Anomalous Hall Crystals in Rhombohedral Multilayer Graphene II: General Mechanism and a Minimal Model

TL;DR

This work addresses how a topological anomalous Hall crystal (AHC) can arise from interactions in rhombohedral multilayer graphene at zero magnetic field. It introduces a minimal three-patch approach, where three high-symmetry patches and two quantum-geometric phases control the Chern-ordering and energetics, yielding a solvable framework that predicts $C=1$ AHC in broad parameter regimes. The authors validate the mechanism by matching the three-patch predictions to self-consistent Hartree-Fock results on a microscopic RMG Hamiltonian, including a density-driven transition from $C=0$ to $C=1$ and characteristic real-space charge patterns. The results provide a tractable route to identify AHC candidates via spinor geometry and density tuning, with concrete experimental implications such as pressure-induced shifts in the phase boundary and a clear path beyond mean-field to explore correlated and possibly fractional AHC states.

Abstract

We propose a minimal "three-patch model" for the anomalous Hall crystal (AHC), a topological electronic state that spontaneously breaks both time-reversal symmetry and continuous translation symmetry. The proposal for this state is inspired by the recently observed integer and fractional quantum Hall states in rhombohedral multilayer graphene at zero magnetic field. There, interaction effects appear to amplify the effects of a weak moiré potential, leading to the formation of stable, isolated Chern bands. It has been further shown that Chern bands are stabilized in mean field calculations even without a moiré potential, enabling a realization of the AHC state. Our model is built upon the dissection of the Brillouin zone into patches centered around high symmetry points. Within this model, the wavefunctions at high symmetry points fully determine the topology and energetics of the state. We extract two quantum geometrical phases of the non-interacting wavefunctions that control the stability of the topologically nontrivial AHC state. The model predicts that the AHC state wins over the topological trivial Wigner crystal in a wide range of parameters, and agrees very well with the results of full self-consistent Hartree-Fock calculations of the rhombohedral multilayer graphene Hamiltonian.

Figures (14)

• Figure 1: Setup of the three-patch model. (a) Real space geometry of the superlattice. The three inequivalent Wyckoff positions are labeled by circles, squares, and triangles. (b) Mini Brillouin zone (mBZ) and associated high symmetry points. The points $\boldsymbol{\kappa}_1$, $\boldsymbol{\kappa}_3$, $\boldsymbol{\kappa}_5$ span the $\kappa$ point, while $\boldsymbol{\kappa}_2$, $\boldsymbol{\kappa}_4$, and $\boldsymbol{\kappa}_6$ span the $\kappa'$ point. (c) Schematic folded band structure of the model in the mBZ. The $\kappa$ and $\kappa'$ points have three-fold degeneracies. (d) Band gaps are opened up along mBZ boundaries by interactions, which crucially depend on (e) the nontrivial quantum geometry of multi-component wavefunctions.
• Figure 2: Predicted phase diagram by the three-patch model. (a) and (b) show the scattering processes contributing to the Hartree and Fock terms $E_{H}(C)$ and $E_F(C)$, and their associated form factors. (c) The angles of the form factors, $\theta_H$ and $\theta_F$, determine the Chern number favored by the Hartree and Fock terms respectively. Colored regions are where the Hartree and Fock ground states agree. Curves correspond to the evolution of $\theta_H$ and $\theta_F$ as a function of $0\leq \eta \leq 1$, which tunes the RMG-inspired spinors $\chi^{(N_L)}$ for different number of components $N_L$,. The transition points where the $C=1$ state becomes the ground state in the three-patch model for $N_L=3,4,5$ are very close to each other, and are marked by the yellow star in the phase diagram. The orange dot marks the $\eta=0$ point. (d) The energy difference between the $C=1$ state and the lowest-energy $C\neq 1$ state in the three-patch model for different number of components $N_L$ as a function of $\eta$. Transition points to the $C=1$ states are marked by yellow stars.
• Figure 3: Hartree and Fock energies of different $C$ states. The rows correspond to different number of components. The first two columns correspond to the Hartree and Fock terms $E_H(C)$ and $E_F(C)$, plotted with $V_{\boldsymbol{g}} = V_{\boldsymbol{\kappa}} = 1$. "HF, contact" is the sum of Hartree and Fock energies with $V_{\boldsymbol{g}} = V_{\boldsymbol{\kappa}} = 1$, while "HF, Coulomb" is computed with $V_{\boldsymbol{\kappa}} = 1, V_{\boldsymbol{g}} = 1/\sqrt{3}$. The yellow star marks the phase transition point between $C=0$ and $C=1$ using Coulomb interaction.
• Figure 4: Charge densities of competing candidate states of the three patch model. (left): the charge densities at the $\kappa$ point. The size of the circles corresponds to different amplitudes of charge densities, which come from different components of the spinor. Different angular momentum states labeled by $\ell$ are related to each other by translation. (right): the charge densities in the ${\kappa}'$ point. (center): the total charge densities of states with different Chern numbers.
• Figure 5: (a) The choice of mBZ for folding. The ${\gamma}$ point is chosen to coincide with the $K$ point of RMG. (b) Schematic side view of the structure of RMG, showing the staircase-like structure. We simplify the RMG model such that only the hoppings $t_{0,1}$ along the staircase are taken into account. (c) The band structure of the simplified Hamiltonian of RMG at $N_L=5$ (Eq. \ref{['eq:RMG_fullham']}) along the $k_x$ axis at $u_D = 40meV$. Vertical lines correspond to the position of $\boldsymbol{\kappa}$ point after folding the band structure at densities $(29,9.2,2.9)\times 10^{11}\mathrm{cm}^{-2}$ from outside to inside. (d), (e), (f) The RMG band structure after folding according to (c). Electronic densities are $(9.2,2.9,29)\times 10^{11}\mathrm{cm}^{-2}$ respectively. Gray shading corresponds to Coulomb interaction strengths at the interparticle distance corresponding to these densities. Only the electronic density in (d) is suitable for the three-patch model. (e) is below the optimal range of density for the three-patch model due to the degeneracy at $\boldsymbol{\gamma}$. (f) is above the optimal range of density because the bandwidth is too large.