We systematically construct flatbands (FB) for tight-binding models on simple Bravais lattices in space dimension $d\ge 2$ in the presence of a static uniform DC field. Commensurate DC field directions yield irreducible Wannier-Stark (WS) bands in perpendicular dimension $d-1$ with $d$-dimensional eigenfunctions. The irreducible bands turn into dispersionless flatbands in the absence of nearest neighbor hoppings between lattice sites in any direction perpendicular to the DC field. The number of commensurate directions which yield flatbands is of measure one. We arrive at a complete halt of transport, with the DC field prohibiting transport along the field direction, and the flatbands prohibiting transport in all perpendicular directions as well. The anisotropic flatband eigenstates are localizing at least factorially (faster than exponential).

中文翻译：

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Bravais晶格中的Wannier-Stark平带

我们在空间维度上的简单Bravais格上系统地为紧绑定模型构建平坦带（FB） $d\ge 2$在静态的均匀直流场中。相称的直流磁场方向会在垂直方向产生不可还原的Wannier-Stark（WS）波段$d-\mathrm{1个}$ 和 $d$维本征函数。在垂直于直流场的任何方向上，在晶格位点之间没有最近的相邻跃点时，不可约带变成无色散平坦带。产生平坦带的相称方向的数量为一。我们完全停止了传输，DC场禁止沿场方向传输，而平坦带也禁止在所有垂直方向上传输。各向异性平带本征态至少局部分解（比指数快）。