Symmetry of Hamiltonians of quantum two-component systems: condensate of composite particles as an exact eigenstate

A class of quantum many-body models of arbitrary dimension and arbitrary statistics of particles, for which exact eigenstates may be obtained, is found. It is assumed that: (i) models contain two (or 2m) kinds of particles with 'symmetric' matrix elements of pairwise interaction (all potentials coincide with each other to within a sign and wavefunctions of free particles of two components coincide to within a phase factor; pairwise interactions are otherwise arbitrary); (ii) there exists the degeneracy of (the sum) of free-particle spectra. Exact many-body eigenstates correspond to a condensation of non-interacting composite particles ('excitons') which are not exactly bosons, into a single quantum state, and to excitations over the condensate. The origin of the possibility of exact solution is the symmetry under the continuous rotations in the isospin space of two components, to which Bogolubov canonical transformations with parameters u, nu independent of momentum correspond. The class of such models comprises, in particular, two-dimensional electron-hole systems in a strong magnetic field.