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 analog of the Casimir effect. We show that this interaction is uniformly attractive and falls off as $r^{-2d+1\mathrm{}}$ in $d$ dimensions. For “stripes,” i.e., parallel $(d-1\mathrm{})$-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.