Nucleation of domain walls by $Z_2$ symmetry breaking transition in $p_x+ip_y$ superconductors

We show that time reversal symmetry breaking $p_x+ip_y$ wave superconductors undergo several phase transitions subjected to external magnetic field or supercurrent. In such system the discrete $Z_2$ symmetry can recover before the complete destruction of the order parameter. The topological defects associated with $Z_2$ symmetry - domain walls can be created in a controllable way by magnetic field or current sweep according to the Kibble-Zurek scenario. Such domain wall generation can take place in exotic superconductors like $Sr_2RuO_4$ and some heavy fermion compounds.