Finite temperature quantum condensations in the space of states: general proof

We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently intro- duced and discussed at zero temperature (Ostilli and Presilla 2021 J. Phys. A: Math. Theor. 54 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space H into two subspaces, Hcond and Hnorm, such that, in the thermodynamic limit, dimHcond/dimH → 0 and the free 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 those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temper- ature of the system. We also show that the critical surface has universal features at high and low temperatures.