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.Comment: LaTeX2e, 40 pages, 2 figures, submitted to Nucl. Phys. B [FS
