SU(3) Quantum Hall Ferromagnetism in SnTe

The (111) surface of SnTe hosts one isotropic $\overline{\mathrm{\Gamma }}$-centered and three degenerate anisotropic $\overline{M}$-centered Dirac surface states. We predict that a nematic phase with spontaneously broken ${\mathcal{C}}_3$ symmetry will occur in the presence of a perpendicular magnetic field when the $N=0\overline{M}$ 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 threefold $\overline{M}$ Landau-level degeneracy, yielding a ground state energy with $2\pi /3$ periodicity as a function of Zeeman-field orientation angle.

Figure 1

(a) Typical equal-energy contours for gapless Dirac states on the (111) surface of SnTe. The three mirror invariance lines are indicated by dotted lines. (b),(c) Single-particle LL structures of (a). The energy difference between $\overline{\mathrm{\Gamma }}$ and $\overline{M}$ Dirac points is 30 meV in (b) and $-20\mathrm{meV}$ in (c). Brown and blue lines represent the nondegenerate $\overline{\mathrm{\Gamma }}$ LLs and the threefold degenerate $\overline{M}$ LLs, respectively. The integer labels in (b) and (c) give LL filling factors in spectral gaps.

Figure 2

(a) In-plane Zeeman energies $m_{\lambda }$ in units of $\sqrt{2}{\eta }^{2}g{\mu }_B{B}_{\parallel }/9$ as a function of $B_{\parallel }$ orientation. (b) Phase diagram for the state at $1/3$ filling of the $N=0$ LL triplet for $\epsilon g{B}_{\parallel }=300\mathrm{T}$ and $U=0.85\mathrm{eV}{\mathrm{nm}}^2$. The red dot denotes the critical field $B_{\perp }^c$ at which a first-order transition occurs between a valley-polarized and a three-valley symmetric state occurs at $B_{\parallel }=0$. The solid blue lines are first-order phase boundaries between valley-polarized states, and the dashed red lines are continuous transition boundaries between states with coherence between different numbers of valleys. The phases labeled by ${\overline{M}}_i$, ${\overline{M}}_{ij}$, and ${\overline{M}}_{123}$ have a full LL spinor that is a coherent superposition of components involving one, two, and three valleys, respectively.