Since 2006 it has been discovered experimentally that the

superconducting state spontaneously breaks time-reversal symmetry

(TRS) in several materials, such as Sr2RuO4, UPt3, URu2Si2, PrOs4Sb12,

and Bi/Ni bilayers. This dissertation studies three physical phenomena

related to time-reversal symmetry breaking (TRSB) in these

superconductors.

The experimental evidence for TRSB comes from the magneto-optical

polar Kerr effect, which is determined by the high frequency ac Hall

conductivity. However, these superconductors are also expected to

exhibit a spontaneous dc Hall effect in the absence of an applied

magnetic field. In the first part of this dissertation we propose a

method for measuring the low frequency Hall conductivity in

superconductors with TRSB. The method is based on a Corbino disk

geometry where an oscillating co-axial magnetic field induces circular

electric field, which, in turn, induces radial charge oscillations due

to the Hall conductivity.

In the second part, we propose an explanation for the polar Kerr

effect observed in the Hidden-Order phase of the heavy-fermion

superconductor URu2Si2. Using a Ginzburg-Landau model for a complex

order parameter, we show that the system can have a metastable

ferromagnetic state, which produces the Kerr signal, even if the

Hidden-Order state respects TRS. We predict that applying a reversed

magnetic field should reset the system to the non-magnetic ground

state, resulting in zero Kerr signal.

In the third part of the dissertation, we investigate the conditions

for the existence of a Majorana bound state on a vortex in a 2D d+id

superconductor with strong spin-orbit coupling. This TRSB pairing was

proposed earlier for the Ni/Bi bilayer. We find that the Majorana

bound state can exist for a d+id pairing under conditions similar to

those for s-wave pairing.