Random phase approximation for the 1 D anti - ferromagnetic Heisenberg model

The Hartree-Fock-RPA approach is applied to the 1D anti-ferromagnetic Heisen-berg model in the Jordan-Wigner representation. Somewhat contrary to expectation , this leads to reasonable results for spectral functions and sum rules in the symmetry unbroken phase. In a preliminary application of Self-Consistent RPA to finite size chains strongly improved results are obtained. The 1D Anti-Ferromagnetic Heisenberg Model (AFHM) [1] belongs to one of the most classic many body research fields. It was the first many-body model to be solved exactly by Bethe with his famous Bethe ansatz solution [2] and it has been a playground for testing and developing many body approaches ever since. In spite of tremendous progress in the understanding of many facets of the model, it continues to be a very active field of interest and research [3, 4]. In an attempt to apply extended RPA-like theories, as e.g. the recently developed Self-Consistent RPA (SCRPA) approach [5, 6], we found out, somewhat to our surprise, that even the standard RPA theory has so far not fully been developed in its application to the 1D-AFHM. This probably stems from the fact that the HF-RPA scheme is usually considered as unreliable in low dimensions [7]. However, it will be found that RPA shows interesting features even in this 1D-version of the model. On the other hand SCRPA shows very promising results in a number of cases [5] and in a first application for a small numbers of sites we here get very close agreement with results from exact diagonalization. We mostly consider the isotropic Heisenberg model with anti-ferromagnetic coupling, however some consideration will also be given to the anisotropic case, for instance to the XXZ-anisotropic Heisenberg Hamiltonian