Charge-density waves and stripes in quarter metals of graphene heterostructures

Motivated by recent experiments, here we identify valley-coherent charge-density wave (VC-CDW) order in the non-degenerate quarter-metal for the entire family of chirally-stacked $n$ layer graphene, encompassing rhombohedral multi-layer, Bernal bilayer, and monolayer cousins. Besides the hallmark broken translational symmetry, yielding a modulated charge-density over an enlarged unit-cell with a characteristic $2\mathbf{K}$ periodicity, where $\pm \mathbf{K}$ are the valley momenta, this phase lacks the three-fold ($C_3$) rotational symmetry but only for even integer $n$. The VC-CDW then represents a stripe order, as observed in hexalayer graphene [arXiv:2504.05129], but preserves the $C_3$ symmetry for odd $n$ as observed in trilayer graphene [Nat. Phys. 20, 1413 (2024) and arXiv: 2411.11163]. From a universal Clifford algebraic argument, we establish that the VC-CDW and an anomalous Hall order can lift the residual valley degeneracy of an antiferromagnetically ordered spin-polarized half-metal, when these systems are subject to perpendicular displacement fields, with only the latter one displaying a hysteresis in off-diagonal resistivity, as observed in all the systems with $2\le n\le 6$. We showcase a confluence of VC-CDW and anomalous Hall orders within the quarter-metal, generically displaying a regime of coexistence, separating the pure phases.