Finite-size prethermalization at the chaos-to-integrable crossover

TL;DR

The paper addresses how infinite-temperature dynamics evolve across the chaos-to-integrable crossover in a mass-deformed SYK model formed by combining SYK$_4$ and SYK$_2$ terms. By combining analytical scaling arguments with exact numerical simulations, the authors identify a finite-time prethermal plateau on the chaotic side and quantify a finite-size crossover to integrable behavior, including a scaling form for the thermalization time $t_{th} \propto 2^{a/\lambda^{2/5}}$ (with $a$ near 2.2) and a critical coupling $\lambda_c \propto 1/N^{5/2}$. The study also reveals distinct Fock-space dynamics: in the integrable limit the FS distribution decays as a stretched exponential with distance, while in the SYK$_4$ limit the distribution becomes uniform across FS, signaling ergodicity. These results illuminate how prethermalization and FS structure emerge in finite-size quantum dots as one tunes between chaotic and integrable dynamics, with potential relevance to MBL-like phenomena in finite systems.

Abstract

We investigate the infinite temperature dynamics of the complex Sachdev-Ye-Kitaev model (SYK$_4$) complimented with a single particle hopping term (SYK$_2$), leading to the chaos-to-integrable crossover of the many-body eigenstates. Due to the presence of the all-to-all connected SYK$_2$ term, a non-equilibrium prethermal state emerges for a finite time window $t_{th}\propto 2^{a/λ^{2/5}}$ that scales with the relative interaction strength $λ$, between the SYK terms before eventually exhibiting thermalization for all $λ$. The scaling of the plateau with $λ$ is consistent with the many-body Fock space structure of the time-evolved wave function. In the integrable limit, the wavefunction in the Fock space has a stretched exponential dependence on distance. On the contrary, in the SYK$_4$ limit, it is distributed equally over the Fock space points characterizing the ergodic phase at long times.

Figures (5)

• Figure 1: Qualitative dynamical phase diagram of the model Hamiltonian \ref{['ham']} as a function of the relative interaction strength $\lambda$ between the SYK terms. $N$ quantifies the number of the sites in the quantum dot. $t_\text{H}$ denotes the Heisenberg time, and $t_\text{th}$ indicated via the dashed line is the thermalizing time, which depends on $\lambda$.
• Figure 2: Density-density correlator of the SYK model with random single particle perturbations as in Eq. \ref{['ham']}. (upper left) $\mathcal{C}(t)$ for the bare SYK case, i.e., $\lambda=1$, for system sizes $N=10,12,14,16,18$. The curves saturate at a plateau for $t\to\infty$, the saturation value $\mathcal{C}(t=\infty)=\langle \mathcal{C}(t>t_{\text{thresh}})\rangle_t$ is shown in the inset as a function of the system size $N$. (upper right) $\mathcal{C}(t)$ for the bare single particle case, i.e. $\lambda=0$, for the same system sizes. Also, here a plateau forms, whose saturation value is shown in the inset and scales linearly with system size $N$. (lower left) $\mathcal{C}(t)$ for different values of $\lambda=0.0,0.02,0.03,0.04,0.05,0.06,0.07,0.08,0.1$ for system size $N=18$. As a function of $\lambda$ an intermediate plateau forms, which eventually drops to the bare SYK saturation value as long as $\lambda$ is big enough. The dashed horizontal line marks the threshold value $1.5\mathcal{C}(\infty)$ below which we consider the system thermalized, i.e., we extract the thermalization time $t_{th}$ from the intersection of the fitted data (black dashed lines, for details see Appendix). (lower right) Scaling collapse of $\mathcal{C}(t)$ for different system sizes $N=14,16,18$ and $\lambda=0.03,0.04,0.05,0.06,0.07,0.08,0.1$ at long times. The inset shows the thermalization time $t_{th}$ extracted as in the previous figure, together with the estimated behavior as a function of $\lambda$ resulting from the approximate scaling collapse.
• Figure 3: Time dependent probability density as a function of the FS distance to the initial state. The colors indicate the time steps (colorbar), corresponding to the time steps shown in Fig. \ref{['f2']} in the single particle case ($\lambda=0$, left) and in the fully interacting case ($\lambda=1$, right). The yellow data shows the infinite time extrapolation. The inset shows the infinite time extrapolation for different $\lambda$ (blue to red). $N=16$.
• Figure 4: The first moment $\Delta x(t)$ of the distribution $\mathcal{P}_d(t)$ as a function of time for different interaction strengths $\lambda$ (color bar indicates the strength). The curves closely resemble the crossover behavior observed in the density-density correlator, Fig \ref{['f2']}. The colored arrows indicate the thermalization time extracted from Fig. \ref{['f2']} (inset). The system size is $N=18$.
• Figure 5: Finite size integrable-to-chaos crossover at infinite time. The extrapolated $\Delta x_\infty$ is a function of the rescaled interaction strength $\lambda\cdot N^{5/2}$ for different system sizes. The curves intersect approximately at the vertical dashed line (expected crossover point).