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

Abstract

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 $2{\bf K}$ periodicity, where $\pm {\bf K}$ 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 $2 \leq n \leq 6$. 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.

Figures (3)

• Figure 1: (a) Lattice structure of chirally-stacked $n$ layer graphene, where $A_j$ (magenta) and $B_j$ (green) correspond to the sites belonging to the $A$ and $B$ sublattice of the honeycomb lattice on the $j$th layer. In a system with $n$ layers, the low-energy two-band model near each valley and for each spin projection is constituted by the Wannier states localized on $A_1$ and $B_n$ sites. Hopping processes are denoted by double-headed arrows. The emergent lattice, constituted by low-energy sites, form (b) a honeycomb lattice when $n=2,4,6, \cdots$ and (c) a prism-like lattice for $n=3,5,\cdots$, where the two-site unit cells are shown by dashed closed loops.
• Figure 2: (a) Lattice realization of the valley-coherent charge-density wave of real amplitude $\Delta_0$ with (i) in-phase and (ii) out-of-phase modulations between the sites from the $A_1$ and $B_n$ sublattices (Fig. \ref{['fig:lattice']}). (b) Intra-sublattice/layer circulating currents in the direction of arrows, yielding the anomalous Hall order. The high-energy inert $B_1$ ($A_n$) sites in the bottom (top) layer are represented by open black circles. For $n>2$, all the sites in intermediate layers are inert.
• Figure 3: Solutions of the mean-field gap equation [obtained from Eq. \ref{['eq:freeenergy']}] at a temperature $T=10^{-5}$ displaying a competition between the anomalous Hall order with amplitude $\Delta_1$ (blue) and valley-coherent charge-density wave with amplitude $\Delta_2$ (red) with numbers on left vertical axes as a function of the ratio of the corresponding coupling constants $g_{_1}$ and $g_{_2}$ (black dashed curve with numbers on right vertical dashed axes), respectively, over a range of chemical potential $\mu$ in the quarter-metal of chirally-stacked $n$ layer graphene (see legends) SM. For smallest (largest) value of $\mu$ the system yields annular (simply connected) Fermi ring when $2 \leq n \leq 6$. For $n=1$ the Fermi ring is always simply connected. Appearance of anomalous Hall (valley-coherent charge-density wave) order for large (small) chemical potential is chosen to qualitatively mimic the experimental observation in rhombohedral trilayer ($n=3$) and hexalayer ($n=6$) graphene. Here, $\Delta_{1}$, $\Delta_{2}$, and $T$ are measured in units of $t_0$ (Fig. \ref{['fig:lattice']}).