Single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians

We study the matrix elements of local and nonlocal operators in the single-particle eigenstates of two paradigmatic quantum-chaotic quadratic Hamiltonians; the quadratic Sachdev-Ye-Kitaev (SYK2) model and the three-dimensional Anderson model below the localization transition. We show that they display eigenstate thermalization for normalized observables. Specifically, we show that the diagonal matrix elements exhibit vanishing eigenstate-to-eigenstate fluctuations and that their variance is proportional to the inverse Hilbert space dimension. We also demonstrate that the ratio between the variance of the diagonal and the off-diagonal matrix elements is 2, as predicted by the random matrix theory. We study distributions of matrix elements of observables and establish that they need not be Gaussian. We identify the class of observables for which the distributions are Gaussian.

Automated summary

The matrix elements of local and nonlocal operators in the single-particle eigenstates of two quantum-chaotic quadratic Hamiltonians exhibit eigenstate thermalization for normalized observables. Specifically, the diagonal matrix elements show vanishing eigenstate-to-eigenstate fluctuations, with their variance proportional to the inverse Hilbert space dimension. The ratio between the variance of diagonal and off-diagonal matrix elements is 2, as predicted by random matrix theory.