  Energy-filtered quantum states are promising candidates for efficiently
simulating thermal states. We explore a protocol designed to transition a
product state into an eigenstate located in the middle of the spectrum; this is
achieved by gradually reducing its energy variance, which allows us to
comprehensively understand the crossover phenomenon and the subsequent
convergence towards thermal behavior. We introduce and discuss three
energy-filtering regimes (short, medium and long), and we interpret them as
stages of thermalization. We show that the properties of the filtered states
are locally indistinguishable from those of time-averaged density matrices,
routinely employed in the theory of thermalization. On the other hand,
unexpected non-local quantum correlations are generated in the medium regimes
and are witnessed by the R\'enyi entanglement entropies of subsystems, which we
compute via replica methods. Specifically, two-point correlation functions
break cluster decomposition and the entanglement entropy of large regions
scales as the logarithm of the volume during the medium filter time.
