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CeRh₂As₂ has a unique superconducting phase diagram with two superconducting states and a critical temperature of T_c = 0.3 K. In our current understanding, a magnetic field drives a transition from a superconducting state of even parity at low fields to one of odd parity at high fields. This phenomenon is likely based on the crystal structure with locally broken inversion symmetry in the Ce layers and two Ce layers related by global inversion symmetry. Even in the case of pure singlet pairing within the layer an odd-parity superconducting gap function can be reached by a sign change of the gap between the two Ce layers. This way of creating odd-parity superconductivity is different from the traditional one - where odd parity comes from the pairing function itself - and might be a direction of finding more candidate materials of odd-parity superconductivity. Remarkably, the odd-parity state is predicted to be a topological crystalline superconducting state. In order to guide the search for new topological superconductors with higher transition temperatures, we need to understand, which material parameters favour is appearance.

Here, I will give an overview on the current experimental progress on CeRh₂As₂ including phase diagrams covering the normal state and the angle dependence. I will also show results where we use pressure to control the superconducting states by tuning the normal state properties. We find that superconductivity is fostered by correlations in proximity to a quantum critical point. Furthermore, pressure effectively shifts the parity transition to lower fields.