On the Inequivalence of the Three Band, Kondo Heisenberg and t-J Models
We compare the spectrum for the three band (Emery) Model, the Kondo Heisenberg model and the t-J model within the lowest order self consistent approximation. The band minima for the three band model is at (π/2,π/2) and its bandwidth is proportional to J, as in the case with the other two models. The quasiparticle wavefunction is calculated to the same level of approximation, and used to demonstrate that the quasiparticle in the three band model cannot be described as a moving singlet.
The claim by Zhang and Rice  of the equivalence of the three band (Emery) model in the strong coupling limit with the t-J model, due to the formation of bound singlet pairs between oxygen and copper holes on small systems  together with numerical simulations supporting that claim, have served to focus attention on the t-J model as the “simplest” model for the CuO2 planes that was thought to contain the essential physics. Indeed, it seems the natural model to describe the electron doped systems, so that a principle of using the minimum necessary model would commend it. In a series of papers, however, Emery and Reiter  showed that the ZR truncation of the original strong coupling Hamiltonian was an approximation, and that there was no small parameter available to justify the approximation. We wish to answer the question of how good is the ZR approximation for the hole doped system, by treating in detail the single hole in an antiferromagnetic background for the strong coupling limit of the Emery model and the t-J model. We will use the approximation scheme, introduced in this context by Kane, Lee and Read , and Schmitt, Rink, Varma and Ruckenstein .
The approximation consists of treating the Cu spin background using spin wave theory, and calculating the lowest order self energy self consistently. The first approximation is likely to be excellent , and the second is found empirically to be very good. The quasiparticle dispersion relations derived in this way are an excellent approximation to exact results on small systems.  Ramsak and Prelovsek (RP)  observing that the means of deriving the coupling between the charge and the background Cu spins using Schwinger boson methods was suspect, used the approximation for the Kondo Heisenberg (KH) model, which goes over to the t-J model in the limit of strong Kondo coupling. They showed that different methods of coupling the spin to the charge gave very similar dispersion relations for the quasiparticle, but that there was a non-analyticity introduced by scattering from low energy magnons in the KH model regularization of the t-J model that was not present in the Schwinger boson treatment. While we share RP’s suspicions of the Schwinger boson treatment of the coupling, we find that it reproduces the bandwidths as a function of J obtained by exact diagonalization of small systems much better than the KH treatment. We will make comparison with the KH model anyway, as it will enable us to make clear the differences between the t-J and Emery models, and in particular, why there is a non-zero value for the average spin on the oxygen hole in the latter case. We find the non-analyticity in the Emery model as well.
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