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 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 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.