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We consider a tight-binding model on the honeycomb lattice in a magneticfield. For special values of the hopping integrals, the dispersion relation islinear in one direction and quadratic in the other. We find that, in this case,the energy of the Landau levels varies with the field B as E_n(B) ~[(n+\gamma)B]^{2/3}. This result is obtained from the low-field study of thetight-binding spectrum on the honeycomb lattice in a magnetic field (Hofstadterspectrum) as well as from a calculation in the continuum approximation at lowfield. The latter links the new spectrum to the one of a modified quarticoscillator. The obtained value $\gamma=1/2$ is found to result from thecancellation of a Berry phase.

A new magnetic field dependence of Landau levels on a graphene like structure