The statistical properties of eigenstates in chaotic many-body quantum systems

We consider the statistical properties of eigenstates of the Hamiltonian or of the time-evolution

operator in chaotic quantum many-body systems. Our focus is on correlations between eigenstates

that are specific to spatially extended systems and that lie outside the standard framework estab-

lished by the eigenstate thermalisation hypothesis (ETH). We propose a maximum-entropy Ansatz

for the joint distribution of n eigenvectors. In the case n = 2 this Ansatz reproduces ETH. For

n = 4 it captures both the growth in time of entanglement between subsystems, as characterised

by the purity of the time-evolution operator, and also operator spreading, as characterised by the

behaviour of the out-of-time-order correlator. We test these ideas by comparing results from Monte

Carlo sampling of our Ansatz with exact diagonalisation studies of Floquet quantum circuits.