Observable-manifested correlations in many-body quantum chaotic systems

In this paper, we investigate the distinctions between realistic quantum chaotic systems and random models from the perspective of observable properties, particularly focusing on the eigenstate thermalization hypothesis (ETH). Through numerical simulations, we find that for realistic systems, the envelope function of off-diagonal elements of observables exhibits an exponential decay at large $\Delta E$, while for randomized models, it tends to be flat. We demonstrate that the correlations of chaotic eigenstates, originating from the delicate structures of Hamiltonians, play a crucial role in the non-trivial structure of the envelope function. Furthermore, we analyze the numerical results from the perspective of the dynamical group elements in Hamiltonians. Our findings highlight the importance of correlations in physical chaotic systems and provide insights into the deviations from RMT predictions. These understandings offer valuable directions for future research.