SU(3) Quantum Hall Ferromagnetism in SnTe

The (111) surface of SnTe hosts one isotropic G-centered and three degenerate anisotropic M- centered Dirac surface states. We predict that a nematic phase with spontaneously broken C3 symmetry will occur in the presence of an external magnetic field when the N$=$0 M Landau levels are 1/3 or 2/3 filled. The nematic state phase boundary is controlled by a competition between intravalley Coulomb interactions that favor a valley-polarized state, and weaker intervalley scattering processes that increase in relative strength with magnetic field. An in-plane Zeeman field alters the phase diagram by lifting the three-fold M Landau level degeneracy, yielding a ground state energy with 2$\pi $/3 periodicity as a function of Zeeman-field orientation angle.