Interlayer-engineering of charge order wavevector in kagome metals

Charge orders in the kagome metals AV$_3$Sb$_5$ sit at the center of a rich phase diagram that also includes superconductivity, nematicity, and signatures of time-reversal-symmetry breaking. Yet even the basic question of which charge ordering wave vectors are intrinsic, and which are selected by dimensionality and lattice coupling, remains unsettled. Importantly, the microscopic origin of different charge orders and, in particular, the relationship between the robust bulk $2 \\times 2$ charge order and the controversial $4 \\times 1$ modulation, which is primarily resolved by surface probes, remains unresolved. Here, we use first-principles calculations to study the role of interlayer coupling in CsV$_3$Sb$_5$ by tuning the interlayer separation from the monolayer limit to the bulk limit. In the monolayer AV$_3$Sb$_5$ (A = Rb, Cs), the phonon spectrum exhibits no instability at the M point; instead, the dominant lattice instability occurs at q = (1/4, 0, 0), consistent with a $4 \\times 1$ modulation. As interlayer coupling increases in CsV$_3$Sb$_5$, an M-point phonon progressively softens and becomes unstable already near c = 12.24 Å, evolving into the strong $2 \\times 2$ instability characteristic of the bulk. These results identify interlayer coupling as a control parameter at a fixed stoichiometry that links competing $4 \\times 1$ and $2 \\times 2$ tendencies, providing a unified framework for understanding why multiple charge-order wave vectors coexist and compete in kagome metals.