Modern semiclassical theory of magnetic transport and breakdown

TL;DR

This paper advances semiclassical theory for magnetic transport in metals by developing a multi-component wavefunction, connection formulae for complex band crossings, and a graph-theoretic framework, incorporating geometric phases and topological invariants.

Contribution

It introduces a comprehensive semiclassical framework with phase corrections, symmetry analysis, and a graph approach to describe magnetic transport and Landau levels, including tunneling effects and topological invariants.

Findings

Extended WKB wavefunction with Berry phase and magnetic moment corrections.

Formulated connection formulae for saddlepoints and Dirac points.

Developed a graph-theoretic method for quantization and topological invariants.

Abstract

The Bohr-Sommerfeld quantization rule lies at the heart of the modern semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and quantum oscillations for conventional metals, as well as for a host of topological metals which have emerged in the recent intercourse between band theory, crystalline symmetries and topology. The essential ingredients in any quantization rule are connection formulae that match the semiclassical (WKB) wavefunction across regions of strong quantum fluctuations. Here, we propose (a) a multi-component WKB wavefunction that describes transport within degenerate-band subspaces, and (b) the requisite connection formulae for saddlepoints and type-II Dirac points, where tunneling respectively occurs within the same band, and between distinct bands. (a-b) extend previous works by incorporating phase corrections that are subleading in powers of the field; these corrections include the geometric Berry phase, and account for the orbital magnetic moment and the Zeeman coupling. A comprehensive symmetry analysis is performed for such phase corrections occurring in closed orbits, which is applicable to solids in any (magnetic) space group. We have further formulated a graph-theoretic description of semiclassical orbits. This allows us to systematize the construction of quantization rules for a large class of closed orbits (with or without tunneling), as well as to formulate the notion of a topological invariant in semiclassical magnetotransport -- as a quantity that is invariant under continuous deformations of the graph. Landau levels in the presence of tunneling are generically quasirandom, i.e., disordered on the scale of nearest-neighbor level spacings but having longer-ranged correlations; we develop a perturbative theory to determine Landau levels in such quasirandom spectra.