Importance of single-boson and single-fermion mappings in the thermal boson expansion

In the context of the boson expansion theory, it is usually the case that the bosonization of single-boson (fermion) states is ignored. Although this is tolerable to some extent in cold systems, it causes serious difficulties in the treatment of thermal ensembles where the single-boson (fermion) density of states plays an important role. In the framework of the thermo-field dynamics it is shown how extended forms of the Holstein-Primakoff mapping for both bosons and fermions can lead to consistent thermal-boson expansions. Applications to the O(N) anharmonic oscillator and the Lipkin model are presented.