In this work, we focus on the Eigenstate Thermalization Hypothesis (ETH) [1] as a leading explanation for the emergence of thermal behavior in many-body quantum systems. We discuss recent advances in the field, particularly the work of Silvia, Laura, and Jorge [2,3], who extended the ETH to include multipoint thermal correlation functions. Their findings revealed a direct connection between these correlations and the cumulants of free probability [4]. This connection, along with the free independence condition, suggests the existence of subleading contributions not captured by the generalized ETH and free probability. To support this idea, we analytically computed the subleading contributions for the two-point correlation functions in a system of random free fermions; for that, we used quantum information and random matrix theory techniques that include quantum channels, Gaussian ensembles and Haar averages. Our results confirmed our initial hypothesis: the subleading contributions contain relevant information about the presence or absence of time-reversal symmetry in the system considered.
In this work, we focus on the Eigenstate Thermalization Hypothesis (ETH) [1] as a leading explanation for the emergence of thermal behavior in many-body quantum systems. We discuss recent advances in the field, particularly the work of Silvia, Laura, and Jorge [2,3], who extended the ETH to include multipoint thermal correlation functions. Their findings revealed a direct connection between these correlations and the cumulants of free probability [4]. This connection, along with the free independence condition, suggests the existence of subleading contributions not captured by the generalized ETH and free probability. To support this idea, we analytically computed the subleading contributions for the two-point correlation functions in a system of random free fermions; for that, we used quantum information and random matrix theory techniques that include quantum channels, Gaussian ensembles and Haar averages. Our results confirmed our initial hypothesis: the subleading contributions contain relevant information about the presence or absence of time-reversal symmetry in the system considered.
The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell. This mathematical framework allows one to reduce in a straightforward way higher-order correlation functions to a decomposition given by minimal blocks, identified as free cumulants, for which we give an explicit formula. This perspective naturally incorporates the consistency property that local functions of ETH operators also satisfy ETH. The present results uncover a direct connection between the eigenstate thermalization hypothesis and the structure of free probability, widening considerably the latter's scope and highlighting its relevance to quantum thermalization.
The eigenstate thermalization hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention toward the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.
We analyze thermalization and eigenstate thermalization in the random free fermion model, a quadratic Hamiltonian with a Gaussian Orthogonal Ensemble hopping matrix, by an exact random-matrix calculation. For a single-particle observable A we compute the ensemble average and fluctuations of Tr(Aρ(t)): we derive the relaxation envelope g2(t/τλ) with τλ=ℏ/(2ηN) and its oscillatory ∣t∣−3 tail, both properties of the semicircular limiting density, and we show that the variance of the fluctuations is O(1/N), so that the relaxation profile self-averages at fixed scaled time. Incorporating particle-number conservation, we prove a factorization theorem for the energy-shell statistics of the many-body eigenstates: the energy-independent ETH structure follows from the statistical independence of the eigenvector matrix and the spectrum, and at fixed realization the energy constraint enters only through a covariance subtraction of relative order 1/N. A controlled eigenvector--eigenvalue correlation deformation H(ε)=H0+εH1 then generates, at first order in ε, the energy dependence of the diagonal energy-resolved slope and of the off-diagonal spectral function, a single-particle diagnostic of the ETH structure in the allowed particle-hole sector. Correlation-function fluctuations and the single-particle spectral form factor finally identify the subleading O(1/N) late-time fingerprints of two-level spectral correlations. The model thus separates eigenbasis chaos from spectral chaos and identifies the statistical origin of energy-independent versus energy-dependent ETH.
The Eigenstate Thermalization Hypothesis (ETH) has been established as a cornerstone for understanding thermalization in quantum many-body systems. Recently, there has been growing interest in the full ETH, which extends the framework of the conventional ETH and postulates a smooth function to describe the multi-point correlations among matrix elements. Within this framework, free cumulants play a central role, and most previous studies have primarily focused on closed systems. In this paper, we extend the analysis to a system-bath setup, considering both an idealized case with a random-matrix bath and a more realistic scenario where the bath is modeled as a defect Ising chain. In both cases, we uncover a universal scaling of the microcanonical free cumulants of observables associated with the central system Hamiltonian with respect to the interaction strength. Furthermore we establish a connection between this scaling behavior and the thermalization dynamics of the thermal free cumulants of corresponding observables.
The thermalizing dynamics of many-body systems is often described through the lens of the eigenstate thermalization hypothesis (ETH). ETH postulates that the statistical properties of observables, when expressed in the energy eigenbasis, are described by smooth functions, which also describe correlations among the matrix elements. However, the form of these functions is usually left undetermined, constituting a key missing component of the ETH framework. In this Letter, we investigate the structure of such smooth functions by focusing on their Fourier transform, recently identified as free cumulants. Using nonlinear hydrodynamics, we provide a prediction for the universal scaling of the late-time behavior of time-ordered free cumulants in the thermodynamic limit. The prediction is further corroborated by large-scale numerical simulations of several nonintegrable one-dimensional spin models that exhibit diffusive transport behavior. Good agreement is observed in both infinite and finite-temperature regimes and for a collection of local observables. Our results indicate that the smooth multipoint correlation functions within the ETH framework admit a universal hydrodynamic description at low frequencies.