  Most of our current understanding of phase separation is based on ideas that
disregard correlaions. Here we illuminate unexpected effects of correlations on
the structure and thermodynamics of interfaces and in turn phase separation,
which are decisive in systems with strong interactions. Evaluating the
continuum limit of the Ising model on the Bethe-Guggenheim level, we derive a
Cahn-Hilliard free energy that takes into account pair correlations. For a
one-dimensional interface in a strip geometry these are shown to give rise to
an effective interface broadening at interaction strengths near and above the
thermal energy, which is verified in the Ising model. Interface broadening is
the result of an entropy-driven interface delocalization, which is not
accounted for in the widely adopted mean field theory. Pair correlations are
required for thermodynamic consistency as they enforce a thermodynamically
optimal local configuration of defects and profoundly affect nucleation and
spinodal decomposition at strong coupling.
