Extended superconducting fluctuation region and 6e and 4e flux quantization in a kagome compound with a normal state of 3Q order

The superconducting state with the usual $2e$ flux quantization formed from a normal state with $3Q$ charge density or loop-current order is a linear combination of 3 different paired states with an overall gauge-invariant phase and two internal phases such that the phases in equilibrium are at $2\ensuremath{\pi}/3$ with respect to each other. In the fluctuation regime of such a 3-component superconductor, internal phase fluctuations are of the same class as for frustrated classical $\mathit{XY}$ spins on a triangular lattice. The fluctuation region is known therefore to be abnormally extended below the mean-field or the Kosterlitz-Thouless transition temperature. A $6e$ flux and a $4e$ flux quantized state can be constructed which are also eigenstates of the BCS Hamiltonian and stationary points of the Ginzburg-Landau free energy with a transition temperature above that of the renormalized $2e$ flux quantized state. Such states have no internal phases and so no frustrating internal phase fluctuations. These states however cannot acquire long-range order because their free energy is higher than the co-existing fluctuating state of $2e$ flux quantization. $6e$ as well as $4e$ flux quantized Little-Parks oscillations however occur in which the resistivity increases periodically with field above that of the $2e$ fluctuating state in its extended fluctuation regime, as is observed, followed at low temperatures to a condensation of the time-reversal odd $2e$ quantized state.