Effects of Pauli paramagnetism on superconducting vortex phase diagram in strong fields

The Ginzburg-Landau (GL) functional and the resultant phase diagram in strong fields where the Pauli paramagnetic depairing is not negligible are examined in details by assuming the weak coupling BCS model with a $d_{x^2-y^2}$-like pairing. In contrast to previous works on the same subject for spin-singlet pairings in which the orbital depairing effect was not treated consistently with the paramagnetic depairing, the temperature at which the mean field (MF) transition at $H_{c2}$ changes into a discontinuous one lies much above another temperature at which the $H_{c2}(T)$ and a second order transition curve $H_{\rm FFLO}(T)$ between an FFLO-like and the ordinary vortex solids branch, and $H_{\rm FFLO}(T)$ decreases upon cooling. Based on these MF results, details of a real phase diagram near $H_{c2}(T)$-line are examined in terms of Monte Carlo simulation, and it is found that the MF discontinuous transition is changed due to the fluctuation into a crossover which is nearly discontinuous in systems with weak enough fluctuation. These results induced by the paramagnetic depairing are consistent both with recent observations of the MF discontinuous behavior and $H_{\rm {FFLO}}(T)$ in the heavy fermion superconductor CeCoIn$_5$ with weak fluctuation and with their absence in organic materials with strong fluctuation.