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Integer and fractional Chern insulators have been extensively explored incorrelated flat band models. Recently, the prediction and experimentalobservation of fractional quantum anomalous Hall (FQAH) states with spontaneoustime-reversal-symmetry breaking have garnered attention. While thethermodynamics of integer quantum anomalous Hall (IQAH) states have beensystematically studied based on models of twisted bilayer graphene, ourtheoretical knowledge on finite-temperature thermodynamic properties of FQAHstates has been severely limited. In this study, we delve into thethermodynamic response and collective excitations of both IQAH and FQAH stateswithin the paradigmatic correlated flat band model on the checkerboard lattice.Our key findings include: i) In both $\nu = 1$ IQAH and $\nu = 1/3$ FQAHstates, it is the charge-neutral collective excitations that determine theonset temperature of these topological states, a value significantly smallerthan the charge gap, aligning with recent experimental observations; ii) Thelowest collective excitations manifest as the zero-momentum excitons in theIQAH state, whereas in the FQAH state, they take the form of magneto-rotonswith finite momentum; iii) The excitons and rotons exhibit distinctexperimental signatures, which we propose to detect in future experiments.

Thermodynamic Response and Neutral Excitations in Integer and Fractional Quantum Anomalous Hall States Emerging from Correlated Flat Bands