We discuss twisted bilayer graphene (TBG) based on a theorem of flat-band ferromagnetism put forward by Mielke and Tasaki. According to this theorem, ferromagnetism occurs if the single-particle density matrix of the flat-band states is irreducible and we argue that this result can be applied to the quasi-flat-bands of TBG that emerge around the charge-neutrality point for twist angles around the magic angle $\theta \sim 1.{05}^{\circ }$. We show that the density matrix is irreducible in this case, thus predicting a ferromagnetic ground state for neutral TBG ($n=0$). We then show that the theorem can also be applied only to the flat conduction or valence bands, if the substrate induces a single-particle gap at charge neutrality. Also in this case, the corresponding density matrix turns out to be irreducible, leading to ferromagnetism at half filling ($n=\pm 2$).

中文翻译：

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双层石墨烯中的平带铁磁性

我们根据Mielke和Tasaki提出的平带铁磁定理讨论双分子层石墨烯（TBG）。根据该定理，如果平带态的单粒子密度矩阵不可约，则铁磁性发生，并且我们认为该结果可以应用于绕电荷中性点出现的TBG的准平带进行扭曲围绕魔角的角度$\theta 〜\mathrm{1个}。{05}^{\circ }$。我们证明在这种情况下密度矩阵是不可约的，因此可以预测中性TBG的铁磁基态（$ñ=0$）。然后，我们证明，如果底物在电荷中性处诱导出单粒子间隙，则该定理也只能应用于平坦的导带或价带。同样在这种情况下，相应的密度矩阵变成不可约的，导致半填充时的铁磁性（$ñ=\pm 2$）。