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

Abstract

Abstract 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 ⊕ F norm where F cond is the subspace spanned by the eigenstates of V with minimal eigenvalue and F norm = F cond ⊥ . 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, ϵ, coincides with the smaller between ϵ cond and ϵ norm, the ground state energies per particle of the system restricted to the subspaces F cond and F norm , respectively: ϵ = min{ϵ cond, ϵ norm}. It may then happen that, as a function of the parameter g, the energies ϵ cond and ϵ 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.