Particle-hole symmetry and bifurcating ground-state manifold in the quantum Hall ferromagnetic states of multilayer graphene

The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc. stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor $\nu$ is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at $\nu =\pm 3,\pm 1$ in the bilayer and at $\nu =\pm 2,\pm 6$ in the ABCA four-layer, there is a transition to a Z${}_2\times$ U(1) manifold. For weak interaction, the structure in the zero-energy Landau band compensates for the different exchange interaction on the sublattices in the Landau orbitals; on the other side, the ground state comes in two copies that distribute charge on the sublattices differently. We expect a sequence of similar bifurcations in multilayers of Bernal stacking.

Figure 1

In the integer quantum Hall effect at $\nu =-2s$ of chiral $s$-layer graphene, the mixing of orbitals of index $\pm m$ produces an unequal charge distribution $w^{\pm }({\theta }_m,{\varphi }_m)=(1\pm sin{\theta }_{m}cos{\varphi }_m)/2$ on the sublattices that belong to the top spinorial component (empty symbols) and the bottom one (filled symbols). At $\nu =2s$ the sublattices are interchanged.

Figure 2

Bifurcation of the ground-state manifold. (a) In the bilayer graphene at $\overline{\nu }=1$, the orbitally coherent ground state comes in two symmetry-related copies if the interaction is strong enough, ${\beta }^{\left(2\right)}>{\beta }_{\mathrm{c}}^{\left(2\right)}\approx 3.56$. The insets show the circle(s) the orbital isospin $z_0\equiv cos\frac{\Theta }{2}{e}^{i\Phi /2}$, $z_1\equiv sin\frac{\Theta }{2}{e}^{-i\Phi /2}$ maps out in the two regions. (b) In the ABCA four-layer at $\overline{\nu }=2$ the ground state no longer belongs to the balanced subspace for ${\beta }^{\left(4\right)}>{\beta }_{\mathrm{c}}^{\left(4\right)}\approx 2.6$; thus it must come in symmetry-related pairs.

Figure 3

The orbital amplitude parameters $|z_i|$ of the QHF state $\overline{\nu }=1,s-1$ in (a) ABC trilayer graphene and (b) ABCA four-layer graphene.