Thermalization in the mixed-field Ising model: An occupation number perspective

TL;DR

The paper addresses how thermalization emerges in a chaotic quantum many-body spin-1 system and establishes a quantitative bound on the quantum relaxation rate by leveraging a large-scale classical limit. It shows GOE-like spectral statistics and ETH validity in the quantum model, while demonstrating that classical ergodicity is approached algebraically with system size, providing a robust lower bound for the quantum case. The occupation-number observable serves as a transparent probe for both quantum and classical thermalization, with a quantitative quantum–classical correspondence evidenced by matched central-spin dynamics and energy distributions. Overall, the study bridges microscopic eigenstate structure with macroscopic thermodynamic behavior and highlights a power-law route to equilibrium instead of the often-anticipated exponential convergence in chaotic quantum systems.

Abstract

The occupation number is a key observable for diagnosing thermalization, as it connects directly to standard statistical laws such as Fermi--Dirac, Bose--Einstein, and Boltzmann distributions. In the context of spin systems, it represents the population of the sublevels of the magnetization in the $z$-direction. We use this quantity to probe the onset of thermalization in the isolated quantum and classical one-dimensional spin-1 Ising model with transverse and longitudinal fields. Thermalization is achieved when the long-time average of the occupation number converges to the microcanonical prediction as the chain length $L$ increases, consistent with the emergence of ergodicity. However, the finite-size scaling analysis in the quantum model is challenged by the exponential growth of the Hilbert space with $L$. To overcome this limitation, we turn to the corresponding classical model, which enables access to much larger system sizes. By tracking the dynamics of individual spins on their three-dimensional Bloch spheres and employing tools from random matrix theory, we establish a quantitative criterion for classical ergodicity in interacting spin systems. We find that deviations from classical ergodicity decay algebraically with system size. This power-law scaling then provides a quantitative bound on the approach to thermal equilibrium in the quantum model.

Figures (10)

• Figure 1: (a)-(b) Analysis of level statistics using (a) ratio of consecutive levels and (b) level number variance, in the odd parity sector. (c)-(f) Participation ratio of the eigenstates in the $z$-basis as a function of the eigenvalues, in both parity sectors. The parameters in (b)-(f) are indicated as $(g,h)$. All panels: $L=9$.
• Figure 2: Study of the eigenstate thermalization in the chaotic regime: $g=0.1$ and $h=0.65$. (a) Eigenstate expectation values of $S^z_{c}$ at the center of the chain, $c= \lceil L/2 \rceil$, as a function of the eigenvalues for $L=7$ (cyan) and $L=9$ (blue). (b) Distribution of the off-diagonal elements of $S^z_{c}$ for 200 eigenstates in the middle of the spectrum, $L=9$; the red curve represents a Gaussian distribution. (c)-(d) Extremal fluctuations of $\langle \alpha |S^z_{c}|\alpha \rangle$ and $\langle \alpha |S^x_{c}|\alpha \rangle$ as a function of system size for (c) a fixed energy window $\delta E = 0.1$ and (d) 40 eigenstates in the middle of the spectrum. Both parity sectors in (a), (c)-(d); odd parity sector in (b).
• Figure 3: (a) Main panel and inset: Time evolution of the single particle occupation number for $m=-1,0,1$ (green, red and blue curves, respectively ); $L=9$. Initial state is $\ket{\Psi(0)}= |000010000\rangle$. Horizontal dashed lines are the time averages (diagonal ensemble). (b)-(c): Normalized extremal fluctuations of the eigenstate expectation values of $n_m^z$ as a function of system size for (b) a fixed energy window $\delta E = 0.1$ and (c) $\mathcal{N} = 40$ eigenstates in the middle of the spectrum from both symmetry sectors. All panels: $g=0.1$, $h=0.65$.
• Figure 4: (a) Average maximal Lyapunov exponent as a function of $h$ for different values of $g$; $L=9$. The vertical arrow indicates the values chosen for the studies of the quantum and classical models. (b) Average maximal Lyapunov exponent as a function of $L$ for $g=0.1$ and $h=0.65$. Averages over $10^3$ initial conditions.
• Figure 5: Distribution of single spin components $S_j^{x,y,z}$ for (a)-(c) $L=9$ and (d)-(f) $L=149$. Each color stands for a spin $j=1,...,L$. As initial conditions, we consider $10^4$ trajectories, all with $S_c^z(0) = \hbar_{\text{eff}} S$ and $S_j^z(0) = 0$ for $j\ne c$ (see text). We sample $10^4$ times in increments $dt=1$ after saturation at $t=10^3$. The number of histogram bins is $M=10^2$ between the spin values -1 and +1 and the parameters are $g=0.1$ and $h=0.65$.