Eigenstate thermalization in the Sachdev-Ye-Kitaev model

TL;DR

This work provides a comprehensive numerical verification of the eigenstate thermalization hypothesis in the complex Sachdev-Ye-Kitaev model, connecting ETH to holographic AdS2 black hole physics. Through exact diagonalization up to N=17, it demonstrates that diagonal operator matrix elements are thermally organized, off-diagonal elements exhibit random-matrix-like fluctuations with an energy scale set by the coupling (E_T ~ J^2), and eigenstate correlation functions closely reproduce their thermal counterparts. The study extends to spectral form factors and four-point OTO functions, showing scrambling behavior consistent with maximal chaos and canonical thermalization in pure states, including special superposition states. The results illuminate how ETH underpins thermalization in holographic contexts and offer nuanced implications for the bulk dual, notably regarding the interpretation of interiors and the role of ensemble averaging. Overall, the paper strengthens the bridge between microscopic quantum chaos, thermodynamics of closed systems, and holographic duality, with concrete metrics like E_T, f_O, and OTO scrambling serving as diagnostic tools.

Abstract

The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS$_2$, the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of the dual theory, i.e. the AdS$_2$ black hole.

Figures (11)

• Figure 3.1: Absolute values of matrix elements $|{\cal O}_{nm}| = \left|\langle n | {\cal O} | m \rangle\right|$ for the single-site number operator ${\cal O} = \hat{n}_N$ at half filling $\nu = \frac{1}{2}$. Left panel: we show the absolute values of matrix elements against their energies $E_n/J$ labelled along horizontal and vertical axes for a single realization at $N=10$. We have checked this behavior for higher values of $N$, and found excellent agreement with ETH expectations. Right panel: Histogram of the remainders $R_{mn}$ for $1000$ realizations at $N=12$. As we see these are accurately fit by a unit width Gaussian with zero mean confirming the ETH ansatz (\ref{['eq.ETHHypothesis']}). Again we have verified this for other accessible values of $N$. Similar results are obtained for models with short-range interactions in beugeling2015off.
• Figure 3.2: Diagonal expectation values for the single-site number operator at site $N$, that is $\hat{n}_N$ at half filling $\nu = \frac{1}{2}$. Top panel: we show a single random realization for increasing Hilbert space dimension corresponding to $N=8, 12, 14$. We see that the on-diagonal expectation values of a single realization approach closer and closer to a smooth curve. Left panel: we show the effect of averaging of the random couplings at given fixed Hilbert space dimension, $N=14$. As expected the on-diagonal values of the ensemble approach closer and closer to a smooth curve. Right panel: we show the limiting curves for the model with fixed Hilbert space dimension corresponding to $N=10,12,14$, averaged over $1000$ realizations.
• Figure 3.3: Off-diagonal values of matrix elements ${\cal O}_{nm}= \left|\langle n | {\cal O} | m \rangle\right|$ for the single-site number operator ${\cal O} = \hat{n}_N$ at half filling $\nu = \frac{1}{2}$. We show the off-diagonal matrix elements against their energies $E_n/J$. Top panel: $N=14$ with raw data in light blue and the running average in dark blue. The inset shows a histogram of relative error between raw data and running average. We see that the histogram is peaked around zero. Left panel: the function $f_{\cal O}(\bar{E}, \omega)$ (we show the running average) for varying Hilbert space dimension corresponding to $N=10,11,12,13,14$. The cross-over from constant to non-constant behavior is identified with the Thouless energy $E_T$. Right panel: scaling of the Thouless energy with average coupling strength $J$. A simple fit gives $E_T \propto J^2$.
• Figure 4.1: Spectral form factor at half filling $\nu = \frac{1}{2}$ for varying number of realizations and two different temperatures at $N=14$. In both cases we see the characteristic decay followed by linear ramp and plateau behavior. Left panel: $\beta = 1$. One sees several partial revivals in the decaying region with a power-law envelope. Right panel: $\beta = 10$. The red markers show points we used for a fit in the slope region, where we find a decay of $\propto t^{-4.53\ldots}$, which is consistent with the value reported in Davison:2016ngz.
• Figure 4.2: Two-point function of the hopping operator at half filling $\nu = \frac{1}{2}$. Top left panel: Two-point correlation function, $G^n_c(t)$, in one eigenstate for $N=8,9,10,11,12$. The initial decay is followed by late time fluctuations around zero of typical size $\sim e^{-S/2}$. Top right panel: Comparison of microcanonical $G^{\bar{E}}_c(t)$ with eigenstate $G^n_c(t)$ at the same energy for $N=10$. One can appreciate the excellent agreement, which would only become better as the number of realizations is increased. Bottom left panel: Comparison of $G^\beta(t), G^n(t), G^\ell(t)$ with parameters $\beta, E(\beta)$ and $2\ell = \beta$ at $N=6$. Bottom right panel: Comparison of $G^\beta_c(t), G^n_c(t), G^\ell_c(t)$ with parameters $\beta$, $E(\beta)$ and $2\ell = \beta$ at $N=6$ (inset $N=10$).