Field-induced transition within the superconducting state of CeRh$2$As$2$

We report the discovery of two-phase unconventional superconductivity in CeRh$_2$As$_2$. Using thermodynamic probes, we establish the highly anisotropic critical field phase diagram. The superconducting state appearing at low fields is characterised by an enhanced and anisotropic Pauli limit. Furthermore, a c-axis field drives a transition to a second superconducting state that seems to be purely orbitally limited with a critical field as high as 14 T, remarkable in a material whose transition temperature is 0.26 K. In spite of the fact that CeRh$_2$As$_2$ is globally centrosymmetric, local inversion-symmetry breaking at the Ce sites would naturally enable Rashba spin-orbit coupling to dominate the critical fields and drive a transition from even to odd parity superconductivity.