Quantum anomalous Hall effects and emergent $\rm{SU}(2)$ Hall ferromagnets at fractional filling of helical trilayer graphene

Abstract

Helical trilayer graphene realizes a versatile moiré system for exploring correlated topological states emerging from high Chern bands. Motivated by recent experimental observations of anomalous Hall effects at fractional fillings of magic-angle helical trilayers, we focus on the higher Chern number $|C_{band}|=2$ band and explore gapped many-body Hall states beyond the conventional Landau level paradigm. Through extensive exact diagonalization, we predict novel phases unattainable in a single $|C_{band}|=1$ band. At filling $ν=2/3$ and $ν=1/3$, a $\sqrt{3}\times \sqrt{3}$ charge-ordered quantum Hall crystal and a Halperin fractional Chern insulator with Hall conductance $|σ_{H}|=2e^2/3h$ are predicted respectively, indicating strong particle-hole asymmetry of the system. At half-filling $ν=1/2$, an extensively degenerate pseudospin Hall ferromagnet featuring emergent $\rm{SU}(2)$ symmetry is found without the band being flat. Inspired by striking robustness of the ferromagnetic degeneracy, we develop a method to unveil and quantify the emergent symmetry via pseudospin operator construction in the presence of band dispersion and Coulomb interaction, and demonstrate persistence of the $\rm{SU}(2)$ quantum numbers even far away from the chiral limit. Incorporating spin-valley degrees of freedom, we identify an optimal filling regime $ν_{\rm{total}}=3+ν$ for realizing the above states. Notably, inter-flavor interactions renormalize the bandwidth and stabilize all the gapped phases even in realistic sublattice corrugation parameter regimes.

Figures (4)

• Figure 1: Chern bands and qualitative many-body phase diagrams. (a) Single-particle Chern bands (black curve) with $C_{band}=-2$ and mean-field renormalized dispersion (red curve) for $w_{AA}/w_{AB}=0.6$. (b) Schematic flavor polarization at two filling regimes. (c) Phase diagrams for quantum Hall crystal (QHC), quantum Hall ferromagnet (QHF) and fractional Chern insulator (FCI) states at filling regimes $0<\nu<1$ and $\nu_{\rm{total}}=3+\nu$. Results for $\nu=2/3$, $1/2$, $1/3$ are obtained on $N_s=27,28,24$ clusters, respectively.
• Figure 2: Particle-hole asymmetry: Quantum Hall crystal at $\nu_{\rm{total}}=3+2/3$ in (a)-(c) and fractional Chern insulator at $\nu_{\rm{total}}=3+1/3$ in (d)-(f) on $N_s=36$ cluster with $w_{AA}/w_{AB}=0.6$. Three columns show energy spectrum, Chern number and structure factor $S(\mathbf{q})$.
• Figure 3: Quantum Hall ferromagnet at $\nu_{\rm{total}}=3+1/2$ with $w_{AA}/w_{AB}=0.6$. (a) Many-body energy spectrum on the $N_s=28$ cluster. (b) Degeneracy and spectrum gap across different cluster sizes. (c) Typical orbital occupation numbers of the momentum eigenstate (blue) and $\mathbf{Q}$-superposed CDW state (red) on the $N_s=28$ cluster. (d) Emergent $S^z$ quantum numbers for the pseudospin ferromagnetism with $S^{z}\approx -N_p/2,-N_p/2+1,...,N_p/2-1,N_p/2$.
• Figure 4: Energy spectra and quantitative phase diagrams versus $w_{AA}/w_{AB}$. Shaded regions show ranges of gapped ground states. Red curves show the evolution of $N_g$ quasi-degenerate ground states, while the black curves show the lowest excitation energies. System sizes for simulating the three phases are $N_s=27,28,24$, on which the degeneracies are $N_g=3,15,3$, respectively.