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Validity of the two-band model of bilayer and trilayer graphene in a magnetic field

The eigenstates of an electron in the chiral two-dimensional electron gas (C2DEG) formed in an AB-stacked bilayer or an ABC-stacked trilayer graphene is a spinor with $4$ or $6$ components respectively. These components give the amplitude of the wave function on the $4$ or $6$ carbon sites in the unit cell of the lattice. In the tight-binding approximation, the eigenenergies are thus found by diagonalizing a $4\times 4$ or a $6\times 6$ matrix. In the continuum approximation where the electron wave vector $% k<<1/a_{0},$ with $a_{0}$ the lattice constant of the graphene sheets, a common approximation is the two-band model\cite{McCann2006} (2BM)\ where the eigenstates for the bilayer and trilayer systems are described by a two-component spinor that gives the amplitude of the wave function on the two sites with low-energy $\left\vert E\right\vert <<γ_{1}$ where $% γ_{1}$ is the hopping energy between sites that are directly above one another in adjacent layers. The 2BM has been used extensively to study the phase diagram of the C2DEG in a magnetic field as well as its transport and optical properties. In this paper, we use a numerical approach to compute the eigenstates and Landau level energies of the full tight-binding model in the continuum approximation and compare them with the prediction of the 2BM when the magnetic field or an electrical bias between the outermost layers is varied. Our numerical analysis shows that the 2BM is a good approximation for bilayer graphene in a wide range of magnetic field and bias but mostly for Landau level $M=0.$ The applicability of the 2BM in trilayer graphene, even for level $M=0,$ is much more restricted. In this case, the 2BM\ fails to reproduce some of the level crossings that occur between the sub-levels of $M=0.$