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We investigate the fine structure in the energy spectrum of bilayer graphenein the presence of various stacking defaults, such as a translational orrotational mismatch. This fine structure consists of four Dirac points thatmove away from their original positions as a consequence of the mismatch andeventually merge in various manners. The different types of merging aredescribed in terms of topological invariants (winding numbers) that determinethe Landau-level spectrum in the presence of a magnetic field as well as thedegeneracy of the levels. The Landau-level spectrum is, within a wide parameterrange, well described by a semiclassical treatment that makes use oftopological winding numbers. However, the latter need to be redefined at zeroenergy in the high-magnetic-field limit as well as in the vicinity of saddlepoints in the zero-field dispersion relation.

Magnetic spectrum of trigonally warped bilayer graphene - semiclassical analysis, zero modes, and topological winding numbers