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 $\mathbb{F}$, we may always write $\mathbb{F}=\mathbb{F}_\mathrm{cond} \oplus \mathbb{F}_\mathrm{norm}$ where $\mathbb{F}_\mathrm{cond}$ is the subspace spanned by the eigenstates of $V$ with minimal eigenvalue and $\mathbb{F}_\mathrm{norm}=\mathbb{F}_\mathrm{cond}^\perp$. If, in the thermodynamic limit, $M_\mathrm{cond}/M \to 0$, where $M$ and $M_\mathrm{cond}$ are, respectively, the dimensions of $\mathbb{F}$ and $\mathbb{F}_\mathrm{cond}$, the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, $\epsilon$, coincides with the smaller between $\epsilon_\mathrm{cond}$ and $\epsilon_\mathrm{norm}$, the ground state energies per particle of the system restricted to the subspaces $\mathbb{F}_\mathrm{cond}$ and $\mathbb{F}_\mathrm{norm}$, respectively: $\epsilon=\min\{\epsilon_\mathrm{cond},\epsilon_\mathrm{norm}\}$. It may then happen that, as a function of the parameter $g$, the energies $\epsilon_\mathrm{cond}$ and $\epsilon_\mathrm{norm}$ cross at $g=g_\mathrm{c}$. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace $\mathbb{F}_\mathrm{cond}$) and a normal phase (system spread over the large subspace $\mathbb{F}_\mathrm{norm}$)....