Charge ordering in half-doped Pr(Nd)$_{0.5}$Ca$_{0.5}$MnO$_3$ under magnetic field

Recent experiments in Pr(Nd)$_{0.5}$Ca$_{0.5}$MnO$_3$ thin films exhibited (multiple) reentrant charge ordering (CO) transitions with change of {\it temperature} T under fixed magnetic field H, which are in contrast to the results for the corresponding bulk materials. To explain the experimental findings, a model including the double-exchange mechanism, intersite Coulomb interaction and electron-phonon coupling is proposed, for which the reentrant CO is naturally obtained due to the temperature dependence of the band-type coherent polaron hopping. Various results for the CO in the (H, T) plane are extensively discussed. The theory is considered to be valid for both thin films and bulk materials.