A class of exactly solvable many-particle models

A class of many-particle quantum models with arbitrary space dimension and arbitrary particle statistics for which it is possible to construct a number of exact eigenstates is found. In the models, the following assumptions are made: 1) the presence of two (or $2m$) components with “symmetric” matrix elements of the two-body interactions (equality up to the sign of all the interaction potentials and equality up to a phase factor of the wave functions of particles of two species; otherwise the two-body interactions are arbitrary), 2) degeneracy of the (total) spectrum of the free particles. The exact states correspond to a condensate of noninteracting composite particles (“excitons”) that are not precisely bosons and to excitations over the condensate. The possibility of exact solution rests on the symmetry with respect to continuous rotations in the isospin space, this corresponding to Bogolyubov transformations with momentum-independent parameters $u$, $v$. The class includes, in particular, two-dimensional electron-hole systems in a strong magnetic field.