Phases of three-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-wave superconductors with two attractive interactions

Three-dimensional $d$-wave ($j=2$) superconducting state may result from Cooper pairing in channels with different angular momenta and spin, such as $(l=0,s=2)$ or $(l=2,s=0)$. We consider spin-3/2 Luttinger fermions in the limit of small spin-orbit coupling parameter and with weak attractive interactions in both of these channels and demonstrate that the stable $d$-wave superconducting phase of the system below critical temperature depends critically on the ratio of two interactions. When a weak inversion-breaking term in the kinetic energy is included, all three of the real, ferromagnetic, and cyclic phases can be realized within a narrow range around unit ratio. The result shows how models of multicomponent superconductors can be surprisingly sensitive to precise values of some of its parameters and display radically different superconducting ground states with their variation.