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We examine weak-coupling theory for unconventional superconducting states ofcubic or tetrahedral symmetry for arbitrary order parameters and Fermi surfacesand identify the stable states in zero applied field. We further examine thepossibility of having multiple superconducting transitions arising from theweak breaking of a higher symmetry group to cubic or tetrahedral symmetry.Specifically, we consider two higher symmetry groups. The first is a weakcrystal field theory in which the spin-singlet Cooper pairs have an approximatespherical symmetry. The second is a weak spin orbit coupling theory for whichspin-triplet Cooper pairs have a cubic orbital symmetry and an approximatespherical spin rotational symmetry. In hexagonal UPt_3, these theories easilygive rise to multiple transitions. However, we find that for cubic materials,there is only one case in which two superconducting transitions occur withinweak coupling theory. This sequence of transitions does not agree with theobserved properties of PrOs_4Sb_{12}. Consequently, we find that to explain twotransitions in PrOs_4Sb_{12} using approximate higher symmetry groups requiresa strong coupling theory. In view of this, we finally consider a weak couplingtheory for which two singlet representations have accidentally nearlydegenerate transition temperatures (not due to any approximate symmetries). Weprovide an example of such a theory that agrees with the observed properties ofPrOs_4Sb_{12}.Comment: 11 pages,1 figur

physical review. b, condensed matter and materials physics

Microscopic theories for cubic and tetrahedral superconductors: Application to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">Pr</mml:mi><mml:msub><mml:mi mathvariant="normal">Os</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">Sb</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:mrow></mml:math>

Microscopic theories for cubic and tetrahedral superconductors: Application toPrOs4Sb12