In search of multipolar order on the Penrose tiling

We use Monte Carlo calculations to analyse multipolar ordering on the Penrose tiling, relevant for two-dimensional molecular adsorbates on quasicrystalline surfaces and for nanomagnetic arrays. Our initial investigations are restricted to multipolar rotors of rank one through four–described by spherical harmonics Y lm with l = 1, …, 4 and restricted to m = 0–positioned on the vertices of the rhombic Penrose tiling. At first sight, the ground states of odd-parity multipoles seem to exhibit long-range order, in agreement with previous investigations of dipolar systems. Yet, careful analysis performed here establishes that, despite earlier claims, long-range order is absent for all types of rotors, and only short-range order exists. Nevertheless, we show here that short-range order suffices to yield a superstructure in the form of the decagonal Hexagon–Boat–Star tiling. Our results should be taken as a warning for any future analysis of order in either real or simulated arrangements of multipoles on quasiperiodic templates.