Fluctuation effects in a two-component $p$-wave superconductor

For a tetragonal material, order parameters of $p_x$ and $p_y$ symmetry are related by rotation and hence have the same $T_c$. This degeneracy can be lifted by a symmetry-breaking field, like (uniaxial) in-plane strain, such that at $T_c$, the order parameter is only of $p_x$ or $p_y$ symmetry. Only at a lower temperature also the respective other order parameter condenses. We analyze consequences of (thermal) fluctuations on these transition temperatures within a Ginzburg-Landau approach to obtain a comprehensive strain-temperature phase diagram. We find that the fluctuations can both enhance or suppress the effect of the symmetry breaking field, and even drive the system into a preemptive chiral phase. Possible consequences for the spin-triplet superconductor Sr$_2$RuO$_4$ will be discussed.