Beyond the 32: Superconducting Pairing Channels of the Icosahedral Group — Classification, Approximant Branching, and Ginzburg–Landau Theory for Quasicrystals

A symmetry-based classification of superconducting order parameters is known for the 32 crystallographic point groups. The icosahedral group is the sole additional finite point group in three-dimensional space that occurs as an exact physical symmetry—in icosahedral quasicrystals—and it is uniquely distinguished by having four- and five-dimensional irreducible representations. This paper develops the corresponding classification. The approach avoids explicit momentum dependence: pairing channels are indexed by irreducible representations of Iₕ (and, with spin–orbit coupling, of its double group 2I), and are represented both as functions on the pseudo-Fermi sphere relevant to the Hume-Rothery icosahedral phases and as amplitudes on the shells of Bergman and Tsai clusters; the exactness of the scheme follows from the indistinguishability-based definition of quasicrystal symmetry. All of ℓ = 2 forms one five-component channel, and any gap function in the four-component channel G has symmetry-enforced point nodes on the twelve five-fold axes. The Ginzburg–Landau theory has two, three and five independent quartic invariants for the three-, four- and five-component channels; in weak coupling the three- and five-component channels are degenerate on the time-reversal-breaking null cone, while the four-component channel selects a single C₃-symmetric time-reversal-breaking state. Passing from the quasicrystal to its cubic approximants splits the four- and five-component channels and predicts two superconducting transitions in the approximant against one in the quasicrystal. The same representation-theoretic data enumerate the eigenmodes of the Poincaré dodecahedral space S³/2I, with the trivial-channel counting function given by the Molien series (1 + t¹⁵)/[(1 − t⁶)(1 − t¹⁰)]. The dodecagonal group of the layered superconductor Ta₁.₆Te is treated by the same method. The companion code, which regenerates every table and equation of the paper in exact arithmetic, is available at github.com/psldm/beyond32 and archived at Zenodo under doi:10.5281/zenodo.22335534. Ver. 1.14 ---Ξυα Mσςς