We propose a general mechanism to realize nematic superconductivity (SC) and reveal its exoticvestigial phases in the quasicrystal (QC). Starting from a Penrose-Hubbard model, our microscopic studiessuggest that the Kohn-Luttinger mechanism driven SC in the QC is usually gapless due to violation ofAnderson’s theorem, rendering that both chiral and nematic SCs are common. The nematic SC in the QCcan support novel vestigial phases driven by pairing phase fluctuations above its Tc. Our combinedrenormalization group and Monte Carlo studies provide a phase diagram in which, besides the conventionalcharge-4e SC, two critical vestigial phases emerge, i.e., the quasinematic (QN) SC and QN metal. In thetwo QN phases, discrete lattice rotation symmetry is counterintuitively “quasibroken” with power-lawdecaying orientation correlation. They separate the phase diagram into various phases connected viaBerezinskii-Kosterlitz-Thouless (BKT) transitions. These remarkable critical vestigial phases, whichresemble the intermediate BKT phase in the q state (q ≥ 5) clock model, are a consequence of the fivefold(or higher) anisotropy field brought about by the unique QC symmetry, which are absent in conventionalcrystalline materials.

DOI: 10.1103/PhysRevLett.133.136002