First-order quantum phase transitions as condensations in the space of states

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H = K + gV, where K and V are two non commuting operators acting on the space of states F, we may always write F = F-cond circle plus F-norm where F-cond is the subspace spanned by the eigenstates of V with minimal eigenvalue and F-norm = F-cond(perpendicular to). If, in the thermodynamic limit, M-cond/M -> 0, where M and M-cond are, respectively, the dimensions of F and F-cond, the above decomposition of F becomes effective, in the sense that the ground state energy per particle of the system, epsilon, coincides with the smaller between epsilon(cond) and epsilon(norm), the ground state energies per particle of the system restricted to the subspaces F-cond and F-norm, respectively: epsilon = min{epsilon(cond), epsilon(norm)}. It may then happen that, as a function of the parameter g, the energies epsilon(cond) and epsilon(norm) cross at g = g(c). In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace F-cond) and a normal phase (system spread over the large subspace F-norm). Since, in the thermodynamic limit, M-cond/M -> 0, the confinement into F-cond is actually a condensation in which the system falls into a ground state orthogonal to that of the normal phase, something reminiscent of Anderson's orthogonality catastrophe (Anderson 1967 Phys. Rev. Lett. 18 1049). The outlined mechanism is tested on a variety of benchmark lattice models, including spin systems, free fermions with non uniform fields, interacting fermions and interacting hard-core bosons.