Instability of charge ordered states in doped antiferromagnets

We analyze the induced interactions between localized holes in weakly-doped Heisenberg antiferromagnets due to the modification of the quantum zero point spin wave energy; i.e. the analogue of the Casimir effect. We show that this interaction is uniformly attractive and falls off as $r^{-2 d+1}$ in d dimensions. For ``stripes'', i.e parallel $d-1$ dimensional hypersurfaces of localized holes, the interaction energy per unit hyperarea is attractive and falls, generically, like $r^{- d}$. We argue that, in the absence of a long-range Coulomb repulsion between holes, this interaction leads to an instability of any charge-ordered state in the dilute doping limit.