Landau quantization of nearly degenerate bands and full symmetry classification of 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.