In search of multipolar order on the Penrose tiling

Abstract

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.

Keywords:

• multipolar order
• adsorption
• magnetic order
• magnetism
• order
• quasicrystals
• surface physics

Acknowledgements

Financial support from the Deutsche Forschungsgemeinschaft in the framework of the part project A11 of the SFB 668 is gratefully acknowledged. This research is also supported by the Israel Science Foundation through Grant 684/06.

Notes

Note

1. Some related results exist for studies of quantum magnetic models on quasicrystals 29–34. See also the discussion of magnetism in quasicrystals in this issue 35.