Thermodynamics of the interplay between magnetism and high-temperature superconductivity

The copper-oxide based high temperature superconductors have complex phase diagrams with multiple ordered phases. It even appears that the highest superconducting transition temperatures for certain cuprates are found in samples which display simultaneous onset of magnetism and superconductivity. We show here how the thermodynamics of fluid mixtures - a touchstone for chemists as well as hard and soft condensed matter physics - accounts for this startling observation, as well as many other properties of the cuprates in the vicinity of the instability towards ``striped'' magnetism.