  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.
