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

We obtain the exact ground state and a part of the excitation spectrum in one dimension on a line and the exact ground state on a circle in a case where N particles are interacting via nearest- and next-to-nearest neighbour interactions. Further, 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 and show that there is no long-range order in the thermodynamic limit. However, quite remarkably, we prove the existence of an off-diagonal long-range order in the symmetrised version of the related many-body theory. Generalisation 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.