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Effects of Magnetic Order on the Upper Critical Field of UPt$_3$

I present a Ginzburg-Landau theory for hexagonal oscillations of the upper critical field of UPt$_3$ near $T_c$. The model is based on a $2D$ representation for the superconducting order parameter, $\vecη=(η_1,η_2)$, coupled to an in-plane AFM order parameter, $\vec{m}_s$. Hexagonal anisotropy of $H_{c2}$ arises from the weak in-plane anisotropy energy of the AFM state and the coupling of the superconducting order parameter to the staggered field. The model explains the important features of the observed hexagonal anisotropy [N. Keller, {\it et al.}, Phys. Rev. Lett. {\bf 73}, 2364 (1994).] including: (i) the small magnitude, (ii) persistence of the oscillations for $T\rightarrow T_c$, and (iii) the change in sign of the oscillations for $T> T^{*}$ and $T< T^{*}$ (the temperature at the tetracritical point). I also show that there is a low-field crossover (observable only very near $T_c$) below which the oscillations should vanish.