Effective $su_q(2)$ Models and Polynomial Algebras for Fermion-Boson Hamiltonians

We show that schematic $su(2)\oplus h_3$ interaction Hamiltonians, where $su(2)$ plays the role of the pseudospin algebra of fermion operators and $h_3$ is the Heisenberg algebra for bosons, are closely related to certain nonlinear models defined on a single quantum algebra $su_q(2)$ of quasifermions. In particular, $su_q(2)$ analogues of the Da Providencia–Schütte and extended Lipkin models are presented. We analyze the connection between $q$ and the physical parameters of the fermion-boson system and, using polynomial algebras, discuss the integrability properties of the interaction Hamiltonians.