Rare collapse of fermionic quasiparticles upon coupling to local bosons

We study the stability of the fermionic quasiparticle in a fermion-boson model on a Bethe lattice, with fermions interacting with local bosons via a polaronic-type coupling. We solve the problem by mapping it onto a non-interacting chain with site-dependent potential. We show that, despite a finite number of bosonic excitations costing zero energy, among the many analyzed cases, the occurrence of a complete collapse of the quasiparticle is rare. The quasiparticle disappearance becomes easier with an increase in: (i) the total number of bosons with zero energy, and (ii) the relative strength of the coupling between bosons and fermions. The postulated model can, among other things, be applied to study systems in which fermions are introduced into antiferromagnetic (or antiferro-orbital) domains surrounded by ferromagnetic (or ferro-orbital) ordered states. This might take place in the overdoped cuprates or upon doping manganese or vanadium oxides. Finally, we show how this model leads to an in-depth understanding of the onset of quasiparticles in the 1D and 2D $t$-$J^z$ model.