几乎交换矩阵、定域Wannier函数与量子霍尔效应
Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
摘要：
对于非相互作用费米子在三维空间中的表面上排列的位置内移动的模型，可能会有障碍来找到局部化的Wannier函数。我们发现，这样的障碍是$K$理论的障碍，近似几乎可交换的，复值矩阵的交换矩阵，我们证明了数值存在这种障碍的量子霍尔效应的晶格模型在一个球形几何形状。障碍物的数值计算是直接的，并且不需要平移不变性或引入通量环面。我们进一步证明了用完全可交换的自对偶矩阵近似几乎可交换的自对偶矩阵存在一个$Z_2$指数障碍，并给出了用完全可交换的实自对偶矩阵近似几乎可交换的实自对偶矩阵的一些新的证明.考虑这个问题的动机是具有额外的反酉对称性的物理系统的情况，例如时间反演或粒子-空穴共轭。最后，在球的情况下-数学上说，三个几乎通勤厄米派的平方和附近的身份-我们给出了第一个定量结果表明，这个指数是唯一的障碍，找到通勤近似。我们回顾已知的非定量结果的环面。
英文摘要：
For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We further show that there is a $Z_2$ index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices, and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as time reversal or particle-hole conjugation. Finally, in the case of the sphere--mathematically speaking three almost commuting Hermitians whose sum of square is near the identity--we give the first quantitative result showing this index is the only obstruction to finding commuting approximations. We review the known non-quantitative results for the torus.
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