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Ground state of one-dimensional fermion-phonon systems

We propose a new framework of the reflection positivity in terms of order preserving operator inequalities. From the viewpoint of the new framework, we study one-dimensional fermion-phonon systems. Especially, properties of ground states are examined in detail. It is clarified that the reflection positivity consistently describes the charge-density-wave (CDW) order in this system. Furthermore, if the Coulomb interaction is long ranged, then we prove the existence of the CDW long range order in the ground state.