Quantum Phase Transitions induced by Infinite Dilution in the Fock Space: a General Mechanism. Proof and discussion

We prove that lattice quantum systems may undergo a first-order quantum phase transition through a general mechanism which consists in an infinite dilution of the states associated to (or, more in general, near to) the lowest energy levels. The equation giving the critical point is derived and discussed at several degrees of generalizations: given an infinitesimal portion of the Fock space, i.e., the cavity space, and its complement, i.e., the reservoir space, in the thermodynamic limit, a first-order quantum phase transition takes place when the cavity and reservoir energies are equal. A comparison with some particular exact solvable cases as, e.g., the Quantum Rem model, is made.