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An alternative view on random tilings

We apply a framework for the description of random tilings without height representation, which was proposed recently, to the special case of quasicrystalline random tilings. Several important examples are discussed, thereby demonstrating the consistency of this alternative description with the conventional one. We also clarify the latter by deriving a group theoretic criterion for the validity of the first random tiling hypothesis.


Introduction
The systematic study of random tilings arose from the insight that they may serve as models for entropically stabilized quasicrystals [2,3]. As such, they are an important and interesting alternative to the perfect tilings based upon the projection method. Although random tilings were used intensively to study transport properties and dynamical properties of real quasicrystals, not much effort was spent to deduce random tiling properties from general grounds [6,7]. We mention the two random tiling hypotheses stated by Henley in 1991, which serve as a starting point to infer diffraction properties, using ideas of the Landau theory of phase transitions. Recently, a more general approach was investigated [16,17,8], mainly for two reasons: On the one hand, the concept of entropic stabilization applies also to crystalline solids which therefore should be included in a more general description. On the other hand, the quasicrystalline random tilings share a special feature which stems from the fact that these tilings are derived from the perfect ones: They have a height representation -each tiling can be embedded as a surface in a higherdimensional space. Since tiling surfaces with equal (mean) density of prototiles have equal (mean) slope, the slope parameters can be used to describe the random tiling ensemble. This is the so-called phason strain parametrization which governs the conventional description.
This description has a number of shortcomings: First of all, it only allows the description of those random tilings which do allow a height representation -being the vanishing (though interesting) minority of all possible tilings. Second, symmetry analysis in this framework is constrained to geometric symmetries, whereas there may be other relevant symmetries of the tiling ensemble. An example are colour symmetries modelling chemical disorder, which is shown to play an important role for real quasicrystals [9,10]. Third, it was expected that a more general analysis could also clarify the origin of the random tiling hypotheses mentioned above.
This was the viewpoint which led to the random tiling description proposed in [16]. There, the grand-canonical tiling ensemble was considered where prototiles are energetically degenerate, and the chemical potentials of the different prototiles or their (mean) densities are the only macroscopic parameters. It was explained how to perform a symmetry analysis in this framework. This led to a proof of a generalization of the first random tiling hypothesis which asserts that the point of maximum entropy is a point of maximum symmetry. In addition, the range of validity of the second hypothesis could be analyzed using the grand-canonical setup, and an exactly solved (crystallographic) counterexample could be given. A rigorous treatment of diffraction is also possible using this setup [1].
The examples presented so far were taken only from the crystallographic case. It remained to apply the proposed description scheme to quasicrystallographic random tilings and to show the consistency of the two approaches. This is where this paper aims at: The next two chapters are devoted to the description of two important examples of quasicrystallogrphic random tilings, the Ammann-Beenker tiling [14] and the square-triangle tiling [15,18,11,12]. We perform a symmetry analysis along the lines of [16] and compare the approach with the conventional one, thereby showing the consistency of the two approaches. In the Appendix, we shortly review and clarify the conventional description. We analyze the connection between maximal symmetry and vanishing phason strain and derive a group theoretic criterion. If the criterion is valid -which is the case in all examples we met -the first random tiling hypothesis is fulfilled in its original formulation: The point of maximum entropy is a point of maximum symmetry [16], and maximum symmetry in turn implies vanishing phason strain. A comparing discussion of the two approaches concludes the paper.

The Ammann-Beenker Random Tiling
The prototiles of the Ammann-Beenker random tiling are two squares and four 45 • -rhombi as shown in Fig. 1. There is no further matching rule apart from the face-to-face tiling condition, which imposes the relation ρ i + σ i = 1 on the prototile densities. Tilings can alternatively be viewed as world lines of particles of two different kinds, as is indicated by the decoration. It was shown in [14] that there is a nonlinear constraint among the different prototile densities which reduces the number of independent variables to four. This constraint was obtained using the height representation of this model, which is given below. It should be noted that it can be independently derived relating the tile description to the line interpretation. This method was introduced in [5]. The constraint reads The derivation imposes periodic boundary conditions. It is highly plausible that the constraint holds (asymptotically) also in the case of free boundary conditions, though there is no proof of this assumption. Complementary, the line representation suggests four independent parameters since it is possible to fix the densities of different lines as well as their mean direction independently. Moreover, care has to be taken in choosing a set of independent parameters: The four rhombi densities, for example, allow the determination of the square densities only up to a permutation, due to the nonlinear constraint. This subtlety does not arise if, for example, three rhombi densities and one square density are chosen. The quadratic invariants given below are insensitive to this because only absolute values of square density differences occur. At the point of maximum symmetry, the four rhombi densities and the two square densities have the same value. The nonlinear constraint fixes them to This is the point where the squares and the rhombi each occupy half of the tiled area. We determine the second order expansion of the entropy density according to D 8 -symmetry. We introduce reduced prototile densities such that the point of maximum symmetry is the origin. The vector space relevant for the symmetry analysis is obtained through linearization of the constraints. It is four-dimensional and determined by r 1 + r 2 = 0, The analysis of the symmetries in the vector space of the independent, reduced prototile densities leads to a second order entropy expansion of the form There are three (positive) elastic constants λ i with invariants This can be obtained by decomposing the representation of the symmetry group in the vector space of the reduced, independent density parameters according to [16], in full analogy to the height representation treatment which is presented in the Appendix. We now derive the relation to the elastic constants defined from the height representation approach. This is done by expressing the expansion (5) in terms of the slope parameters of the height representation. The positions of tiling vertices are integer linear combinations of vectors of the regular 8-star. They therefore belong, viewed as complex numbers, to the module Z[ξ] with ξ = e 2πi/8 . We characterize the height function algebraically. To this end, we recall some facts concerning the associated cyclotomic field [13]. ξ is a root of the eighth cyclotomic polynomial P 8 (x) = x 4 + 1. The other roots are ξ 3 , ξ 5 = −ξ and ξ 7 =ξ. The automorphism group on this set of primitive roots of unity is the Galois group G ≃ C 2 × C 2 . The automorphisms can be uniquely extended to automorphisms of the corresponding module. The usual height function h on the set of possible vertex positions is defined by with α ∈ Z[ξ], G ∋ σ : ξ → ξ 5 . The set of all embedded vertex positions constitutes the fourdimensional primitive hypercubic lattice Z 4 , and each tiling corresponds to a two-dimensional surface. This is the geometric origin of the height representation. It is possible to define a height function on the prototiles by linear extrapolation of the height function on the corresponding vertices. The new macroscopic parameters to describe the tiling ensemble are the components of the phason strain tensor in cartesian coordinates. The phason strain is defined as (mean) slope of the height function. This leads to the relations We observe E = 0 at the point of maximal symmetry, which already follows from group theoretical considerations given in the Appendix. In order to write the entropy expansion (5) in terms of the phason strain, the relation between phason strain and independent tile densities has to be invertible at the point of maximum symmetry. This is not the case if the four rhombi are taken as parameters. If three rhombi densities and one square densities are taken instead, the relation (8) is invertible in the whole phase space. This leads to an expansion This expression is known from the literature [14] and is rederived in the Appendix using the symmetry analysis in the height representation. The relation between the elastic constants is 3 The Square-Triangle Random Tiling The prototiles of the square-triangle random tiling are three squares and four equilateral triangles as shown in Fig. 2. There is no further matching rule apart from the face-to-face tiling condition, σ 1 σ 3 σ 2 ρ 1 ρ 2 ρ 3 ρ 4 Figure 2: Prototiles of the square-triangle random tiling which imposes the relation ρ i + σ i = 1 on the prototile densities. Tilings can alternatively be viewed as world lines of particles of two different kinds, as is indicated by the decoration. Using this representation, it is easy to see that triangles always occur in pairs. Between the (mean) prototile densities we therefore have the relations ρ 1 = ρ 3 and ρ 2 = ρ 4 . It was shown in [4] that there is a nonlinear constraint among the different prototile densities which reduces the number of independent variables to three. The constraint reads It should be mentioned that the derivation imposes periodic boundary conditions. The same constraint is believed to hold (asymptotically) in the case of free boundary conditions. This is suggested from the complementary observation that the line representation allows three independent parameters: the densities of the lines of different type and the frequency of one type of crossing. Due to the nonlinear constraint, care has to be taken in choosing a set of independent parameters: The three square densities, for example, allow the determination of the triangle densities only up to a permutation. This subtlety does not arise if two square densities and one triangle density are chosen instead. The quadratic invariants given below are insensitive to this because only absolute values of triangle density differences occur. At the point of maximum symmetry, the four triangle densities and the three square densities have the same value. The nonlinear constraint fixes their values to This corresponds to the situation where the squares (resp. triangles) occupy half of the tiled area. We determine the second order expansion of the entropy according to D 12 -symmetry. We introduce reduced prototile densities such that the point of maximum symmetry is the origin. The vector space relevant for the symmetry analysis is obtained through linearization of the constraints. It is three-dimensional and determined by r 1 + r 2 = 0, The analysis of the symmetries in the vector space of the independent, reduced prototile densities leads to an entropy expansion of the following form, up to second order: There are two (positive) elastic constants λ i with invariants The invariants are obtained from the decomposition of the representation of the symmetry group into its irreducible components, according to [16]. We now derive the relation to the elastic constants of the height representation approach. The positions of tiling vertices are integer linear combinations of vectors of the regular 12-star. They therefore belong, viewed as complex numbers, to the module Z[ξ] with ξ = e 2πi/12 . We characterize the height function algebraically. Therefore, we remind of some facts concerning the associated cyclotomic field [13]. ξ is a primitive root of unity, annihilating the 12th cyclotomic polynomial P 12 (x) = x 4 − x 2 + 1. The other roots are ξ 5 , ξ 7 = −ξ and ξ 11 =ξ. The automorphism group on this set of primitive roots of unity is the Galois group G ≃ C 2 × C 2 . The automorphisms can be uniquely extended to automorphisms of the corresponding module. The usual height function h on the set of possible vertex positions is defined by h(α) = σ(α) (17) with α ∈ Z[ξ], G ∋ σ : ξ → ξ 7 . It is possible to define a height function on the prototiles by linear extrapolation of the height function on the corresponding vertices. The new macroscopic parameters to describe the tiling ensemble are the components of the phason strain tensor in cartesian coordinates. The phason strain is defined as (mean) slope of the height function. This leads to the relations We find E = 0 at the point of maximal symmetry, which already follows from group theoretical considerations given in the Appendix. Since the model has three degrees of freedom, as follows from the density analysis, we have to expect a special property of the height function which reduces the number of independent slope parameters, in contrast to the previous example. It is indeed easily seen that each height function has the irrotationality property [15] ℜ h(z) dz = 0.
The equality of the off-diagonal elements is a consequence of the irrotationality property (19). (It is also possible to define a height function by G ∋ σ : ξ → ξ 5 . In this case, the irrotationality property reads ℜ h(z) dz = 0.) In order to write the entropy expansion (15) in terms of the phason strain, the relation between phason strain and independent tile densities has to be invertible, at least at the point of maximum symmetry. If the three squares are taken as parameters, this is not the case in the phase with C 4 -symmetry. If two square densities and one triangle densities are taken instead, the relation (18) is invertible in the whole phase space. This leads to an expansion which is known from the literature [15] and is rederived in the Appendix using the symmetry analysis in the height representation. The relation between the elastic constants is

Conclusion
We performed a symmetry analysis of well-known planar quasicrystalline random tilings using the prototile density approach. We were able to show the consistency to the conventional approach via the height representation of these models. For the square-triangle tiling, the approach illuminated the irrotationality property whose appearance was rather unexpected before. Whereas it seems to be natural to describe random tilings using the prototile densities as macroscopic parameters, the approach is commonly hard to follow because it requires the knowledge of all (possibly nonlinear) constraints in order not to overestimate the number of elastic constants. The height representation, in turn, has the disadvantage that is is not obvious whether the slope parameters are independent and whether they constitute a sufficient set to describe all degrees of freedom of the ensemble. Furthermore, to describe other than geometric symmetries, one has to derive the relation between phason strain and the prototile densities explicitly. Therefore, the two approaches are in fact complementary in deriving properties of random tilings. The exactly solved eightfold random tiling, for example, must possess a constraint additional to the ones derived in [5], since the height representation and the density description yield different numbers of elastic constants otherwise, as can be shown. Research in this direction is in progress.
We now focus on the point of maximum symmetry. E is invariant at this point, since this is by definition true of the (mean) prototile densities, which determine E uniquely. If the representations D || and D ⊥ are irreducible and not equivalent, Schur's lemma yields E = 0. In the general case we have • The phason strain tensor vanishes at maximal symmetry if no irrep in internal space is equivalent to an irrep in physical space.
If this criterion is fulfilled, the first random tiling hypothesis in the formulation of Henley is satisfied since the point of maximum entropy is always a point of maximum symmetry, as was shown in [16]. On the other hand, the criterion is fulfilled in all situations we met due to the special construction of the height function. Symmetry analysis may alternatively be performed in the tensor product V ⊗ V ⊥ , which is most advantageous when discussing the influence of symmetries on the entropy density. In this case, E is regarded as a vector with components of the matrix E : V → V ⊥ . The geometric symmetries are represented viaẼ Since the point of maximum symmetry is by definition a fixed point, we conclude: • The phason strain at the point of maximum symmetry consists of components in direction to the trivial 1d-irreps of the representation in V ⊗ V ⊥ .
• In particular, the phason strain vanishes at maximal symmetry, if the representation in V ⊗ V ⊥ does not contain trivial parts.
This criterion is equivalent to the statement given above, as follows from a closer look at the Clebsch-Gordan decomposition of the product representation. We give an expansion of the entropy density of the random tiling ensemble in terms of the phason strain at E = 0. Without loss of generality we assume that the symmetries are represented by orthogonal mappings in V ⊗ V ⊥ . The entropy density is invariant under symmetry transformations, s(Ẽ) = s(E). Since E = 0 is a fixed point of the action of the symmetry group, the symmetries commute with the Hessian H, the second derivative of the entropy density, According to Schur's lemma, H| E=0 acts trivially on the irreducible components in V ⊗ V ⊥ . Therefore, H| E=0 is a linear combination of projectors P (i) onto the irreducible components, The (positive) λ i are the elastic constants of the random tiling, arising from the height representation. The second order term of the entropy expansion is of the form where I (i) (E, E) denotes the quadratic invariants ·, · denotes the scalar product in V ⊗ V ⊥ and || · || the induced norm. We treat as example two-dimensional random tilings with dihedral symmetry D n . This comprises all models discussed above. In the physical space, generators of the symmetries may be given by a rotation R and a reflection S . We assume that the height representation induces a representation in the internal space with generators R ⊥ = −R and S ⊥ = S . (This is the canonical choice for n even.) Since the corresponding representations D and D ⊥ are irreducible and not equivalent, we immediately conclude E = 0 at maximal symmetry. Let us determine the quadratic invariants. The representation of the symmetry group in V ⊗ V ⊥ decomposes into three parts, a two-dimensional rotation and two one-dimensional reflections. If we introduce standard bases, this yields three elastic constants with invariants [7] Mind that the square-triangle tiling is described by only two elastic constants, since the off-diagonal elements in E are equal, due to the irrotationality property (19).