Moiré Fractional Chern Insulators III: Hartree-Fock Phase Diagram, Magic Angle Regime for Chern Insulator States, the Role of the Moiré Potential and Goldstone Gaps in Rhombohedral Graphene Superlattices

TL;DR

This work analyzes the ν=+1 phase diagram of L-layer rhombohedral graphene aligned to hBN (R$L$G/hBN) using self-consistent Hartree-Fock and time-dependent Hartree-Fock methods, incorporating a 3D Coulomb interaction, two stacking configurations ξ=0,1, valence-band moiré charge backgrounds, and two interaction schemes (average and CN). It demonstrates that the average scheme transmits moiré physics from filled valence bands into the conduction bands, producing stacking-dependent topologies (e.g., |C|=1 for ξ=1) at large displacement fields, while the CN scheme largely decouples valence-band moiré effects, yielding similar phase behavior for both stackings. The study reveals a potential magic-angle regime where correlated topological phases are favored and provides a detailed TDHF analysis of collective modes, finding a low-energy valley magnon mode and moiré-pinned pseudophonons whose gaps depend sensitively on κ$_{\text{hBN}}$ and the chosen interaction scheme. These results underscore the necessity of careful microscopic modelling of both single-particle and interaction terms to understand FCIs and Chern insulators in rhombohedral graphene moiré systems and offer falsifiable predictions for experimental tests of interaction schemes.

Abstract

We investigate in detail the $ν=+1$ displacement-field-tuned interacting phase diagram of $L=3,4,5,6,7$ layer rhombohedral graphene aligned to hBN (R$L$G/hBN). Our calculations account for the 3D nature of the Coulomb interaction, the inequivalent stacking orientations $ξ=0,1$, the effects of the filled valence bands, and the choice of `interaction scheme' for specifying the many-body Hamiltonian. We show that the latter has a dramatic impact on the Hartree-Fock phase boundaries and the properties of the phases, including for pentalayers (R5G/hBN) with large displacement field $D$ where recent experiments observed a Chern insulator at $ν=+1$ and fractional Chern insulators for $ν<1$. In this large $D$ regime, the low-energy conduction bands are polarized away from the aligned hBN layer, and are hence well-described by the folded bands of moiréless rhombohedral graphene at the non-interacting level. Despite this, the filled valence bands develop moiré-periodic charge density variations which can generate an effective moiré potential, thereby explicitly breaking the approximate continuous translation symmetry in the conduction bands, and leading to contrasting electronic topology in the ground state for the two stacking arrangements. Within time-dependent Hartree-Fock theory, we further characterize the strength of the moiré pinning potential in the Chern insulator phase by computing the low-energy $\mathbf{q}=0$ collective mode spectrum, where we identify competing gapped pseudophonon and valley magnon excitations. Our results emphasize the importance of careful examination of both the single-particle and interaction model for a proper understanding of the correlated phases in R$L$G/hBN.

Figures (55)

• Figure 1: (a) The moiré Brillouin zone (mBZ). $\text{K}_G$ and $\text{K}_{hBN}$ are the K point of rhombohedral graphene and hBN, respectively. The red hexagon is the mBZ, the red arrow is $\mathbf{q}_1 = \text{K}_G - \text{K}_{hBN}$, and $\theta$ is the twist angle. (b) Schematics of the two stacking configurations $\xi$ related by rotating only hBN by 180$^\circ$. C$_\text{A}$, C$_\text{B}$, B, N refer to the carbon atom at A sublattice, the carbon atom at B sublattice, boron atom in hBN, and nitrogen atom in hBN, respectively.
• Figure 2: The band structures of the R5G/hBN single-particle model in $\text{K}$ valley (Eq. \ref{['eq_main:H_K_nohBN']}) and $\theta=0.77^\circ$. Top (bottom) row shows $\xi=0$ ($\xi=1$) stacking. See Tab. \ref{['tab:parameters_full']} in App. \ref{['app:SP_model']} for a list of parameters in the continuum model.
• Figure 3: The dimensionless density fluctuation $\Delta\rho(\boldsymbol{r})$ (Eq. \ref{['eq_main:density_profile']}, color bar) of all valence bands (up to the momentum cutoff) of the continuum model (Eq. \ref{['eq_main:H_K_nohBN']}) in the $\text{K}$ valley for (a) $\xi=0, V=24$meV, (b) $\xi=0,V=48$meV, (c) $\xi=1,V=24$meV and (d) $\xi=1,V=48$meV. We choose $L=5$ layers, $\theta=0.77^\circ$, and 4 shells of reciprocal lattice vectors (in total 19 reciprocal lattice vectors, which give $95$ valence bands per valley per spin).
• Figure 4: HF phase diagram at $\nu=1$ for R5G/hBN in the average interaction scheme. (a) Results for $\xi=0$ stacking. The HF gap indicates the energy difference of the HF eigenvalues between the lowest unoccupied and highest occupied orbital, and is equivalent to the indirect gap for an insulating state. A non-zero $\max_\mathbf{k}[n(\mathbf{k})]-\min_\mathbf{k}[n(\mathbf{k})]$, where $n(\mathbf{k})$ counts the occupation number at momentum $\mathbf{k}$, rules out an insulating state. $C$ is the Chern number of the HF state. (b) Results for $\xi=1$ stacking. (c) Results for zero hBN coupling $\kappa_\text{hBN}=0$, which are independent of $\xi$. The HF calculations are performed with $(4+4)$ bands per spin/valley using the screened basis projection, average interaction scheme, $\theta=0.77^\circ$, and system size $N_1\times N_2=12\times 12$.
• Figure 5: Band structures of the self-consistent $\nu=1$ HF ground states for $\xi=1$ R5G/hBN as a function of $V$. We only show the fully polarized spin-$\uparrow$ sector. Black (red) shows the dispersion in microscropic valley $K$ ($K'$). The HF calculations are performed with $(4+4)$ bands per spin/valley using the screened basis projection, average interaction scheme, $\theta=0.77^\circ$, $\epsilon_r=6.25$, and system size $N_1\times N_2=12\times 12$.