Chaotic-Integrable Transition in the Sachdev-Ye-Kitaev Model

Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions in $0+1$ dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.

Figure 1

(Upper) $P(s)$ for $N=34$ and different $\kappa$’s with $\kappa$ in units of $J=1$. We clearly observe a crossover from Wigner-Dyson (WD) to Poisson statistics as $\kappa$ increases. (Lower) Finite-size scaling analysis of the averaged adjacent gap ratio $⟨r⟩$ [Eq. (4)] as a function of $\kappa$ for different $N$’s. For sufficiently large $N$, we observe a crossing at ${\kappa }_c\approx 66$, which suggests the existence of a chaotic-integrable transition. Results for larger $N$ would be necessary to confirm it. See the main text for an explanation of the absence of crossing for small $N$. Both $P(s)$ and $⟨r⟩$ were computed by using ensemble and spectral average in a window comprising 10% of the eigenvalues around the center of the spectrum. Results are robust to changes in the percentage of eigenvalues provided that the spectrum edges are avoided.

Figure 2

(Upper) Connected spectral form factor $g_c(t)$ for the unfolded spectrum from Eq. (5) for $N=30$, $J=\kappa =1$, and different $\beta$’s. For small $\beta$, we observe the correlation hole [48, 49] followed by a ramp typical of quantum chaotic systems. As $\beta$ increases, which probes the tail of the spectrum, results are not conclusive. (Lower) A finite-size scaling analysis, also with $J=1$, of the average adjacent gap ratio $⟨r{⟩}_{\beta }$ [Eq. (4)], where, unlike the previous figure, the average is weighted by the function $\exp [-\beta (E_i+2E_{i+1}+E_{i+2})/4]$, but the spectrum is not unfolded. Here we have excluded the ten smallest eigenvalues from the analysis because $⟨r⟩$ for such eigenvalues at the spectral tail is anomalous high even in the large-$\kappa$ limit. For details, see the Supplemental Material [41]. For $\beta =0.2$, which probes the low energy part of the spectrum, we observe a crossing at ${\kappa }_c\approx 25$ that seems to indicate a chaotic-integrable transition. However, the size dependence is too weak to confirm the existence of the transition.

Figure 3

Lyapunov exponent ${\lambda }_L$ for the model Eq. (1) with $J=1$ as a function of the inverse temperature $\beta =1/T$ and $\kappa$. From top to bottom, $\kappa =0$, 0.2, 0.5, 1, 2. A finite ${\lambda }_L$, which is a signature of quantum chaos, is observed for a fixed $\kappa$ and not too low temperature. For sufficiently low temperatures, and $\kappa >0$, we identify a $T^{*}(\kappa )$ such that ${\lambda }_L=0$ for any $T<T^{*}(\kappa )$, which signals a chaotic-integrable transition. (Inset) Critical value of $\kappa ={\kappa }_c$ at which the transition takes place as a function of $\beta$. Dots result from fitting the numerical data of the main plot near the transition. The solid line is the analytical expression from Eq. (15) valid in the large-$q$ limit. Agreement with the numerical data is reasonable for $\kappa /J\ll 1$.