Energy quantization at the three-quarter Dirac point in a magnetic field

The quantization of the energy in a magnetic field (Landau quantization) at a three-quarter Dirac point is studied theoretically. The three-quarter Dirac point is realized in the system of massless Dirac fermions with the critically tilted Dirac cone in one direction, where a linear term disappears and a quadratic term α2q2x with a constant α2 plays an important role. The energy is obtained as En∝α352(nB)45, where n=1,2,3,⋯, by means of numerically solving the differential equation. The same result is obtained analytically by adopting an approximation. The result is consistent with the semiclassical quantization rule studied previously. The existence of the n=0 state is studied by introducing the energy gap due to the inversion-symmetry-breaking term, and it is obtained that the n=0 state exists in one of a pair of three-quarter Dirac points, depending on the direction of the magnetic field when the energy gap is finite