In a many-atom molecule, the Jahn-Teller effect inducing distortions in the molecular structure and the accompanying Berry phase on the electronic wave function are well known. Koizumi, Hotta, and Takada have paid attention for the first time to the role of the Berry phase in the conduction electrons in a crystal composed of such Jahn-Teller molecules and they have obtained a modified Bloch theorem in the system [1,2]. 
  We are now trying to apply our result to the manganites, a system drawing much attention due to the colossal magnetoresistance (CMR), in order to investigate the microscopic origin of the orbital ordering and stripe structure.