Gap opening in the spectrum of some Dirac-like pseudo-differential operators

In this paper, we study the opening of a spectral gap for a class of 2-dimensional periodic Hamiltonians which include those modelling multilayer graphene. The kinetic part of the Hamiltonian is given by $\sigma \cdot F(-i\nabla)$, where $\sigma$ denotes the Pauli matrices and $F$ is a sufficiently regular vector-valued function which equals 0 at the origin and grows at infinity. Its spectrum is the whole real line. We prove that a gap appears for perturbations in a certain class of periodic matrix-valued potentials depending on $F$, and we study how this gap depends on different parameters.