Wigner crystal in the manganese oxidesR1−xAxMnO3

A path-integral approach for the double-exchange model with long-range Coulomb interaction $[V_{\mathrm{ij}}{=V}_0{a}_0{/r}_{\mathrm{ij}}$ with $a_0$ the lattice constant] is used to investigate the electronic phase transition in $R_{1-x}{A}_x{\mathrm{MnO}}_3$ compounds. This method corresponds to the extended random-phase approximation and can treat the metallic phase and charge-ordered phase of the system on an equal footing. We show that when the interaction strength $V_0$ is larger than a critical value, the uniform ferromagnetic state becomes unstable near $x=0.5\mathrm{}$ and a Wigner-crystal state is energetically more favorable. This phase transition is first order. It is estimated that the interaction strength in the Mn oxides may be larger than the critical value, yielding a possible interpretation of the charge orderings observed in experiments.