If an isolated quantum system admits a partition of its space of states
$\mathcal{H}$ into two subspaces, $\mathcal{H}_\mathrm{cond}$ and
$\mathcal{H}_\mathrm{norm}$, such that, in the thermodynamic limit, $\dim
\mathcal{H}_\mathrm{cond}/ \dim \mathcal{H} \to 0$ and the ground state
energies of the system restricted to these subspaces cross each other for some
value of the Hamiltonian parameters, then, the system undergoes a first-order
quantum phase transition driven by that parameter. A proof of this general
class of phase transitions, which represent a condensation in the space of
quantum states, was recently provided at zero temperature. It is reasonable to
extend the above condensation quantum phase transitions to finite temperature
by substituting the ground state energies with the corresponding free energies.
Here, we illustrate this criterion in two different systems. We derive,
analytically, the phase diagram of the paradigmatic Grover model and,
numerically, that of a system of free fermions in a one-dimensional
inhomogeneous lattice, where the condensation realizes as a spatial
condensation. These phase diagrams are structurally similar, in agreement with
the universal features of the present class of phase transitions. Finally, we
suggest an experimental realization of the fermionic system in terms of
heterostructure superlattices.
