Ginzburg-Landau theory of the spin-charge-separated system

The phenomenological Ginzburg-Landau theory is developed for the resonating-valence-bond state where the spin and charge degrees of freedom are separated. We have the two order parameters corresponding to the fermion pairing and the Bose condensation, respectively, which are coupled with the gauge field. We find only one transition temperature which is the superconducting ${\mathit{T}}_{\mathit{c}}$. Above ${\mathit{T}}_{\mathit{c}}$, a crossover occurs from the spin-charge-separated phase to the Fermi liquid as the concentration of holes is increased. Below ${\mathit{T}}_{\mathit{c}}$, the penetration length λ is given by ${\lambda }^2$=${\lambda }_{\mathit{F}}^2$+${\lambda }_{\mathit{B}}^2$. The coherence length ξ, on the other hand, is complicated but is predicted to increase rapidly as the concentration approaches the overdoped region. There are two types of the vortex structure with the flux quantization hc/2e and hc/e, respectively. In the mean-field theory and the type-II limit, the former is stable in almost all the cases, but the latter becomes stable near ${\mathit{T}}_{\mathit{c}}$ in the low-hole-concentration region.