Predicted Nucleation of Domain Walls in $p_x\mathbf{+}i{p}_y$ Superconductors by a $Z_2$ Symmetry-Breaking Transition in External Magnetic Fields

We show that time reversal symmetry-breaking $p_x+i{p}_y$ wave superconductors undergo several phase transitions subjected to an external magnetic field or supercurrent. In such a system, the discrete $Z_2$ symmetry can recover before a complete destruction of the order parameter. The domain walls associated with $Z_2$ symmetry can be created in a controllable way by a magnetic field or current sweep according to the Kibble-Zurek scenario. Such domain wall generation can take place in exotic superconductors like ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$, thin films of superfluid ${}^3\mathrm{He}\mathrm{\text{−}}A$, and some heavy fermion compounds.

Figure 1

Phase transitions in a thin film of the $p_x+i{p}_y$ superconductor. (a),(b) Second-order $Z_2$ and $U(1)$ transitions under the action of an external magnetic field and (c),(d) first order transitions in an external current. By red solid and dashed lines, the order parameter amplitudes ${\eta }_{+}$ and $-{\eta }_{-}$ are shown in the $U(1)\times Z_2$ phase. The energetically equivalent state is obtained by interchanging values of ${\eta }_{+}$ and ${\eta }_{-}$. The dotted blue line corresponds to the nondegenerate $U(1)$ phase with order parameter components ${\eta }_{+}=-{\eta }_{-}$. The magnetic field and current are normalized to $H_0=H_{cm}d/(2\sqrt{3}\lambda )$ and $j_0=(c/4\pi )H_{cm}/\sqrt{2}\lambda$, correspondingly.