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The advent of moiré platforms for engineered quantum matter has led to discoveries of integer and fractional quantum anomalous Hall effects, with predictions for correlation-driven topological states based on electron crystallization. Here, we report an array of trivial and topological insulators formed in a moiré lattice of rhomobohedral pentalayer graphene (R5G). At a doping of one electron per moiré unit cell ( $\nu =1$), we see a correlated insulator with a Chern number that can be tuned between $C=0$and $+1$by an electric displacement field. This is accompanied by a series of additional Chern insulators with $C=+1$originating from fractional fillings of the moiré lattice— $\nu =1/4$, $1/3$, and $2/3$—associated with the formation of moiré-driven topological electronic crystals. At $\nu =2/3$the system exhibits an integer quantum anomalous Hall effect at zero magnetic field, but further develops hints of an incipient $C=2/3$fractional Chern insulator in a modest field. Our results establish moiré R5G as a fertile platform for studying the competition and potential intertwining of integer and fractional Chern insulators. Published by the American Physical Society2025 

Downlink reconfigurable intelligent surface (RIS)-assisted multi-input-multi-output (MIMO) systems are considered with far-field, near-field, and hybrid-far-near-field channels. According to the angular or distance information contained in the received signals, 1) a distance-based codebook is designed for near-field MIMO channels, based on which a hierarchical beam training scheme is proposed to reduce the training overhead; 2) a combined angular-distance codebook is designed for hybrid-far-near-field MIMO channels, based on which a two-stage beam training scheme is proposed to achieve alignment in the angular and distance domains separately. For maximizing the achievable rate while reducing the complexity, an alternating optimization algorithm is proposed to carry out the joint optimization iteratively. Specifically, the RIS coefficient matrix is optimized through the beam training process, the optimal combining matrix is obtained from the closed-form solution for the mean square error (MSE) minimization problem, and the active beamforming matrix is optimized by exploiting the relationship between the achievable rate and MSE. Numerical results reveal that: 1) the proposed beam training schemes achieve near-optimal performance with a significantly decreased training overhead; 2) compared to the angular-only far-field channel model, taking the additional distance information into consideration will effectively improve the achievable rate when carrying out beam design for near-field communications.