Motivated by the observation of a pair density wave (PDW) in the kagome metal $\mathrm{CsV}{}_3\mathrm{Sb}{}_5$, we consider the fate of electrons near a $p$-type Van Hove singularity (vHS) of the kagome lattice with on-site Hubbard $U$ and the nearest-neighbor repulsion $V$. We study the effect of such interactions on Fermi surface “patches” at the vHS. We show how a feature unique to the kagome lattice known as sublattice interference crucially affects the form of the interactions among the patches. The renormalization group (RG) flow of such interactions results in a regime where the nearest-neighbor interaction $V$ exceeds the on-site repulsion $U$. We identify this condition as being favorable for the formation of charge-density-wave (CDW) and PDW orders. In the weak-coupling limit, we find a complex CDW order as the leading instability, which breaks time-reversal symmetry. Beyond RG, we perform a Hartree-Fock study to a $V$-only model and find the pair-density-wave order indeed sets in at some intermediate coupling.

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• Charge density waves
• FFLO
• Kagome lattice
• Mean field theory
• Renormalization group