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 hopping. In lowest order of the recursion (i.e., for all processes with only one pair of empty/doubly occupied sites) we obtain an effective anisotropic pseudospin $1∕2$ Heisenberg Hamiltonian $H_{\mathit{eff}}$ 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 ${[E-H_{\mathit{eff}}\left(E\right)]}^{-1}$ has an infinite number of poles, even for a single site. $H_{\mathit{eff}}$ is subject to a crossover from an isotropic Heisenberg model (weak electron-phonon coupling and isotropic hopping) to an Ising model (strong electron-phonon coupling).

Figure 1

(Color online) The schematic structure of the recursive projection method that reduces the Hilbert space (Russian doll approach). The actual reduction is due to the recursion relation $\left(R\right)$, whereas the projection $\left(P\right)$ separates a part of the Hilbert space through the Hamiltonian $H_{j+1,j}$. The low-lying poles of the resolvent $G_{2j}$ (indicated by the levels) are removed recursively by the method.

Figure 2

Coupling coefficients of the effective pseudospin-$1∕2$ Hamiltonian of Eq. (21) for isotropic hopping $t_{\downarrow }=t_{\uparrow }\equiv t$ as a function of $y=2g^2∕{\omega }_0^2$: $a_{\uparrow \uparrow }$ (upper set of curves) and $a_{\uparrow \downarrow }$ (lower set of curves) in units of $t^2∕{\omega }_0$ for $(U-E)∕{\omega }_0=1.0,1.2,1.5$ (from top to bottom in each set of curves).