Theory of charge-6e condensed phase in Kagome lattice superconductors

We develop a Ginzburg-Landau theory for commensurate pair density wave (PDW) states in a hexagonal lattice system, relevant to the kagome superconductors $\mathrm{A}{\mathrm{V}}_3\mathrm{S}{\mathrm{b}}_5$. Compared to previous theoretical frameworks, the commensurate wave vectors permit additional symmetric terms in the free energy, altering the system’s ground state and its degeneracy. In particular, we analyze topological defects in the energetically favorable ${\psi }_{\text{kagome}}$ ground state and find that kinks on domain walls can carry $1\mathrm{/}3$ topological charges. We further establish a correspondence between the SC fluctuations in these states and an effective $J_1-J_2$ frustrated XY model on the emergent kagome lattice. By employing a state-of-the-art numerical tensor network method, we rigorously solve this effective model at finite temperatures and confirm the existence of a vestigial phase characterized by $1\mathrm{/}3$ vortex-antivortex pairs in low temperatures with the absence of phase coherence of Cooper pairs, which is dual to the charge-$6e$ condensed phase. Our theory provides a potential explanation for the vestigial charge-$6e$ magnetoresistance oscillations observed in recent experiments [J. Ge, et. al., Phys. Rev. X 14, 021025 (2024)].