A class of N-body problems with nearest- and next-to-nearest-neighbour interactions

Abstract

We obtain the exact ground state and part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in the case where the N particles are interacting via nearest- and next-to-nearest-neighbour interactions. Furthermore, using the exact ground state, we establish a mapping between these N-body problems and the short-range Dyson models introduced recently to model intermediate spectral statistics. Using this mapping we compute the one- and two-point functions of a related many-body theory in the thermodynamic limit and show the absence of long-range order. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrized version of the related many-body theory. Generalization of the models to other root systems is also considered. Besides, we also generalize the model on the full line to higher dimensions. Finally, we consider a model in two dimensions in which all the states exhibit novel correlations.