Geometric Phase and Orbital Moment in Quantization Rules for Magnetic Breakdown

The modern semiclassical theory of a Bloch electron in a magnetic field encompasses the orbital magnetization and geometric phase. Beyond this semiclassical theory lies the quantum description of field-induced tunneling between semiclassical orbits, known as magnetic breakdown. Here, we synthesize the modern semiclassical notions with quantum tunneling—into a single Bohr-Sommerfeld quantization rule that is predictive of magnetic energy levels. This rule is applicable to a host of topological solids with unremovable geometric phase, that also unavoidably undergo breakdown. A notion of topological invariants is formulated that nonperturbatively encode tunneling, and is measurable in the de Haas–van Alphen effect. Case studies are discussed for topological metals near a metal-insulator transition and overtilted Weyl fermions.

Figure 1

(a) Illustration of a region in ${\mathit{k}}^{\perp }$ space with significant tunneling. (b)–(c) Constant-energy band contours of two distinct orbits (above the saddle point) that merge into one (below). Black arrows indicate the orientation determined by Hamilton’s equation. (d)–(e) Plot of ${\theta }_{\eta }$ vs $|\mathcal{T}|=(1+e^{-2\pi \mu }{)}^{-1/2}$ for the conventional metal (blue line) and the topological metal (red). (f)–(g) Band dispersion of a conventional metal (left), and a topological metal (right).

Figure 2

An overtilted Weyl point that is not linked by tunneling to other Weyl points. (a) Band contours at fixed nonzero energy. (b) Zero-energy band contour of $H_{II}-|t|(1-{\sigma }_3)k_x^3≔d_1({\mathit{k}}^{\perp }){\sigma }_1+d_3({\mathit{k}}^{\perp }){\sigma }_3$; the $\mathit{d}$-vector is illustrated by blue arrows, with $d_1$ ($d_3$) the vertical (horizontal) component of each arrow. (c) Band dispersion. (d) Landau spectrum for the tight-binding model in Ref. [19], employing their units for energy and $|\mathit{B}|$.