Spontaneous valley ordering in SnTe (001) surface states

In this work we study the spontaneous valley ordering in SnTe surface states. SnTe is a topological mirror insulator in which the bulk mirror symmetry allows robust surface states to appear on certain surfaces like (001), (111) and (110). These surface states provide a new platform to study multi-flavor Landau level physics in the quantum Hall regime. Attention has been drawn to the (001) surface by a recent experimental observation of a four-fold degenerate Landau level formed at zero energy [Science 341, 1496 (2013)]. This degeneracy arises because the single-particle bandstructure at low energies consists of four surface Dirac cones. In the presence of electron-electron interactions, however, this four-fold degeneracy is expected to be lifted. In this work we construct a theory to show that at all integer filling factors between -2 and $+$2 the ground state is a gapped quantum Hall state. We further demonstrate that these ground states are valley nematic phases, and discuss some experimental consequences.