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Multimechanism quantum anomalous Hall and Chern number tunable states in germanene (silicene, stanene)/ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mi>Bi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Te</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math> heterostructures
By constructing germanene (silicene, stanene)/$M{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ ($M=3d$-transition elements) heterostructures, we discovered and designed multimechanism quantum-anomalous-Hall (QAH) systems, including $\mathrm{\ensuremath{\Gamma}}$-based QAH, $K\ensuremath{-}{K}^{\ensuremath{'}}$-connected QAH, and valley-polarized $K$- or ${K}^{\ensuremath{'}}$-based QAH states via first-principle computations. The unique systems possess a global gap and tunable Chern number. The coexisting conventional $\mathrm{\ensuremath{\Gamma}}$-based QAH state of $M{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}$ …