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Hyperuniformity and Anti-Hyperuniformity in One-Dimensional Substitution Tilings

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Hyperuniformity and Anti-Hyperuniformity in One-Dimensional Substitution Tilings Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings Erdal C. O˘guz School of Mechanical Engineering and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv, Israel Joshua E. S. Socolar Physics Department, Duke University, Durham, NC 27708, USA Paul J. Steinhardt Department of Physics & Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA and Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY, 10003, USA Salvatore Torquato Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08540, USA (Dated: June 29, 2018) We consider the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long wavelength fluctuations in a broad class of one-dimensional substitution tilings. We present a simple argument that predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and confirm with numerical computations that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra, and limit-periodic tilings. Tilings with quasiperiodic or singular continuous spectra can be con- structed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law. I. INTRODUCTION type in the same way. We consider here only one- dimensional (1D) tilings. Although generalization to higher dimensions would be of great interest, the 1D Recent work has shown that spatial structures case already reveals important conceptual features. with density fluctuations weaker at long wavelengths than those of a typical random point set may have Substitution rules are known to produce a vari- desirable physical properties, and such structures are ety of structures with qualitatively different types said to be hyperuniform [1]. Crystals and quasicrys- of structure factors S(k). Some rules generate pe- tals are hyperuniform, as are a variety of disordered riodic or quasiperiodic tilings, in which case S(k) systems, including certain equilibrium structures, consists of Bragg peaks on a reciprocal space lat- products of nonequilibrium self-assembly protocols, tice supported at sums and differences of a (small) and fabricated metamaterials. (For examples, see set of basis wavevectors, which in the quasiperiodic Refs. [2{8].) One approach to generating point sets case form a dense set. Others produce limit-periodic with nontrivial spatial fluctuations is to use substitu- structures consisting Bragg peaks located on a dif- tion tilings as templates. Our aim in this paper is to ferent type of dense set consisting of wavenumbers m characterize the degree of hyperuniformity in such of the form ±k0n=p , where n, m, and p are posi- arXiv:1806.10641v1 [cond-mat.stat-mech] 20 Jun 2018 systems and thereby provide design principles for tive integers [10, 11]. Still others produce structures creating hyperuniform (or anti-hyperuniform) point for which S(k) is singular at a dense set of points sets with desired scaling properties. but does not consist of Bragg peaks [12{14], for Substitution tilings are self-similar, space-filling which S(k) is absolutely continuous [15], or for which tilings generated by repeated application of a rule the nature of the spectrum has not been clearly de- that replaces each of a finite set of tile types with scribed. The precise natures of the spectra are in scaled copies of some or all of the tiles in the many cases not fully understood. set [9]. We are interested in the properties of point In this paper, we present a simple ansatz that sets formed by decorating each tile of the same predicts the scaling properties relevant for assess- 2 ing the hyperuniformity (or anti-hyperuniformity) of In both cases, α may be defined by the relation [16] 1D substitution tilings. We illustrate the validity of 1+α the ansatz via numerical computations for a variety Z(k) ∼ k as k ! 0 : (3) of example tilings that fall in different classes with respect to hyperuniformity measures. We also delin- Systems with α > 0 have long wavelength spatial eate the full range of behaviors that can be obtained fluctuations that are suppressed compared to Pois- using the substitution construction method, includ- son point sets and are said to be hyperuniform [1]. ing a novel class in which Z(k) decays faster than Prototypical strongly hyperuniform systems (with any power. α > 1) include crystals and quasicrystals. We refer Section II reviews the definition of the scaling ex- to systems with α < 0 as anti-hyperuniform [8]. Pro- ponent α associated with both the integrated Fourier totypical examples of anti-hyperuniformity include intensity Z(k) and the variance σ2(R) in the num- systems at thermal critical points. ber of points covered by a randomly placed inter- An alternate measure of hyperuniformity is based val of length 2R. We then review the classification on the local number variance of particles within a of tilings based on the value of α. Section III re- spherical observation window of radius R (an inter- val of length 2R in the 1D case), denoted by σ2(R). views the substitution method for creating tilings, 2 using the well-known Fibonacci tiling as an illus- If σ (R) grows more slowly than the window vol- trative example. The substitution matrix M is de- ume (proportional to R in 1D) in the large-R limit, the system is hyperuniform. The scaling behavior fined and straightforward results for tile densities 2 are derived. Section IV presents a heuristic discus- of σ (R) is closely related to the behavior of Z(k) sion of the link between density fluctuations in the for small k [1, 16]. For a general point configuration in one dimension with a well-defined average num- tilings and the behaviors of S(k) and Z(k), which 2 leads to a prediction for α. The prediction is shown ber density ρ, σ (R) can be expressed in terms of to be accurate for example tilings of three quali- S(k) and the Fourier transformµ ~(k; R) of a uniform tatively distinct types [8]: strongly hyperuniform density interval of length 2R: (Class I), weakly hyperuniform (Class III), and anti- h 1 Z 1 i hyperuniform. Section V shows, based on the heuris- σ2(R) = 2Rρ S(k)~µ(k; R)dk (4) 2π tic theory, that the range of possible values of α pro- −∞ duce by 1D substitution rules is [−1; 3] and that this with interval is densely filled. Section VI considers sub- stitutions that produce limit-periodic tilings. Ex- sin2(kR) µ~(k; R) = 2 ; (5) amples are presented of four distinct classes: loga- k2R rithmic hyperuniform (Class II), weakly hyperuni- form (Class III), anti-hyperuniform, and an anoma- where ρ is the density. (See Ref. [1] for the general- lous class in which Z(k) approaches zero faster than ization to higher dimensions.) One can express the any power law. Finally, Section VII provides a sum- number variance alternatively in terms of the inte- mary of the key results, including a table showing grated intensity [16]: which types of tilings can exhibit the various classes " # 1 Z 1 @µ~(k; R) of (anti-)hyperuniformity. σ2(R) = −2Rρ Z(k) dk : (6) 2π 0 @k II. CLASSES OF HYPERUNIFORMITY For any 1D system with a smooth or quasicrys- talline structure factor, the scaling of σ2(R) for large For systems having a structure factor S(k) that is R is determined by α as follows [1, 8, 17]: a smooth function of the wavenumber k, S(k) tends 8 0 to zero as k tends to zero [1], typically scaling as a R ; α > 1 (Class I) 2 < power law σ (R) ∼ ln R; α = 1 (Class II) : (7) : R1−α; α < 1 (Class III) S(k) ∼ kα: (1) For hyperuniform systems, we have α > 0, and the In one dimension, a unified treatment of standard 2 cases with smooth S(k) and quasicrystals with dense distinct behaviors of σ (R) define the three classes, but discontinuous S(k) is obtained by defining α in which we refer to as strongly hyperuniform (Class I), terms of the scaling of the integrated Fourier inten- logarithmic hyperuniform (Class II), and weakly hy- sity peruniform (Class III). Systems with α < 0 are called \anti-hyperuniform." Z k The bounded number fluctuations of Class I Z(k) = 2 S(q) dq : (2) 0 occur trivially for one-dimensional periodic point 3 � � ��� The associated rule may be the following: S ! SS : : : S LL : : : L;L ! SS : : : S LL : : : L ; � � � � ��� � | {z } | {z } | {z } | {z } a c b d (11) FIG. 1. The Fibonacci substitution rule. The tiling on the upper line is uniformly stretched, then additional but different orderings of the tiles in the substituted points are added to form tiles congruent to the originals. strings are possible, and the choice can have dra- matic effects. Note, for example, that the rule sets (crystals) and are also known to occur for S ! SL; L ! SLSL (12) certain quasicrystals, including the canonical Fi- produces the periodic tiling : : : SLSLSL : : :, while bonacci tiling described below [16]. Other quasiperi- the rule odic point sets (not obtainable by substitution) are known to belong to Class II [16, 18, 19]. S ! SL; L ! SLLS (13) produces the more complicated sequence discussed III. SUBSTITUTION TILINGS AND THE below in Section VI. SUBSTITUTION MATRIX Defining the substitution tiling requires assign- ing finite lengths to S and L. We let ξ denote A classic example of a substitution tiling is the the length ratio L=S, and we consider only cases one-dimensional Fibonacci tiling composed of two where the substitution rule preserves this ratio (i.e., intervals (tiles) of length L and S.
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