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

Abstract

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.

Figures (11)

• 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 $\langle r \rangle$ Eq. (\ref{['eq:agr']}) 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 main text for an explanation of the absence of crossing for small $N$. Both $P(s)$ and $\langle r \rangle$ 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. (\ref{['eq:gt']}) for $N = 30$, $J = \kappa = 1$ and different $\beta$'s. For small $\beta$, we observe the correlation hole alhassid1992Torres-Herrera2017 followed by a ramp typical of quantum chaotic systems. As $\beta$ increases, that 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 $\langle r \rangle_\beta$ Eq. \ref{['eq:agr']}, 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 $\langle r \rangle$ for such eigenvalues at the spectral tail is anomalous high even in the large $\kappa$ limit. For details, see the supplemental material. 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. \ref{['hami']} 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.\ref{['eq:Lyapunov_kappa']} valid in the large-$q$ limit. Agreement with the numerical data is reasonable for $\kappa/J\ll1$.
• Figure 4: Entropy $S$ as a function of temperature $T$ from the exact diagonalisation of Eq.(\ref{['hami']}) for different $N$'s and $\kappa$ where $k_B$ stands for the Boltzmann constant. The number of eigenvalues employed is mentioned in the main text. The weak dependence on $N$ is consistent with a vanishing zero temperature entropy.
• Figure 5: Specific heat $C(T)$ as a function of temperature, in units of $J/\kappa_B$, obtained from Eq.(\ref{['c2']}) and the exact diagonalisation of Eq.(\ref{['hami']}) for $N = 34$ and different $\kappa$'s. The specific heat is clearly linear in the low temperature limit with a slope that it is close to the $\kappa = 0$ prediction $c =\pi/6$ for any $\kappa \leq 1$.