Topological constraints on quasicrystal transformations

Abstract

The notion of a perfect crystal in d-dimensional physical space may be generalized to the ‘‘deterministic’’ quasicrystal generated by a cut by a d-dimensional ‘‘physical’’ plane through a periodic array of (D-d)-dimensional hypersurfaces in a D-dimensional space. (The quasicrystal tilings generated by projection from a D-dimensional lattice are a special case.) When these surfaces are smooth, and we impose a noncrystallographic symmetry (e.g., a fivefold axis), then they must intersect, which is unphysical. We explore the topology of the transformations induced on these structures by transverse displacements of the physical plane, which correspond to the phase degree of freedom in incommensurate structures. For the projected structures, the atoms undergo a nontrivial permutation when the physical plane is transported in a small closed loop about a vertex of the D-dimensional lattice. We see no way to build a deterministic model for the quasicrystals of physical interest in which the individual atoms move continuously in response to transverse displacements of the physical plane. Thus the conventional spontaneously broken continuous symmetry arguments for a hydrodynamic ‘‘phason’’ mode no longer apply. We suggest that real quasicrystals are nondeterministic and speculate about possible glassy properties of these materials.