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

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.

💡 Research Summary

The paper addresses a central question in quantum statistical mechanics: how isolated many‑body quantum systems approach thermal equilibrium despite unitary dynamics. While the conventional Eigenstate Thermalization Hypothesis (ETH) successfully explains thermalization of few‑body observables through diagonal matrix elements and exponentially small off‑diagonal fluctuations, it fails to capture higher‑order correlations because the Gaussian assumption forces all higher cumulants to vanish. To overcome this limitation, the authors employ the “full ETH,” which treats the random variables R_{ij} in the ETH ansatz as general, non‑Gaussian objects and formulates the hypothesis directly in terms of averaged products of matrix elements. In the language of free probability theory, these products are expressed through free cumulants k_q.

The study focuses on concrete chaotic spin‑chain models, primarily the mixed‑field Ising chain with Hamiltonian H = Σ Z_i Z_{i+1} + w Σ X_i + h Σ Z_i (J=1, w=1.05, h=√5−1/2). Open boundary conditions and a parity‑symmetry sector reduce the Hilbert space, allowing exact diagonalization up to system sizes L≈15–17 spins. The authors consider both single‑site operators (central Pauli‑Z) and two‑site operators, and they work in the infinite‑temperature ensemble (ρ = 1/D).

Correlation functions C^{(q)}(t_1,…,t_{q‑1}) are defined as infinite‑temperature expectation values of time‑ordered products of an operator O. Using the full‑ETH ansatz, the authors derive theoretical predictions for these correlators in terms of free cumulants k_q and identify the finite‑size correction terms that arise when the exact diagonalization data are compared with the ETH predictions. For the two‑point function, the correction is F_{11} = (1/D) Σ_i O_{ii}^2 –