Superconductors with broken time-reversal symmetry: Spontaneous magnetization and quantum Hall effects

Broken time-reversal symmetry (BTRS) in d + id′ as well as in d + is superconductors is studied and is shown to yield current carrying surface states. We evaluate the temperature and thickness dependence of the resulting spontaneous magnetization and show a marked difference between weak and strong BTRS. We also derive the Hall conductance which vanishes at zero wave vector q and finite frequency ω; however, at finite q, ω it has an unusual structure. The chirality of the surface states leads to quantum Hall effects for spin and heat transport in d + id′ superconductors.