Coexistence of order and disorder and reentrance in an exactly solvable model

We show in this Letter exact results for the Ising model on the two-dimensional Kagomé lattice with nearest- and next-nearest-neighbor interactions ${\mathrm{J}}_1$ and ${\mathrm{J}}_2$. In some regions of phase space, we find a nonzero critical temperature despite a finite zero-point entropy. For a narrow range of ${\mathrm{J}}_2$/${\mathrm{J}}_1$ we find successive transitions with a reentrance at low temperature. We studied the nature of ordering by Monte Carlo method and found that in these regions one sublattice remains disordered below the transition and down to zero temperature except in the reentrant region. Thus disorder can coexist with order at equilibrium.