Numerical Verification of the Fluctuation-Dissipation Theorem for Isolated Quantum Systems

Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU(2) symmetry

We investigate the eigenstate thermalization properties of the spin-1/2 XXZ model in two-dimensional rectangular lattices of size L_{1}×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 4×7 and of the energy eigenstates up to 4×6. 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 the spin interaction is isotropic, the XXZ model Hamiltonian conserves the total spin and has SU(2) symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace where the total spin is a good quantum number.

Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field

We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a crossover scaling law when the longitudinal field is weak. The crossover scaling confirms that the uniform longitudinal field makes the system chaotic at any nonzero value. We also discuss the implication of the crossover scaling on the thermalization dynamics and the effect of a nonuniform local longitudinal field.

Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions

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 decay of dynamical correlations between local operators which can be computed exactly from the elementary building blocks of the dual-unitary circuits. Comparing the asymptotic expression with results obtained by exact diagonalization we find excellent agreement. Small fluctuations at finite system size are explicitly related to dynamical correlations at intermediate times and the deviations from their asymptotical dynamics. Moreover, we confirm the expected Gaussian distribution of the matrix elements by computing higher moments numerically.

Eigenstate thermalization hypothesis and eigenstate-to-eigenstate fluctuations

We investigate the extent to which the eigenstate thermalization hypothesis (ETH) is valid or violated in the nonintegrable and the integrable spin-1/2 XXZ chains. We perform the energy-resolved analysis of statistical properties of matrix elements of observables in the energy eigenstate basis. The Hilbert space is divided into energy shells of constant width, and a block submatrix is constructed whose columns and rows correspond to the eigenstates in the respective energy shells. In each submatrix, we measure the second moment of off-diagonal elements in a column. The columnar second moments are distributed with a finite variance for finite-sized systems. We show that the relative variance of the columnar second moments decreases as the system size increases in the non-integrable system. The self-averaging behavior indicates that the energy eigenstates are statistically equivalent to each other, which is consistent with the ETH. In contrast, the relative variance does not decrease with the system size in the integrable system. The persisting eigenstate-to-eigenstate fluctuation implies that the matrix elements cannot be characterized with the energy parameters only. Our result explains the origin for the breakdown of the fluctuation dissipation theorem in the integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light on the meaning of the ETH.