SU(3) Quantum Hall Ferromagnetism in SnTe

The (111) surface of SnTe hosts one isotropic $\bar{\mathrm{\Gamma }}$-centered and three degenerate anisotropic $\bar{M}$- centered Dirac surface states. We predict that a nematic phase with spontaneously broken $C_3$ symmetry will occur in the presence of an external magnetic field when the $N=0$ $\bar{M}$ Landau levels are $1\mathrm{/}3$ or $2\mathrm{/}3$ 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 $\bar{M}$ Landau level degeneracy, yielding a ground state energy with $2\pi \mathrm{/}3$ periodicity as a function of Zeeman-field orientation angle.