We study flat bands and their topology in 2D materials with quadratic band crossing points under periodic strain. In contrast to Dirac points in graphene, where strain acts as a vector potential, strain for quadratic band crossing points serves as a director potential with angular momentum $\ell =2$. We prove that when the strengths of the strain fields hit certain “magic” values, exact flat bands with $C=\pm 1$ emerge at charge neutrality point in the chiral limit, in strong analogy to magic angle twisted-bilayer graphene. These flat bands have ideal quantum geometry for the realization of fractional Chern insulators, and they are always fragile topological. The number of flat bands can be doubled for certain point group, and the interacting Hamiltonian is exactly solvable at integer fillings. We further demonstrate the stability of these flat bands against deviations from the chiral limit, and discuss possible realization in 2D materials.

中文翻译：

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周期应变下二维材料中的拓扑精确平带

我们研究了在周期性应变下具有二次能带交叉点的二维材料中的平能带及其拓扑结构。与石墨烯中的狄拉克点相反，其中应变作为矢量势，二次能带交叉点的应变作为具有角动量的导向势$\ell =\mathrm{2个}$. 我们证明，当应变场的强度达到某些“神奇”值时，精确的平带$C=\pm \mathrm{1个}$出现在手性极限的电荷中性点，与魔角扭曲双层石墨烯非常相似。这些平带具有实现分数陈绝缘体的理想量子几何结构，并且它们始终是脆弱的拓扑结构。对于某些点群，平带的数量可以加倍，并且相互作用的哈密顿量在整数填充处是完全可解的。我们进一步证明了这些平带对偏离手性极限的稳定性，并讨论了二维材料中的可能实现。