Semi-Dirac point in the Hofstadter spectrum

The spectrum of tight-binding electrons on a square lattice with half a magnetic-flux quantum per unit cell exhibits two Dirac points at the band center. We show that, in the presence of an additional uniaxial staggered potential, this pair of Dirac points may merge into a single one with a topological transition toward a gapped phase. At the transition, the spectrum is linear in one direction and quadratic in the other one (a spectrum recently named (hybrid or semi-Dirac). This transition is studied in the framework of a general Hamiltonian describing the merging of Dirac points. The possibility of creating gauge fields for cold atoms in optical lattices may offer an opportunity to observe this merging of Dirac points and the hybrid dispersion relation.