Finite-Size Scaling of the Full Eigenstate Thermalization in Quantum Spin Chains

TL;DR

The paper analyzes finite-size corrections to the full eigenstate thermalization hypothesis (ETH) in chaotic quantum spin chains using exact diagonalization. By decomposing fluctuations into longitudinal (within-energy-window) and transverse (between-energy-window) contributions, it shows that longitudinal effects decay exponentially with system size while transverse effects decay polynomially, clarifying apparent anomalies in higher-order correlations. For both single-site and two-site observables, the authors quantify scaling of a suite of error terms and demonstrate that, when properly decomposed, the full ETH holds in the thermodynamic limit, including at finite times for OTOCs. The work provides a practical framework for validating full ETH in finite quantum many-body systems and offers guidance on interpreting finite-size data to avoid false indications of ETH violation.

Abstract

Despite the unitary evolution of closed quantum systems, long-time expectation of local observables are well described by thermal ensembles, providing the foundation of quantum statistical mechanics. A promising route to understanding this quantum thermalization is the eigenstate thermalization hypothesis (ETH), which posits that individual energy eigenstates already appear locally thermal. Subsequent studies have extended this concept to the full ETH, which captures higher-order correlations among matrix elements through nontrivial relations. In this work, we perform a detailed exact-diagonalization study of finite-size corrections to these relations in the canonical ensemble. We distinguish two distinct sources of corrections: those arising from energy fluctuations, which decay polynomially with system size, and those originating from fluctuations within each energy window, which decay exponentially with system size. In particular, our analysis resolves the puzzle that, for certain observables, finite-size corrections exhibit anomalous growth with increasing system size even in chaotic systems. Our results provide a systematic and practical methodology for validating the full ETH in quantum many-body systems.

Figures (11)

• Figure 1: An illustration of the decomposition of finite-size corrections introduced in this work. In panel (a), we introduce a series of narrow energy windows with a fixed width $\Delta=2$ that is independent of the system size. In panel (b), we illustrate the distribution of $O_{ii}$ over all eigenstates, where the total variance consists of the variance within each energy window (longitudinal direction, denoted as L) and the variance between different energy windows (transverse direction, denoted as T).
• Figure 2: Numerical results for the error terms $F_{11}$, $F_{111}$, and $F_{1111}$ of the mixed-field Ising model as a function of system size $L$ (odd) for the single-site operator $O=Z_{(L+1)/2}$. Here, we fix the width of the energy window to $\Delta=2$. The results are presented in both log plots and log--log plots. The dashed lines correspond to fits to the last four data points, as described in the main text, while the solid lines serve as guides to the eye.
• Figure 3: Numerical results for the error terms $F_{21}$, $F_{31}$, $F_{211}^{(1)}$, $F_{22}$, and $F_{211}^{(2)}$ of the mixed-field Ising model as a function of system size $L$ (odd) for the single-site operator $O=Z_{(L+1)/2}$ with $\Delta=2$. The results are presented in both log plots and log--log plots. The dashed lines correspond to fits to the last four data points, as described in the main text, while the solid lines serve as guides to the eye.
• Figure 4: Numerical results for the error terms $F_{11}$, $F_{111}$, and $F_{1111}$ of the mixed-field Ising model as a function of the system size $L$ (even) for the two-site operator $O = Z_{L/2} Z_{L/2+1}$. We fix the width of the energy window to $\Delta=2$. The results are presented in both log plots and log--log plots. The dashed lines correspond to fits to the last four data points, while the solid lines serve as guides to the eye.
• Figure 5: Numerical results for the error terms $F_{21}$, $F_{31}$, $F_{211}^{(1)}$, $F_{22}$, and $F_{211}^{(2)}$ of the mixed-field Ising model as a function of the system size $L$ (even) for the two-site operator $O = Z_{L/2} Z_{L/2+1}$, with $\Delta=2$. The results are presented in both log plots and log--log plots. The dashed lines correspond to fits to the last four data points, while the solid lines serve as guides to the eye.