  We consider different effects that arise when time-reversal symmetry breaking
superconductors are subjected to an external magnetic field, thus rendering the
superconductor to be in the mixed state. We focus in particular on two
time-reversal symmetry breaking order parameters which are believed to be
realized in actual materials: $p+\i p'$-wave and $d+\i s$- or $d+\i d'$-wave.
The first order parameter is relevant for Sr$_2$RuO$_4$, while the latter order
parameters have been suggested to exist near surfaces in some of the high-$T_c$
cuprates. We investigate the interplay between surface states and vortex states
in the presence of an external magnetic field and their influence on both the
tunneling conductance and the local density of states. Our findings may be
helpful to experimentally identify the symmetry of unconventional time-reversal
symmetry breaking superconducting states.
