Interplay of Kekul\'{e} bond order and lattice instability in $\mathrm{C}_6\mathrm{Li}$

Understanding the interplay between charge order and lattice instability in quantum materials remains a central challenge, as their coexistence often obscures causal relationships. This work introduces $\mathrm{C}_6\mathrm{Li}$ as a novel platform to investigate charge order mediated by two distinct mechanisms. We show that the hybridization between carbon $\pi$ and lithium $s$ orbitals generates an effective long-range hopping within Li-centered hexagons. This hopping drives a Kekul\'{e} bond order, whose structure varies with charge density and the sign of the hopping. This bond order induces a Kekul\'{e} lattice distortion via electron-phonon coupling. In the limit where lithium atoms are distant from the graphene layer, a Fermi surface nesting-driven Kekul\'{e} bond order emerges, stabilized by the electron-phonon interaction. Our results establish $\mathrm{C}_6\mathrm{Li}$ as a tunable platform for elucidating the causal hierarchy between electronic and structural orders in quantum materials.