Topological properties of ferromagnetic superconductors

A variety of heavy fermion superconductors, such as UCoGe, ${\mathrm{UGe}}_2$, and URhGe exhibit a striking coexistence of bulk ferromagnetism and superconductivity. In the first two materials, the magnetic moment decreases with pressure, and vanishes at a ferromagnetic quantum critical point (qcp). Remarkably, the superconductivity in UCoGe varies smoothly with pressure across the qcp and exists in both the ferromagnetic and paramagnetic regimes. We argue that in UCoGe, spin-orbit interactions stabilize a time-reversal invariant odd-parity superconductor in the high pressure paramagnetic regime. Based on a simple phenomenological model, we predict that the transition from the paramagnetic normal state to the phase where superconductivity and ferromagnetism coexist is a first-order transition.

Figure 1

Schematic phase diagram of a ferromagnetic superconductor in the presence of strong spin-orbit interactions. We claim that in the low-pressure ferromagnetic (FM) regime, the superconductor ${\mathrm{SC}}_1$ is the ${\mathrm{B}}_2$ phase, whereas in the high pressure paramagnetic (PM) regime, the superconductor ${\mathrm{SC}}_2$ is the B phase.

Figure 2

Lowest order Feynman diagrams involving the ferromagnetic (solid external legs) and superconducting (dashed external legs) order parameters. The solid internal loops represent fermion propagators. In the mean-field approximation, the diagrams are evaluated at zero energy and momentum. Diagram (c) is nonzero only for an odd-parity superconductor.

Figure 3

Phase diagrams in the $A_{\mathrm{\Delta }}/A_M$ plane constructed by minimizing the free energy $f=f_0+\delta f_1+\delta f_2$. The parameters used are $B_M=1,B_{\mathrm{\Delta }}=1,C_{\mathrm{\Delta }}=2,{\lambda }_1=0.5$, (a) ${\lambda }_2=-0.05$, and (b) ${\lambda }_2=-0.01$. The blue line denotes the superconducting phase transition and the red line denotes the ferromagnetic phase transition. The four phases of UCoGe are reproduced with this model: the normal phase (N), the ferromagnetic phase (FM), the superconducting phase (SC), and the ferromagnetic superconducting phase (FM+SC). The model predicts the generic existence of first order phase transitions (black line) near the boundary between the N and FM+SC phases.