  This study investigated the extended Holstein--Hubbard model at half-filling
as a model for describing interplay of the electron-electron and
electron-phonon couplings. When the electron-phonon and nearest-neighbor
electron-electron interactions are strong, we prove the existence of long-range
charge order at sufficiently low temperature in three or more dimensions. As a
result, we rigorously justify the phase competition between the
antiferromagnetic and charge orders.
