Helical level structure of Dirac potential wells

In graphene and other massless two-dimensional Dirac materials, Klein tunneling compromises electron confinement, and momentum-space contours can be assigned a Berry phase which is either zero or π. Consequently, in such systems the energy spectrum of circular potential wells exhibits an interesting discontinuity as a function of magnetic field B : for a given angular momentum the ladder of eigen-resonances is split at an energy-dependent critical field B c . Here we show that introducing a mass term Δ in the Hamiltonian bridges this discontinuity in such a way that states below B c are adiabatically connected to states above B c whose principal quantum number differs by unity depending on the sign of Δ. In the B -Δ plane, the spectrum of these circular resonators resembles a spiral staircase, in which a particle prepared in the ∣ n, m ⟩ resonance state can be promoted to the ∣ n ± 1, m ⟩ state by an adiabatic circuit of the Hamiltonian about B c , the sign depending on the direction of the circuit. We explain the phenomenon in terms of the evolving Berry phase of the orbit, which in such a circuit changes adiabatically by 2 π .