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) in the nondegenerate quarter metal for the entire family of chirally-stacked $n$ layer graphene, encompassing rhombohedral multilayer, Bernal bilayer, and monolayer cousins. Besides the hallmark broken translational symmetry, yielding a modulated charge density over an enlarged unit-cell with a characteristic $2{\bf K}$ periodicity, where $\pm {\bf K}$ are the valley momenta, this phase lacks the three-fold ($C_3$) rotational symmetry but only for $n=1$ and even integer $n$. The VC-CDW then represents a stripe order, as observed in hexalayer graphene [\href{https://arxiv.org/abs/2504.05129}{arXiv:2504.05129}], but preserves the $C_3$ symmetry for other odd integer $n$ as observed in trilayer graphene [\href{https://www.nature.com/articles/s41567-024-02560-7}{Nat.\ Phys.\ {\bf 20}, 1413 (2024)} and \href{https://arxiv.org/abs/2411.11163}{arXiv:2411.11163}]. From a \emph{universal} Clifford algebraic argument, we show 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. Only the anomalous Hall order displays a hysteresis in off-diagonal resistivity, as observed in all the systems with $2 \leq n \leq 6$. We showcase a confluence of VC-CDW and anomalous Hall orders within the quarter metal, \emph{generically} displaying a regime of coexistence, otherwise separating the pure phases.