Abstract
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 $\mathbb{Z}_2$ 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.
Highlights
The semiclassical Peierls-Onsager-Lifshitz theory [1,2,3] connects experimentally accessible quantities in magnetic phenomena to Fermi-surface parameters of the solid at zero field
The modern semiclassical theory of a Bloch electron in a magnetic field encompasses the orbital magnetic moment and the geometric phase. These two notions are encoded in the Bohr-Sommerfeld quantization condition as a phase (λ) that is subleading in powers of the field; λ 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
While it is conventionally believed that φB 1⁄4 0 vs π distinguishes between Schrödinger and Dirac systems [22], we propose to view φB=π as a continuous quantity that is sometimes fixed to an integer in certain space groups and for certain types of field-dependent orbits; while φR vanishes for centrosymmetric metals without spin-orbit coupling (SOC), it plays an oft-ignored role in most other space groups
Summary
The semiclassical Peierls-Onsager-Lifshitz theory [1,2,3] connects experimentally accessible quantities in magnetic phenomena to Fermi-surface parameters of the solid at zero field. Our comprehensive symmetry analysis identifies the (magnetic) space groups in which λ=π is robustly integer valued—we will formulate λ as a topological invariant in magnetotransport, which is distinct from the traditional formulation of topological invariance in band insulators [23,24]. We extend our symmetry analysis to the multiband generalization of λ, with envisioned application to bands of arbitrary degeneracy (D); D 1⁄4 2 is exemplified by spin degeneracy. These formulas extend previous works [3] in their applicability to orbits of any energy degeneracy and symmetry, including orbits in magnetic solids. VI; a final remark broadens the applicability of our symmetry analysis to matrix representations of holonomy [25] in the Brillouin torus, known as Wilson loops [26] of the Berry gauge field [13]
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