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 the case where N particles are interacting via nearest and nextto-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 oneand two-point functions of a related many-body theory and show the absence of long-range order in the thermodynamic limit. 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.