On thermalization in the SYK and supersymmetric SYK models

TL;DR

The paper numerically investigates eigenstate thermalization in the SYK model and its N=1 supersymmetric extension using exact diagonalization. It tests few-body operators and finds diagonal elements align with microcanonical averages while off-diagonal fluctuations obey Gaussian statistics, providing strong evidence for ETH in both models. The results have implications for holographic duality, black hole thermalization, and quantum information aspects such as scrambling and random-matrix-like behavior. Additionally, pure-state constructions reproduce thermal correlators, linking ETH to the thermal properties of low-energy pure states in these holographic quantum-mechanical systems.

Abstract

The eigenstate thermalization hypothesis is a compelling conjecture which strives to explain the apparent thermal behavior of generic observables in closed quantum systems. Although we are far from a complete analytic understanding, quantum chaos is often seen as a strong indication that the ansatz holds true. In this paper, we address the thermalization of energy eigenstates in the Sachdev-Ye-Kitaev model, a maximally chaotic model of strongly-interacting Majorana fermions. We numerically investigate eigenstate thermalization for specific few-body operators in the original SYK model as well as its $\mathcal{N}=1$ supersymmetric extension and find evidence that these models satisfy ETH. We discuss the implications of ETH for a gravitational dual and the quantum information-theoretic properties of SYK it suggests.

Figures (12)

• Figure 1: We plot the density of states, on the left for the SYK model and on the right for the $\mathcal{N}=1$ supersymmetric SYK model, for $N =$ 12, 16, 20, 24 and 28 Majoranas, where we take 25600 $(N=12)$, 6400 $(N=16)$, 1600 $(N=20)$, 400 $(N=24)$, and 100 $(N=28)$ realizations of disorder. The above data is collected from the full spectrum.
• Figure 2: Eigenstate thermalization density plots for the particle number operator in the original SYK model. We choose a single realization of the model with $N=$ 12, 16, 20, 24, and 28. The horizontal and vertical axes denote the energy eigenvalues for the corresponding matrix elements, while the value of the density is the complex norm of the matrix element. Given the rapid growth in the number of eigenvalues, we downsample the $N=24$ and $N=28$ data to $512\times512$. Considering the possible degeneracies, we average the doubled data points with the same energy eigenvalues.
• Figure 3: Eigenstate thermalization density plots for the particle number operator in the supersymmetric SYK model. The setup is the same as described in Figure \ref{['syk_cc']}.
• Figure 4: Eigenstate thermalization density plots for single hopping operators in the original SYK model, with the same setup as described in Figure \ref{['syk_cc']}. Note that the microcanonical value of the hopping operator is zero, so agreement with ETH is simply the appearance of the off-diagonal Gaussian fluctuations.
• Figure 5: Eigenstate thermalization density plots for single hopping operators in the supersymmetric SYK model. The setup is the same as described in Figure \ref{['syk_cc']}.