On perturbation theory for the three-band model of cuprates

Abstract

We raise and discuss the following question. Why does the spectrum for the three-band model of Hybertson, Stechel, Schluter, and Jennison, claimed not to be approachable by perturbation theory because of rather large hopping integrals compared to site energy differences, follow precisely what would be expected by low-order perturbation theory? The latter is, for the insulating case, that the low-lying levels are describable by a Heisenberg Hamiltonian with nearest-neighbor interactions plus much smaller next-nearest-neighbor interactions and n-spin terms, n≥4. We first check whether perturbation theory actually does not converge, treating the hopping and p-d exchange terms as perturbations. For the crystal, we find that the first three terms contributing to the nearest-neighbor exchange coupling J (which are of third, fourth, and fifth order) increase in magnitude, and are not of the same sign, i.e., there is no sign of convergence to this order. We also consider the small cluster, Cu2O7, for which we have carried out the perturbation series to 14th order; there is still no sign of convergence. Thus the nonconvergence of this straightforward perturbation theory is convincingly established. Yet the apparent perturbative nature of the spectrum suggests the existence of some perturbation theory that does converge. The possibility of a particular transformation of the Hamiltonian leading to a convergent perturbation series, thereby answering the above question, is discussed.