Generalized Falicov-Kimball models

In this paper we extend the Falicov-Kimball model (FKM) to the case where the "quasi-particles" entering the FKM are not ordinary" fermions. As an example we first discuss how the FKM can be generalized to the case with spin-dependent hopping. Afterward we discuss several cases where the "quasi-particles" entering the FKM are Majorana fermions (Majorana-Falicov-Kimball Model (MFKM). Two examples of MFKM are discussed in detail: (i) a $p$-wave BCS superconductor on a bipartite lattice and (ii) a BCS-Anderson model. We also discuss the most general forms of MFKM, including a brief discussion on the case where the Majorana fermions represent spins, but not real fermion particles.