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The Kohn-Luttinger mechanism for unconventional superconductivity (SC) drivenby weak repulsive electron-electron interactions on a periodic lattice isgeneralized to the quasicrystal (QC) via a real-space perturbative approach.The repulsive Hubbard model on the Penrose lattice is studied as an example, onwhich a classification of the pairing symmetries is performed and a pairingphase diagram is obtained. Two remarkable properties of these pairing statesare revealed, due to the combination of the presence of the point-groupsymmetry and the lack of translation symmetry on this lattice. Firstly, thespin and spacial angular momenta of a Cooper pair is de-correlated: for eachpairing symmetry, both spin-singlet and spin-triplet pairings are possible evenin the weak-pairing limit. Secondly, the pairing states belonging to the 2Dirreducible representations of the $D_5$ point group can betime-reversal-symmetry-breaking topological SCs carrying spontaneous bulk supercurrent and spontaneous vortices. These two remarkable properties are generalfor the SCs on all QCs, and are rare on periodic lattices. Our work starts thenew area of unconventional SCs driven by repulsive interactions on the QC.

Kohn-Luttinger mechanism driven exotic topological superconductivity on the Penrose lattice