Quantum Hall to Charge-Density-Wave Phase Transitions in $ABC$-Trilayer Graphene

$ABC$-stacked trilayer graphene’s chiral band structure results in three ($n=0$, 1, 2) Landau level orbitals with zero kinetic energy. This unique feature has important consequences on the interaction-driven states of the 12-fold degenerate (including spin and valley) $N=0$ Landau level. In particular, at many filling factors ${\nu }_T=\pm 5$, $\pm 4$, $\pm 2$, $\pm 1$ a quantum phase transition from a quantum Hall liquid state to a triangular charge-density wave occurs as a function of the single particle-induced Landau level orbital splitting ${\Delta }_{\mathrm{LL}}$. Experimental signatures of this phase transition are also discussed.

Figure 1

Schematic representation of the interaction-induced energy gaps as a function the magnetic field for interaction-driven states in $ABC$-trilayer graphene’s 12-fold degenerate LL; the numbers represent the Hall conductivity in units of $e^2/h$. For (a) ${\Delta }_{\mathrm{LL}}<{\Delta }_{\mathrm{LL}}^{(c)}$ the presence of insulating triangular CDW states for $\nu =\pm 5$, $\pm 4$, $\pm 2$, $\pm 1$ results in Hall plateaus only at ${\sigma }_{xy}=\pm 6$, $\pm 3$, $0(e^2/h)$. For (b) ${\Delta }_{\mathrm{LL}}>{\Delta }_{\mathrm{LL}}^{(c)}$ interaction-driven QH states appear at all intermediate integer fillings with the sequence determined by Hund’s rule behavior (see text for details).

Figure 2

Collective mode dispersion ${\omega }_q$ for $\nu =-5$, $-2$, 1, 4 in units of interaction strength $e^2/\epsilon l_B=11.2(B[T]{)}^{1/2}\mathrm{meV}$ as a function of $q{l}_B$ at different values of the LL-splitting ${\Delta }_{\mathrm{LL}}$ at a magnetic field of $B=10\mathrm{Tesla}$. Charge density—wave instability is accompanied by a softening of the magnetoroton minima for ${\Delta }_{\mathrm{LL}}\le {\Delta }_{\mathrm{LL}}^{(*)}$. The gapped mode corresponds to the hard direction in LL pseudospin space and is not essential for our discussion (see text for details).

Figure 3

Electronic density and orientation of the in-plane electric dipole field for the triangular crystal at filling factor (a) $\nu =-5$, $-2$, 1, 4 and (b) $\nu =-4$, $-1$, 2, 5.The electrons for (a) and holes for (b) crystallize in a triangular pattern with a lattice constant $a=\sqrt{4\pi /\sqrt{3}}{l}_B$. The crystal state is characterized by a superposition of three LL orbitals $n=0$, 1, 2 pseudospin which manifests itself as a periodic density modulation $n(\mathbf{r})$ (see text for details).