The Eigenstate Thermalization Hypothesis (ETH) explains emergence of the thermodynamic equilibrium by assuming a particular structure of observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by Random Matrix Theory (RMT). To what extent physical operators can be described by RMT, more precisely at which energy scale strict RMT description applies, is however not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible for exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that genuine RMT behavior is absent even for narrow energy windows corresponding to time scales of the order of thermalization time $τ_\text{th}$ of the respective observables. We also demonstrate that residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
We show that macroscopic thermalization and transport impose constraints on matrix elements entering the Eigenstate Thermalization Hypothesis (ETH) ansatz and require them to be correlated. It is often assumed that the ETH reduces to Random Matrix Theory (RMT) below the Thouless energy scale. We show this conventional picture is not self-consistent. We prove that energy scale at which the RMT behavior emerges has to be parametrically smaller than the inverse timescale of the slowest thermalization mode coupled to the operator of interest. We argue that the timescale marking the onset of the RM…
Generic ETH: Eigenstate Thermalization beyond the Microcanonical
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Syracuse University（雪城大学）；Sarah Lawrence College（莎拉劳伦斯学院）
We investigate the eigenstate thermalization properties of the spin-1/2 $XXZ$ model in two-dimensional rectangular lattices of size $L_1\times L_2$ under periodic boundary conditions. Exploiting the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up to system size $4\times 7$ and of the energy eigenstates up to $4\times 6$. Numerical analysis of the Hamiltonian eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis supports that the two-dimensional $XXZ$ model follows the eigenstate thermalization hypothesis. When…
We study the fluctuations of eigenstate expectation values in a microcanonical ensemble. Assuming the eigenstate thermalization hypothesis, an analytical formula for the finite-size scaling of the fluctuations is derived. The same problem was studied by Beugeling et al. [Phys. Rev. E 89, 042112 (2014)]. We compare our results with theirs.
The Eigenstate Thermalization Hypothesis (ETH) has played a major role in understanding thermodynamic phenomena in closed quantum systems. However, its connection to the timescale of thermalization for open system dynamics has remained elusive. This paper establishes a rigorous link between ETH and fast thermalization to the global Gibbs state. Specifically, we demonstrate fast thermalization for a system coupled weakly to a bath of quasi-free Fermions that we refresh periodically. To describe the joint evolution, we derive a finite-time version of Davies' generator with explicit error bounds …
The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi-)energy eigenbasis. Here we study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits in dependence of the frequency associated with the corresponding eigenstates. We provide an exact asymptotic expression for the spectral function, i.e., the second moment of this frequency resolved distribution. The latter is obtained from the…
