Correlation stabilized anomalous Hall crystal in bilayer graphene

When the charge density is sufficiently low, interacting two-dimensional electron gas (2DEG) would undergo a phase transition from homogeneous Fermi liquid to an electronic crystal state, known as Wigner crystal. Besides conventional 2DEG, various topological fermionic excitations may also be realized in 2D materials. For example, ``high-order" Dirac fermions exhibiting nontrivial Berry phases may approximately characterize the low-energy excitations in rhombohedral multilayer graphene (RMG). In this work, we develop a beyond-mean-field theoretical framework to study the interacting ground states and single-particle excitations in slightly charge-doped RMG under vertical electric field. We find that transitions from Fermi liquid to trivial Wigner-crystal states would occur at critical carrier density $\sim 10^{10}\,\textrm{cm}^{-2}$ for all $n$-layer RMG (with $n=2, 3, 4, 5, 6$) which are approximately described by $n$-order Dirac-fermion models. Most saliently, using a more realistic modeling of bilayer graphene including trigonal warping effects, we find that an anomalous Hall crystal state with spontaneous quantized anomalous Hall conductivity would emerge when the carrier density is below $\sim 1\times 10^{11}\,\text{cm}^{-2}$, and it becomes the unique ground state over trivial Wigner crystal when the density is further lower. Counter intuitively, such topological anomalous Hall crystal becomes more stable than the trivial Wigner crystal due to the lower correlation energy gained from dynamical charge fluctuations, which is beyond mean-field description. Our work suggests that slightly carrier-doped bilayer graphene is one of the most promising candidates to realize anomalous Hall crystal. Moreover, the method developed in this work can be readily applied to other interacting 2D systems including moir\'e superlattices.