Effects of nucleation at a first-order transition between two superconducting phases: Application to CeRh$_2$As$_2

Recent experiments observed a phase transition within the superconducting regime of the heavy-fermion system CeRh$_2$As$_2$ when subjected to a $c$-axis magnetic field. This phase transition has been interpreted as a parity switching from even to odd parity as the field is increased, and is believed to be of first order. If correct, this scenario provides a unique opportunity to study the phenomenon of local nucleation around inhomogeneities in a superconducting context. Here, we study such nucleation in the form of sharp domain walls emerging on a background of spatially varying material properties and hence, critical magnetic field. To this end, we construct a spatially inhomogeneous Ginzburg-Landau functional and apply numerical minimization to demonstrate the existence of localized domain wall solutions and study their physical properties. Furthermore, we propose ultrasound attenuation as an experimental bulk probe of domain wall physics in the system. In particular, we predict the appearance of an absorption peak due to domain wall percolation upon tuning the magnetic field across the first-order transition line. We argue that the temperature dependence of this peak could help identify the nature of the phase transition.