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Integer or fractional quantum Hall crystals, states postulating thecoexistence of charge order with integer or fractional quantum Hall effect,have long been proposed in theoretical studies in Landau levels. Inspired byrecent experiments on integer or fractional quantum anomalous Hall (IQAH/FQAH)states in MoTe2 and rhombohedral multilayer graphene, this work examines thearchetypal correlated flat band model on a checkerboard lattice at filling{\nu} = 2/3. Interestingly, at this filling level, we find that thistopological flatband does not stabilize conventional FQAH states. Instead, theunique interplay between smectic charge order and topological order gives riseto two intriguing quantum states. As the interaction strength increases, thesystem first transitions from a Fermi liquid into FQAH smectic (FQAHS) states,where FQAH topological order coexists cooperatively with smectic charge order.With a further increase in interaction strength, the system undergoes anotherquantum phase transition and evolves into a polar smectic metal. Contrary toconventional smectic order and FQAHS states, this gapless state spontaneouslybreaks the two-fold rotational symmetry, resulting in a nonzero electric dipolemoment and ferroelectric order. In addition to identifying the ground states,large-scale numerical simulations are also used to study low-energy excitationsand thermodynamic characteristics. We find that FQAHS states exhibit twodistinct temperature scales: the onset of charge order and the onset of thefractional Hall plateau, respectively. Interestingly, the latter is dictated bycharge-neutral low-energy excitations with finite momenta, known asmagnetorotons. Our studies suggest that these nontrivial phenomena could, inprinciple, be accessed in future experiments with moir\'e systems.

From Fractional Quantum Anomalous Hall Smectics to Polar Smectic Metals:  Nontrivial Interplay Between Electronic Liquid Crystal Order and Topological  Order in Correlated Topological Flat Bands