论文标题
扭曲的双层石墨烯I.矩阵元素，近似值，扰动理论和$ k \ cdot P $ 2频段模型
Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k\cdot p$ 2-Band model
论文作者
论文摘要
我们研究了扭曲的双层石墨烯（TBG）模型，以获得对其多体计算所需的能量和波形的分析理解。我们提供了一个近似方案，该方案首先阐明了为什么仅包含$ 4 $平面波的BM $ K_M $ $点计算为第一个魔术角提供了良好的分析值。近似方案还阐明了为什么投射到活跃带的库仑哈密顿量中的大多数多体矩阵元素可以被忽略。通过以第一个魔术角将我们的近似方案应用于$γ_m$ - 点的6平面波的中心模型，我们可以分析地理解同位素极限$ W_0中的活动和被动带之间的小$γ_m$ - 点差距。此外，我们通过分析计算各向同性极限中被动带的组速度，并表明它们是\ emph {几乎}双重变性，而没有对称性迫使它们成为。此外，我们从$γ_M$和$ k_m $ points不远，我们提供了明确的分析扰动理解，即尽管仅通过消除$ k_m $ $ $ $ point dirac Velocity来定义TBG频段为何以第一个魔术角平面。我们在分析上得出一个连接的“魔术歧管” $ W_1 = 2 \ sqrt {1+w_0^2} - \ sqrt {2+3w_0^2} $，在同时调整了$ w_0 $的频段在同位素（$ w_0 = w_1 $）和chiral（$ w_ilal（$ w_0 = 0）之间。我们分析地显示了为什么通过减少$ W_0 $（但不大于$ W_1 $）增加$γ_M$ $ - 积分和被动频段之间的点差距，从而脱离了各向同性限制。最后，我们提供了一个分析性$γ_m$ $ k \ cdot p $ 2 $ 2 $ 2 $ - 带模型，该模型在特定$ W_0，W_1 $参数范围内重现TBG频段结构和特征。该模型的进一步完善表明，在整个$ W_0，W_1 $参数范围内，可能忠实的$ 2 $ -band $γ_m$ $ k \ cdot p $型号。
We investigate the Twisted Bilayer Graphene (TBG) model to obtain an analytic understanding of its energetics and wavefunctions needed for many-body calculations. We provide an approximation scheme which first elucidates why the BM $K_M$-point centered calculation containing only $4$ plane-waves provides a good analytical value for the first magic angle. The approximation scheme also elucidates why most many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a $Γ_M$-point centered model of 6 plane-waves, we analytically understand the small $Γ_M$-point gap between the active and passive bands in the isotropic limit $w_0=w_1$. Furthermore, we analytically calculate the group velocities of passive bands in the isotropic limit, and show that they are \emph{almost} doubly degenerate, while no symmetry forces them to be. Furthermore, away from $Γ_M$ and $K_M$ points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite it is defined only by vanishing $K_M$-point Dirac velocity. We derive analytically a connected "magic manifold" $w_1=2\sqrt{1+w_0^2}-\sqrt{2+3w_0^2}$, on which the bands remain extremely flat as $w_0$ is tuned between the isotropic ($w_0=w_1$) and chiral ($w_0=0$) limits. We analytically show why going away from the isotropic limit by making $w_0$ less (but not larger) than $w_1$ increases the $Γ_M$- point gap between active and passive bands. Finally, perturbatively, we provide an analytic $Γ_M$ point $k\cdot p$ $2$-band model that reproduces the TBG band structure and eigenstates in a certain $w_0,w_1$ parameter range. Further refinement of this model suggests a possible faithful $2$-band $Γ_M$ point $k\cdot p$ model in the full $w_0, w_1$ parameter range.
