Emergence of unitary symmetry of microcanonically truncated operators in chaotic quantum systems

We study statistical properties of matrix elements of observables written in the energy eigenbasis and truncated to small microcanonical windows. We present numerical evidence indicating that for all few-body operators in chaotic many-body systems, truncated below a certain energy scale, collective statistical properties of matrix elements exhibit emergent unitary symmetry. Namely, we show that below a certain scale the spectra of the truncated operators exhibit universal behavior, matching our analytic predictions, which are numerically testable for system sizes beyond exact diagonalization. We discuss operator and system-size dependence of the energy scale of emergent unitary symmetry and put our findings in the context of previous works exploring the emergence of random-matrix behavior at small energy scales.