Charge-$4e$ superconductivity is an exotic state of matter that may emerge as a vestigial order from a charge-$2e$ superconductor with multicomponent superconducting order parameters. Showing its emergence in a lattice phase model from numerically exact large-scale computations has remained rare. Here, we propose a kagome lattice model with a nematic superconducting ground state and show that it supports a rich set of vestigial phases at elevated temperatures, including a charge-$4e$ phase and a quasi-long-range nematic phase, through large-scale Monte Carlo simulations. Combining theoretical analysis with Monte Carlo simulations, we uncover the nature of these phases and show that the phase transitions are governed by the proliferation of distinct topological defects: the $(\frac{1}{2},0)$ half superconducting vortices, the $(\frac{1}{2},\frac{1}{2})$ vortices, the (0, 1) integer nematic vortices, and domain-wall excitations. In particular, we demonstrate that domain-wall proliferation is crucial for the quasinematic phase and should be carefully accounted for when analyzing phase transitions associated with vestigial charge-$4e$ order.

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• Exotic phases of matter
• Nematic order
• Phase transitions
• Strongly correlated systems
• Superconductors
• Unconventional superconductors