Spontaneous Breakdown of Time-Reversal Symmetry Induced by Thermal Fluctuations

In systems with broken $U(1)$ symmetry, such as superfluids, superconductors or magnets, the symmetry restoration is driven by proliferation of topological defects in the form of vortex loops. Here we discuss that in certain systems the proliferation of topological defects can, by contrast, lead to the breakdown of an additional symmetry. As a particular example we demonstrate that this effect should take place in $s+is$ superconductors, which are widely discussed in connection with the Iron-based materials. In these systems a vortex excitation can create a "bubble" of fluctuating $Z_2$ order parameter. Thermal excitation of vortices then leads to breakdown of $Z_2$ time-reversal symmetry when the temperature is increased.