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 = Fcond Fnorm where Fcond is the subspace spanned by the eigenstates of V with minimal eigenvalue and Fnorm = Fcond. If, in the thermodynamic limit, Mcond/M → 0, where M and Mcond are, respectively, the dimensions of F and Fcond, the above decomposition of F becomes effective, in the sense that the ground state energy per particle of the system, ε, coincides with the smaller between εcond and εnorm, the ground state energies per particle of the system restricted to the subspaces Fcond and Fnorm, respectively: ε = min{εcond, εnorm}. It may then happen that, as a function of the parameter g, the energies εcond and εnorm cross at g = gc. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace Fcond) and a normal phase (system spread over the large subspace Fnorm). Since, in the thermodynamic limit, Mcond/M → 0, the confinement into Fcond 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.