The modern semiclassical theory of a Bloch electron in a magnetic field now encompasses the orbital magnetic moment and the geometric phase. These two notions are encoded in the Bohr-Sommerfeld quantization condition as a phase (\lambda) that is subleading in powers of the field; \lambda is measurable in the phase offset of the de Haas-van Alphen oscillation, as well as of fixed-bias oscillations of the differential conductance in tunneling spectroscopy. In some solids and for certain field orientations, \lambda/\pi are robustly integer-valued owing to the symmetry of the extremal orbit, i.e., they are the topological invariants of magnetotransport. Our comprehensive symmetry analysis identifies solids in any (magnetic) space group for which \lambda is a topological invariant, as well as identifies the symmetry-enforced degeneracy of Landau levels. The analysis is simplified by our formulation of ten (and only ten) symmetry classes for closed, Fermi-surface orbits. Case studies are discussed for graphene, transition metal dichalchogenides, 3D Weyl and Dirac metals, and crystalline and \Z_2 topological insulators. In particular, we point out that a \pi phase offset in the fundamental oscillation should \emph{not} be viewed as a smoking gun for a 3D Dirac metal.

中文翻译：

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从磁量子振荡揭示费米面波函数的拓扑

磁场中布洛赫电子的现代半经典理论现在涵盖了轨道磁矩和几何相位。这两个概念在Bohr-Sommerfeld量化条件下被编码为一个相位（\ lambda），该相位在场强中处于次要地位。λ可以通过de Haas-van Alphen振荡的相位偏移以及隧道光谱中微分电导的固定偏置振荡来测量。在某些固体中，对于某些磁场方向，\ lambda / \ pi由于极值轨道的对称性而具有强整数值，即，它们是磁传输的拓扑不变量。我们全面的对称性分析可以确定\ lambda是拓扑不变量的任何（磁性）空间组中的固体，还可以确定Landau能级的对称性引起的简并性。通过为封闭的费米面轨道建立十个（只有十个）对称类别，简化了分析。讨论了有关石墨烯，过渡金属二卤化物，3D Weyl和Dirac金属以及晶体和\ Z_2拓扑绝缘子的案例研究。特别是，我们指出，基本振荡中的一个\ pi相移应该被视为3D Dirac金属的吸烟枪。