Quasiperiodic Spin Space Groups

An outline is given of the spin space-group classi cation of quasiperiodic arrangements of spins. It is based on an extension to multicomponent quasiperiodic elds of the Fourierspace approach to crystal symmetry. The classi cation of decagonal spin space groups in two dimensions is given as an example. A group theoretic argument is given which is used to determine extinctions in a neutron di raction experiment, associated with a given spin space group. 1. Quasiperiodic Spin Density Fields We consider quasiperiodic arrangements of spins, like the ones shown in Figure 1, described by a spin density eld S(r) which transforms as an axial vector eld under rotations and changes sign under time inversion. The symmetry groups of periodic spin density elds, called spin groups, were treated by Litvin and Opechowski. Janner and Janssen have suggested the possibility of extending the superspace procedure to treat quasiperiodic spin arrangements. Here I outline the symmetry classi cation of quasiperiodic as well as periodic spin density elds using the Fourier-space approach to crystal symmetry which has been extended to deal with multicomponent elds. Figure 1. Examples of Decagonal Spin Arrangements. The left one has a spin point group with G = 10mm, G = 1, = (10)2 020, and e = 1, generated by (10; 10 z); (m; 2 0 x). The right one has a spin point group with G = 10mm, G = 2, = 52 0, and e = 1, generated by (10; 5 3 z); (m; 2 0 x). Both are decorations of the Penrose tiling. Spins are located at the tails of the arrows. Two quasiperiodic elds S(r) and S0(r) are indistinguishable if the positionally averaged autocorrelation functions of S(r) of any order, and for any choice of components, are identical to the corresponding autocorrelation functions of S0(r). As such, S(r) and S0(r) have the same distribution of substructures on any scale. Lifshitz and Mermin prove that an equivalent statement of indistinguishability is that the Fourier coe cients of the two elds are related by S0(k) = e i S(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. The lattice is the set of all integral linear combinations of wave vectors at which at least one component of the eld has a nonvanishing Fourier coe cient. 2. The Classi cation Scheme The point group G of S(r) is the set of all operations g from O(3) that leave it indistinguishable to within a transformation of its components. For every pair (g; ) there exists a gauge function, g (k), called a phase function, which satis es S(gk) = e i g (k) S(k) : (2) The transformations in spin space are proper rotations possibly combined with time inversion. It is enough to consider only proper rotations because the spins are axial vectors. The identity in spin space is denoted by , the time inversion operator by 0, and any spin rotation followed by the time inversion is denoted by . The rotations in spin space are decoupled from those in physical space so there can be many di erent 's associated with each element of G. The possible relations between elements of the point group G and the spin transformations are severely constrained by Eq. (2). If (g; ) and (h; ) both satisfy (2) then so does (gh; ). The set of all the transformations is a group, and the set of pairs (g; ) satisfying (2) is a subgroup of G , called the spin point group GS. The corresponding phase functions must satisfy the group compatibility condition: gh(k) g (hk) + h(k) ; (3) where ` ' denotes equality modulo an integer. When enumerating the possible symmetry classes of a eld S, for a given lattice L and point group G, one rst needs to consider all distinct spin point groups, associating a set g of spin transformations with every point group operation g. These must satisfy the following requirements imposed by Eq. (2): (1) The set of transformations e associated with the identity of G forms an abelian normal subgroup of , where each g is a coset of e in . It follows from (3) that if is in g and is in e then