Integer quantum Hall effect in bilayer Dice lattices

We investigate the quantum Hall effect in bilayer dice lattices with AA-BB-CC and AB-BC-CA stacking configurations, focusing on the interplay between flat band stability and band topology. For the aligned AA-BB-CC stacking, the system exhibits a narrow zero Hall conductivity plateau at the Dirac point and hybrid integer quantum Hall states <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mo>(</a:mo><a:mi>n</a:mi><a:mo>=</a:mo><a:mo>±</a:mo><a:mn>2</a:mn><a:mo>,</a:mo><a:mspace width="0.16em"/><a:mo>±</a:mo><a:mn>4</a:mn><a:mo>)</a:mo></a:mrow></a:math> in some energy regions, originating from the coexistence of spin, valley, and layer degeneracies. Additionally, the flat band generated by AA-BB-CC stacking is more easily destroyed. In contrast, the cyclic AB-BC-CA stacking displays irregular Hall conductivity due to complex band structures lacking complete flat bands. By tuning interlayer coupling strength <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>t</c:mi><c:mi>⊥</c:mi></c:msub></c:math> and hopping parameter <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>α</d:mi></d:math>, the corresponding results show that the position of the Dirac point relative to the high-symmetry point in the Brillouin zone determines the appearance of the conductivity transitions. A mixed integer quantum Hall effect with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>n</e:mi></e:math> = 2 and = 4 appears for specific values of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>α</f:mi></f:math>.