  Quantum states of a novel Bose-Einstein condensate, in which both
fermion-pair and exciton condensations are simultaneously present, have
recently been realized theoretically in a model Hamiltonian system. Here we
identify quantum phase transitions in that model between fermion-pair and
exciton condensations based on a geometric analysis of the convex set of
ground-state 2-particle reduced density matrices (2-RDMs). The 2-RDM set
provides a finite representation of the infinite parameter space of
Hamiltonians that readily reveals a fermion-pair condensate phase and two
distinct exciton condensate phases, as well as the emergence of first- and
second-order phase transitions as the particle number of the system is
increased. The set, furthermore, shows that the fermion-exciton condensate
(FEC) lies along the second-order phase transition between the exciton and
fermion-pair condensate phases. The detailed information about the exciton and
fermion-pair phases, the forces behind these phase, as well as their associated
transitions provides additional insight into the formation of the FEC
condensate, which we anticipate will prove useful in its experimental
realization.
