A quantum phase transition (QPT) occurs strictly speaking only at $T=0$, at a point where the gap in the Hamiltonian closes. At finite temperature, the transition becomes less sharp and a 'quantum critical region(QCR)' in parameter space emerges where the dynamics is governed by long-range quantum correlations.
Now, take for example the 2D transverse Ising model. At finite temperature, it retains a phase transition which is classical in nature (meaning the gap in Free energy closes), on a phase diagram its line bends outside of the QCR (meaning the Hamiltonian gap remains open).
Now my question is, can there exist phase transitions at high-$T$ where both the free energy gap and the Hamiltonan gap vanish simultaneously? What would be its nature?