Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N

TL;DR

The paper analytically determines the spectral density of the q-body SYK model at finite N by exact evaluation of energy moments and mapping to Q-Hermite polynomials with a nontrivial Q = η_{N,q}. It then derives a simple large-N asymptotic form for the bulk density and a distinct edge behavior, validating these results against extensive exact diagonalization up to N ≈ 34. In the infrared, the density exhibits random-matrix–like square-root edges and exponential growth, with level statistics in agreement with GOE/GSE and Tracy-Widom behavior for the ground state. The findings support universal random-matrix correlations as a feature of quantum black holes and suggest holography-based models can capture key aspects of complex nuclear dynamics.

Abstract

We show analytically that the spectral density of the $q$-body Sachdeev-Ye-Kitaev (SYK) model agrees with that of Q-Hermite polynomials with Q a non-trivial function of $q \ge 2$ and the number of Majorana fermions $N \gg 1$. Numerical results, obtained by exact diagonalization, are in excellent agreement with the analytical spectral density even for relatively small $N \sim 8$. For $N \gg 1$ and not close to the edge of the spectrum, we find the macroscopic spectral density simplifies to $ρ(E) \sim \exp[2\arcsin^2(E/E_0)/\log η]$, where $η$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions. This spectral density reproduces the known result for the free energy in the large $q$ and $N$ limit. In the infrared region, where the SYK model is believed to have a gravity-dual, the spectral density is given by $ρ(E) \sim \sinh[2π\sqrt 2 \sqrt{(1-E/E_0)/(-\log η)}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low energy nuclear excitations. Results for level-statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to model certain aspects of the nuclear dynamics.

Figures (5)

• Figure 1: In this figure we compare $Q$-Hermite spectral density $\rho_{\rm QH}(E)$ Eq. (\ref{['eden']}) (black), of the SYK Hamiltonian to two different asymptotic forms, $\rho_{\rm Bethe}(E)$ Eq.(\ref{['rhobethe']}) (red dashed) and $\rho_{\rm asym}(E)$ Eq.(\ref{['rhosim']}) (blue dotted) all normalized to area one. Results are given for $N=18$, $N=24$, $N=32$ and $N=64$. For $N \geq32$ the three curves are barely distinguishable. In all plots the spectral density is normalized to $1$ and $J=2/3$. We note that this is also the value of $J$ in our previous paper garcia2016.
• Figure 2: Comparison of the numerical spectral density (red) of the SYK Hamiltonian Eq. (\ref{['hami']}) for $N=16$, $N=24$, $N=32$ and $N=34$, obtained by exact diagonalization, with the analytical prediction $\rho_{\rm QH}(E)$ Eq. (\ref{['eden']}) (black). The agreement is excellent. Even though there is no free parameters the curves are almost indistinguishable. As in the previous figure the spectral density is normalized to 1 and $J = 2/3$.
• Figure 3: The tail of the spectral density for $N=32$ and $400$ disorder realizations. In the right figure, $E_0-\langle E_0 \rangle$ has been subtracted from all eigenvalues, while in the right figure no subtractions have been made. The agreement is excellent despite the fact finite $N$ effects, not fully captured in our theoretical analysis, should be stronger in this region. Even without this subtraction the agreement is still very good.
• Figure 4: Distribution of the lowest eigenvalue for $N = 24$ (left) and $N=28$ (right) for an ensemble of $50,000$ and $15,000$ disorder realizations, respectively, compared to the random matrix prediction for the Tracy-Widom distribution. The numerical data have been shifted and rescaled to reproduce the average and variance of the Tracy-Widom distribution. The agreement is excellent which confirms that the low energy limit of the SYK model is fully ergodic and well described by random matrix theory.
• Figure 5: Level spacing distribution $P(s)$ resulting from exact diagonalization of the SYK Hamiltonian Eq. (\ref{['hami']}) for $N = 32$ and $400$ realizations (squares) and $N = 24$ and $10000$ realizations (circles). We only consider the infrared part of the spectrum, about $1.5\%$, which is related to the gravity dual of the model. As in the bulk of the spectrum you2016garcia2016, we observe excellent agreement with the Gaussian Orthogonal Ensemble (GOE) result. This strongly suggests that full ergodicity, typical of quantum systems described by random matrix theory, is also a universal feature of quantum black holes.