Color Symmetry of Aperiodic Structures

The Fourier-space approach to crystal symmetry 1;2 is extended to colored quasiperiodic structures as an example of the general case of indistinguishable quasiperiodic multicomponent elds. Examples are given of 2-and 5-color space groups on the 10-fold 2-dimensional lattice. We consider a multicomponent quasiperiodic eld (r) whose components (r) could specify a tensor eld, as in magnetically ordered solids or liquid crystals , or a number of elds unrelated to spatial orientation, such as the components in a Potts model or the elds of diierent colors in an aperiodic colored crystal. We deene two quasiperiodic elds (r) and 0 (r) to be indistinguishable if the positionally averaged autocorrelation functions of (r) of any order and for any choice of components are identical to the corresponding autocorrelation functions of 0 (r). We prove elsewhere 4 that an equivalent statement of indistinguishability is that the Fourier coeecients of the two elds are related by 0 (k) = e 2ii(k) (k) ; (1) where , called a gauge function, is the same for all components and is linear modulo an integer over the lattice L of wave vectors. This (reciprocal) lattice (or Fourier module) is the set of all integral linear combinations of wave vectors at which at least one component of the eld has a non-vanishing Fourier coeecient (k). The (geometric) point group G of the eld is the set of all operations from O(3) that leave it indistinguishable to within an invertible linear transformation of its components. In the case of tensor elds is just the representation of G carried by the components of the eld; in the case of tensor elds associated with quantum-mechanical spins in the absence of spin-orbit coupling, there can be many diierent associated with each element of G, corresponding to independent rotations of the spins; in the case of color groups (on which we concentrate below) the 's associated with each element of G are certain permutations of the colors.