Based on a theoretical argument and Monte Carlo simulations of a Ginzburg-Landau model derived microscopically, it is argued that, in type-II superconductors where { \it both} the paramagnetic { \it and} orbital depairings are important, a strong first-order transition (FOT) at $H_{ c2}$ expected in the mean field (MF) approximation never occurs in real systems and changes due to the fluctuation into a crossover. The present result explains why a { \it nearly} discontinuous crossover at $H_{ c2}$ with { \it no} intrinsic hysteresis is observed only in a clean superconducting material with a singlet pairing and a high condensation energy such as CeCoIn$_5$.
Categories: cond-mat.supr-con cond-mat.str-el