Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model

TL;DR

This work provides a comprehensive analysis of the SYK model’s spectral and thermodynamic properties, showing that its spectrum is Gaussian in the large-N limit with a finite-N semicircular tail, and that level statistics exhibit random-matrix behavior across a broad energy range while revealing a finite-size, Thouless-energy–like regime at larger separations. It connects these quantum-chaotic features to gravity-dual expectations by matching low-temperature thermodynamics to AdS2-like behavior and extracting scalable quantities such as S0, E0, q, and c. The results underscore the model’s chaotic dynamics at all timescales and illuminate the role of Clifford-algebra Bott periodicity in determining symmetry classes. The study also sets the stage for further analytic work on two-level correlations and finite-N corrections relevant for holographic interpretations.

Abstract

We study spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model, a variant of the $k$-body embedded random ensembles studied for several decades in the context of nuclear physics and quantum chaos. We show analytically that the fourth and sixth order energy cumulants vanish in the limit of large number of particles $N \to \infty$ which is consistent with a Gaussian spectral density. However, for finite $N$, the tail of the average spectral density is well approximated by a semi-circle law. The specific heat coefficient, determined numerically from the low temperature behavior of the partition function, is consistent with the value obtained by previous analytical calculations. For energy scales of the order of the mean level spacing we show that level statistics are well described by random matrix theory. Due to the underlying Clifford algebra of the model, the universality class of the spectral correlations depends on $N$. For larger energy separations we identify an energy scale that grows with $N$, reminiscent of the Thouless energy in mesoscopic physics, where deviations from random matrix theory are observed. Our results are a further confirmation that the Sachdev-Ye-Kitaev model is quantum chaotic for all time scales. According to recent claims in the literature, this is an expected feature in field theories with a gravity-dual.

Figures (11)

• Figure 1: The fourth and sixth normalized energy cumulant related to the Hamiltonian (\ref{['hami']}) as a function of the system size $N$. The circles correspond to the numerical results obtained by exact diagonalization after spectral and ensemble average. At least a total of $5\times10^5$ eigenvalues were employed for each $N$. The solid line is the analytical prediction for the fourth (left) Eq.(\ref{['moment4']}) and sixth cumulant (right) Eq.(\ref{['moment6']}).
• Figure 2: Spectral density $\rho(E)$ as a function of the energy $E$. The solid line is the analytical prediction valid in the $N \to \infty$ limit. Circles are the numerical spectral density for the largest size $N=34$ for which we can obtain all eigenvalues of the Hamiltonian. Except for the tails, the agreement with the numerical results is very good.
• Figure 3: Ensemble average of the smallest eigenvalue as a function of the system size $N$. For $N \gg 1$ we observe that it decreases linearly with $N$. This is an expected feature for a system of $N$ interacting fermions.
• Figure 4: The fitted values of $\log a$ (left) and $b$ (right), defined in Eq.(\ref{['denfit']}), versus $N$. The lines are the best fits to the data. In the right figure only the points for $N\ge 28$ have been used for the fitting
• Figure 5: The specific heat as a function of the temperature for $N=28$ (left) and $N=36$ (right). The red dots represent the numerical result for the SYK model when specific heat is calculated relative to ensemble average (see Eq. (\ref{['c1']})) while the blue dots show the results where the free energy is calculated relative to the average energy $\bar{E}_p$ for each realization of the ensemble (see Eq. (\ref{['c2']})). The blue curve is a linear fit to the blue dots on the interval [0.0075, 0.015] for $N=36$ and cubic polynomial fit on [0.025,0.05] for $N=28$.