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 jump discontinuity, can be described as condensation space states. Given system having Hamiltonian $H=K+gV$, where $K$ and $V$ are two non commuting operators acting on states $\mathbb{F}$, we may always write $\mathbb{F}=\mathbb{F}_\mathrm{cond} \oplus \mathbb{F}_\mathrm{norm}$ $\mathbb{F}_\mathrm{cond}$ subspace spanned by eigenstates with minimal eigenvalue $\mathbb{F}_\mathrm{norm}=\mathbb{F}_\mathrm{cond}^\perp$. If, thermodynamic limit, $M_\mathrm{cond}/M \to 0$, $M$ $M_\mathrm{cond}$ are, respectively, dimensions $\mathbb{F}$ $\mathbb{F}_\mathrm{cond}$, above decomposition becomes effective, sense system, $\epsilon$, coincides smaller between $\epsilon_\mathrm{cond}$ $\epsilon_\mathrm{norm}$, energies restricted to subspaces $\mathbb{F}_\mathrm{norm}$, respectively: $\epsilon=\min\{\epsilon_\mathrm{cond},\epsilon_\mathrm{norm}\}$. It then happen that, function parameter $g$, $\epsilon_\mathrm{norm}$ cross at $g=g_\mathrm{c}$. In this case, transition takes place condensed (system small $\mathbb{F}_\mathrm{cond}$) normal spread over $\mathbb{F}_\mathrm{norm}$)....