  We report on a remarkable spectral phenomenon in a generic type of quantum
lattice gas model. As the interaction strength increases, eigenstates
spontaneously reorganize and lead to plateaus of the interaction energy, with
gaps opening akin to continuous phase transitions. Perturbation theory
identifies a hidden structure underlying eigenstates within each plateau,
resulting in a statistical shift in the wavefunction amplitudes described by
extreme value theory. The structured eigenstates manifest themselves naturally
in far-from-equilibrium dynamics proceeding through multiple universal stages.
Our findings reveal a profound connection between emergent properties in
high-energy states and out-of-equilibrium dynamics, providing insights into the
impact of interactions across the entire energy spectrum. The results are
directly relevant to experiments probing equilibration in quantum spin and
lattice gases.
