Jahn-Teller systems at half filling: crossover from Heisenberg to Ising behavior

The Jahn-Teller model with $E\otimes\beta$ electron-phonon coupling and local (Hubbard-like) Coulomb interaction is considered to describe a lattice system with two orbitals per site at half filling. Starting from a state with one electron per site, we follow the tunneling of the electrons and the associated creation of an arbitrary number of phonons due to electron-phonon interaction. For this purpose we apply a recursive method which allows us to organize systematically the number of pairs of empty/doubly occupied sites and to include infinitely many phonons which are induced by electronic tunneling. In lowest order of the recursion (i.e. for all processes with only one pairs of empty/doubly occupied sites) we obtain an effective anisotropic pseudospin 1/2 Heisenberg Hamiltonian $H_{eff}$ as a description of the orbital degrees of freedom. The pseudospin coupling depends on the physical parameters and the energy. This implies that the resulting resolvent $(z-H_{eff}(z))^{-1}$ has an infinite number of poles, even for a single site. $H_{eff}$ is subject to a crossover from an isotropic Heisenberg model (weak electron-phonon coupling) to an Ising model (strong electron-phonon coupling).