Absence of Ginzburg-Landau mechanism for vestigial order in the normal phase above a two-component superconductor

A two-component superconductor may hypothetically support a vestigial order phase above its superconduct-ing transition temperature, with rotational or time-reversal symmetry spontaneously broken while remaining nonsuperconducting. This has been suggested as an explanation for the observed normal state nematicity of the nematic superconductor MxBi2Se3. We examine the condition for this vestigial order to occur within Ginzburg-Landau theory with order parameter fluctuations, on both the nematic and chiral sides of the theory. Contrary to prior theoretical results, we rule out a large portion of parameter space for possible vestigial order. We argue that very extreme anisotropy is one prerequisite for the formation of a stable vestigial phase via this mechanism, which is likely not met in real materials.