Local dynamics and the structure of chaotic eigenstates
Zhengyan Darius Shi, Shreya Vardhan, Hong Liu
Abstract
We identify properties of the energy eigenstates in a family of chaotic spin-chain systems with local interactions, which distinguish them both from integrable systems and from nonlocal chaotic systems. We study the relation between the energy eigenstates of the full system and products of energy eigenstates of two extensive subsystems. The magnitudes of the coefficients relating the two bases have a simple form as a function of $\ensuremath{\omega}$, the energy difference between the full system eigenstate and the product of eigenstates. This form explains the exponential decay with time of the probability for a product of eigenstates to return to itself during thermalization. We also find certain statistical properties of the coefficients. While it is generally expected that the coefficients are uncorrelated random variables, we point out that correlations implied by unitarity are important for understanding the transition probability between two products of eigenstates, and the evolution of operator expectation values during thermalization. Moreover, we find that there are additional correlations resulting from locality, which lead to a slower growth of the second R\'enyi entropy than the one predicted by an uncorrelated random variable approximation. We expect that the above properties are universal in chaotic systems with local interactions.