Pauli-Limited Superconductors with Magnetic Fluctuations

We examine the temperature ($T$) and magnetic field ($B$) phase diagram of a paramagnetically-limited superconductor in the presence of fluctuating magnetic moments. We derive a Ginzburg- Landau free energy functional for the superconducting order from microscopics and include classical fluctuations of the magnetization. We consider the full wave vector ($q$) dependence of the expansion coefficients and examine the onset of both the uniform and spatially modulated superconducting phases along critical magnetic field $B_c(T)$. In the absence of fluctuations, both superconducting transitions are second order in 2D and meet at a tricritical point $B_c (T^\star)$. We find that including magnetic fluctuations introduces first order transitions into the superconducting phase. We find the tricritical points where these transitions meet and determine $B_c(T)$ and $q(T)$ as functions of the coupling between superconducting order and the fluctuating magnetization.