With only a few exceptions that are well understood, conventional superconductivity does not coexist with long-range magnetic order (for example, ref. 1). Unconventional superconductivity, on the other hand, develops near a phase boundary separating magnetically ordered and magnetically disordered phases^{2, 3}. A maximum in the superconducting transition temperature T_c develops where this boundary extrapolates to zero Kelvin, suggesting that fluctuations associated with this magnetic quantum-critical point are essential for unconventional superconductivity^{4, 5}. Invariably, though, unconventional superconductivity masks the magnetic phase boundary when T < T_c, preventing proof of a magnetic quantum-critical point⁵. Here we report specific-heat measurements of the pressure-tuned unconventional superconductor CeRhIn₅ in which we find a line of quantum–phase transitions induced inside the superconducting state by an applied magnetic field. This quantum-critical line separates a phase of coexisting antiferromagnetism and superconductivity from a purely unconventional superconducting phase, and terminates at a quantum tetracritical point where the magnetic field completely suppresses superconductivity. The T 0 K magnetic field–pressure phase diagram of CeRhIn₅ is well described with a theoretical model^{6, 7} developed to explain field-induced magnetism in the high-T_c copper oxides, but in which a clear delineation of quantum–phase boundaries has not been possible. These experiments establish a common relationship among hidden magnetism, quantum criticality and unconventional superconductivity in copper oxides and heavy-electron systems such as CeRhIn₅.

1. Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
   
2. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

Correspondence to: Tuson Park¹ Correspondence and requests for materials should be addressed to T.P. (Email: tuson@lanl.gov).