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

Broken time-reversal symmetry (BTRS) in ${d+id}^{\prime }$ 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 $\omega ;$ however, at finite $q,\omega$ it has an unusual structure. The chirality of the surface states leads to quantum Hall effects for spin and heat transport in ${d+id}^{\prime }$ superconductors.