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

TL;DR

This paper theoretically investigates the unique Landau quantization of massless Dirac fermions at a three-quarter Dirac point with a tilted cone, revealing unconventional energy scaling and the conditions for zero-energy states.

Contribution

It introduces the concept of three-quarter Dirac points with a critically tilted cone and derives the quantization rule and zero-energy state conditions in a magnetic field.

Findings

Energy levels scale as $E_n o (n B)^{4/5}$

Zero-energy state existence depends on magnetic field direction and energy gap

Analytical and numerical methods confirm the unconventional quantization

Abstract

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 $\alpha_2 q_x^2$ with aconstant $\alpha_2$ plays an important role. The energy is obtained as $E_n \propto \alpha_2^{\frac{3}{5}} (n B)^{\frac{4}{5}}$, where $n=1, 2, 3, \dots$, by means of numerically and analytically solving the differential equation, as well as by the semiclassical quantization rule. 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.