  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 certain energy scale, collective statistical
properties of matrix elements exhibit emergent unitary symmetry. Namely, we
show that below 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 context of previous works exploring emergence of random-matrix
behavior at small energy scales.
