Landau quantization of nearly degenerate bands, and full symmetry classification of avoided Landau-level crossings

Abstract

Semiclassical quantization rules compactly describe the energy dispersion of Landau levels, and are predictive of quantum oscillations in transport and thermodynamic quantities. Such rules -- as formulated by Onsager, Lifshitz and Roth -- apply when the spin-orbit interaction dominates over the Zeeman interaction (or vice versa), but does not generally apply when the two interactions are comparable in strength. In this work, we present a generalized quantization rule which treats the spin-orbit and Zeeman interactions on equal footing, and therefore has wider applicability to spin-orbit-coupled materials lacking a spatial inversion center, or having magnetic order. More generally, our rule describes the Landau quantization of any number of nearly degenerate energy bands -- in any symmetry class. The resultant Landau-level spectrum is generically non-equidistant but may contain spin (or pseudospin) degeneracies. To tune to such degeneracies in the absence of crystalline point-group symmetries, three real parameters are needed. We have exhaustively identified all symmetry classes of cyclotron orbits for which this number is reduced from three, thus establishing symmetry-enforced 'non-crossing rules' for Landau levels. In particular, only one parameter is needed in the presence of spatial rotation or inversion; this single parameter may be the magnitude or orientation of the field. Signatures of single-parameter tunability include (i) a smooth crossover between period-doubled and -undoubled quantum oscillations in the low-temperature Shubnikov-de Haas effect, as well as (ii) 'magic-angle' magnetoresistance oscillations. We demonstrate the utility of our quantization rule, as well as the tunability of Landau-level degeneracies, for the Rashba-Dresselhaus two-dimensional electron gas -- subject to an arbitrarily oriented magnetic field.