Spin-glass behavior of frustrated two-dimensional Penrose lattice in the classical planar model

Via extensive Monte Carlo studies we show that the frustrated $\mathrm{XY}$ Hamiltonian on a two-dimensional Penrose lattice admits of a spin-glass phase at low temperature. Studies of the Edwards-Anderson order parameter, spin-glass susceptibility, and the local (linear) susceptibility point unequivocally to a paramagnetic-to-spin-glass transition as the temperature is lowered. The specific heat shows a rounded peak at a temperature above the spin-glass transition temperature, as is commonly observed in spin glasses. Our results strongly suggest that the critical-point exponents are the same as obtained by Bhatt and Young in the $\pm J$ Ising model on a square lattice. However, unlike in the latter case, the critical temperature is clearly finite (nonzero). The results imply that a quasiperiodic two-dimensional array of superconducting grains in a suitably chosen transverse magnetic field should behave as a superconducting glass at low temperature.