  While the ground state phase diagram of the correlated flat-band systems have
been intensively investigated, the dynamic and thermodynamic properties of such
lattice models are less explored, but it is the latter which is most relevant
to the experimental probes (transport, quantum capacitance and spectroscopy) of
the quantum moir\'e materials such as twisted bilayer graphene and transition
metal dichalcogenides. Here we show, by means of momentum-space quantum Monte
Carlo and exact diagonalization, there exists a unique thermodynamic
characteristic for the correlated flat-band models with interaction-driven
quantum anomalous Hall (QAH) ground state, namely, the transition from the QAH
insulator to the metallic state takes place at a much lower temperature
compared with the zero-temperature single-particle gap generated by the
long-range Coulomb interaction. Such low transition temperature comes from the
proliferation of excitonic particle-hole excitations, which "quantum teleport"
the electrons across the gap between different topological bands to restore the
broken time-reversal symmetry and give rise to a pronounced enhancement in the
charge compressibility. Future experiments, to verify such generic
thermodynamic characteristics, are proposed.
