Having analytical instances of the eigenstate thermalization hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generally a hard task, due to the belief that nonlinear interactions are basic ingredients of the thermalization mechanism. In this article we prove that random Gaussian-free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian systems, providing a physically motivated notion of randomness of the microscopic quantum state.
Having analytical instances of the eigenstate thermalization hypothesis (ETH) is of obvious interest, both for fundamental and applied reasons. This is generally a hard task, due to the belief that nonlinear interactions are basic ingredients of the thermalization mechanism. In this article we prove that random Gaussian-free fermions satisfy ETH in the multiparticle sector, by analytically computing the correlations and entanglement entropies of the theory. With the explicit construction at hand, we finally comment on the differences between fully random Hamiltonians and random Gaussian systems, providing a physically motivated notion of randomness of the microscopic quantum state.
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 emergence of statistical mechanics for isolated classical systems comes about through chaotic dynamics and ergodicity. Here we review how similar questions can be answered in quantum systems. The crucial point is that individual energy eigenstates behave in many ways like a statistical ensemble. A more detailed statement of this is named the eigenstate thermalization hypothesis (ETH). The reasons for why it works in so many cases are rooted in the early work of Wigner on random matrix theory and our understanding of quantum chaos. The ETH has now been studied extensively by both analytic and numerical means, and applied to a number of physical situations ranging from black hole physics to condensed matter systems. It has recently become the focus of a number of experiments in highly isolated systems. Current theoretical work also focuses on where the ETH breaks down leading to new interesting phenomena. This review of the ETH takes a somewhat intuitive approach as to why it works and how this informs our understanding of many body quantum states.
In an isolated quantum many-body system undergoing unitary evolution, we study the thermalization of a subsystem, treating the rest of the system as a bath. In this setting, the eigenstate thermalization hypothesis (ETH) was proposed to explain thermalization. Consider a nearly integrable Sachdev-Ye-Kitaev model obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. When the subsystem size is larger than the square root of but is still a vanishing fraction of the system size, we prove thermalization if the system is initialized in a random product state, while almost all eigenstates violate the ETH. In this sense, the ETH is not a necessary condition for thermalization.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
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) 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.