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 $\mathbb{F}={\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}\oplus {\mathbb{F}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ where ${\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ is the subspace spanned by the eigenstates of $V$ with minimal eigenvalue and ${\mathbb{F}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}^{\perp }$. If, in the thermodynamic limit, $M_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}\mathrm{/}M\to 0$, where $M$ and $M_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ are, respectively, the dimensions of $\mathbb{F}$ and ${\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$, the above decomposition of $\mathbb{F}$ becomes effective, in the sense that the ground state energy per particle of the system, $ϵ$, coincides with the smaller between ${ϵ}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ and ${ϵ}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$, the ground state energies per particle of the system restricted to the subspaces ${\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ and ${\mathbb{F}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$, respectively: $ϵ=\min \{{ϵ}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}},{ϵ}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\}$. It may then happen that, as a function of the parameter $g$, the energies ${ϵ}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ and ${ϵ}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ 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 ${\mathbb{F}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$) and a normal phase (system spread over the large subspace ${\mathbb{F}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$)…