right)\rangle$, as a function of the energy window $E$. Symbols (black, box; red, $\gamma =1$; blue, $\gamma =1.5)$ represent the variance with local ensemble unfolding, fitted for larger values by $a+b{\langle n\left(E\right)\rangle }^2$ (where $b=1.78\times {10}^{-5}$ for the box distribution, $b=1.25\times {10}^{-3}$ for $\gamma =1$, and $b=4.4\times {10}^{-3}$ for $\gamma =1.5)$. The cyan line is the GOE prediction $\langle {\delta }^{2}n\left(E\right)\rangle =(2/{\pi }^2)ln\left(\langle n\left(E\right)\rangle \right)+0.44$. Insert: Zoom into smaller values of $\langle n\rangle$. Deviations from the GOE behavior are observed. A curve depicting GOE plus a constant of 0.2 corresponds to the dashed cyan line. Using the intersection between the variance and the dashed cyan line to determine the Thouless energy results in $E_{\mathrm{Th}}\sim 95\mathrm{\Delta }$ for the box distribution, $E_{\mathrm{Th}}\sim 13\mathrm{\Delta }$ for $\gamma =1$, and $E_{\mathrm{Th}}\sim 7\mathrm{\Delta }$ for $\gamma =1.5$.

Figure 3

The scree plot of the singular values for the CSYK model with box and log-normal distributions with $p=0.5$ and $\gamma =1$, 1.5, 2, 2.5, 3, 4, and 5 for $L=16$ and $N=8$, with $M=4096$ realizations and $P=4096$ eigenvalues around the middle of the band. The square amplitudes of the singular values ${\lambda }_k$ are indicated by the symbols. The cyan line corresponds to ${\lambda }_k\sim k^{-\alpha }$, with $\alpha =1$, as expected for a spectrum which follows Wigner-Dyson statistics. The lower $k$ modes which capture the nonuniversal global structure of the spectrum deviate from Wigner-Dyson statistics.

Figure 4

The level spacing for the CSYK model with box distribution and different log-normal distribution with $\gamma =1$, 2, 3, 4, and 5 for $L=16$ and $N=8$, for $M=4096$ realizations and $P=4096$ eigenvalues around the middle of the band. The ensemble-averaged level spacing, ${\mathrm{\Delta }}_i$, for the local unfolding method is the same for all samples and is depicted by the black curve. The SVD unfolded level spacing, ${\mathrm{\Delta }}_i^j$, is realization dependent. Five individual realizations for each distribution are shown (red, green, blue, yellow, and brown curves). As $\gamma$ increases the sample-to-sample fluctuation increases. It is clear that the long-range correlations within a sample are very significant.

Figure 5

The SVD unfolded level number variance, $\langle {\delta }^{2}n\left(E\right)\rangle$, as a function of the energy window $E$ for box and log-normal $\gamma =1$, 1.5, 2, 2.5, 3, 4, and 5 distributions, represented by the symbols. The cyan dashed line is the GOE prediction $\langle {\delta }^{2}n\left(E\right)\rangle =(2/{\pi }^2)ln\left(\langle n\left(E\right)\rangle \right)+0.44$. The SVD unfolded number variance departs from GOE predictions at $E_{{\mathrm{Th}}^{*}}\sim 800\mathrm{\Delta }$ for the box and $\gamma =1$ distributions, at $E_{{\mathrm{Th}}^{*}}\sim 500\mathrm{\Delta }$ for $1.5\le \gamma \le 2.5$ distributions, and at $E_{{\mathrm{Th}}^{*}}\sim 300\mathrm{\Delta }$ for $3\le \gamma \le 5$ distributions.

Figure 6

The level number variance, $\langle {\delta }^{2}n\left(E\right)\rangle$, as a function of the energy window $E$ for the box distribution using local ensemble unfolding (black symbols) and SVD unfolding (red symbols) for $L=15, N=8, P=3000$, and $M=3000$. The cyan dashed line is the GUE prediction $\langle {\delta }^{2}n\left(E\right)\rangle =(1/{\pi }^2)ln\left(\langle n\left(E\right)\rangle \right)+0.35$, while the green dashed line follows $a+b{n}^2$, with $b=2.48\times {10}^{-5}$. The different behavior for the two unfolding methods is clear. Inset: The scree plot of the singular values ${\lambda }_k$ indicated by the symbols as functions of $k$. The purple line corresponds to ${\lambda }_k\sim k^{-\alpha }$, with $\alpha =1$.