Landau level broadening, hyperuniformity, and discrete scale invariance
LLandau level broadening, hyperuniformity, and discrete scale invariance
Jean-Noël Fuchs, ∗ Rémy Mosseri, † and Julien Vidal ‡ Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France
We study the energy spectrum of a two-dimensional electron in the presence of both a perpen-dicular magnetic field and a potential. In the limit where the potential is small compared to theLandau level spacing, we show that the broadening of Landau levels is simply expressed in terms ofthe structure factor of the potential. For potentials that are either periodic or random, we recoverknown results. Interestingly, for potentials with a dense Fourier spectrum made of Bragg peaks (asfound, e.g., in quasicrystals), we find an algebraic broadening with the magnetic field character-ized by the hyperuniformity exponent of the potential. Furthermore, if the potential is self-similarsuch that its structure factor has a discrete scale invariance, the broadening displays log-periodicoscillations together with an algebraic envelope.
I. INTRODUCTION
In the presence of a magnetic field, the energy spec-trum of noninteracting electrons in two dimensions isknown to consist of Landau levels. These discrete en-ergy levels are responsible for many remarkable phenom-ena, among which is the celebrated integer quantum Halleffect . Each Landau level has a macroscopic degen-eracy that is proportional to the strength of the mag-netic field. This degeneracy is expected to be lifted by ageneric perturbation leading to a broadening of Landaulevels that may have important physical consequences.For instance, plateaus observed in the Hall resistance aredirectly related to the broadening induced by disorder,as realized early by Ando et al. . Most studies on Lan-dau level broadening focused on disordered systems (seeRef. 6 for a review), but the role played by periodic po-tentials has also attracted much attention following theoriginal work of Rauh . Based on a free-electron pic-ture, Rauh’s approach also allows one to qualitatively un-derstand the Landau level broadening in the small-fieldlimit of the Hofstadter butterfly for periodic lattices ,although a quantitative analysis requires a semi-classicaltreatment . Recently, the Hofstadter butterfly of somequasiperiodic systems has been investigated, unveiling anunusual broadening of Landau levels different from theone expected for periodic or disordered systems, hencesuggesting a nontrivial mechanism for potentials with adense set of Bragg peaks.The goal of the present paper is to provide a generalframework to compute the broadening of Landau levels inthe presence of an arbitrary potential. Our main result,given in Eq. (11), relates the variance of the lowest Lan-dau level (LLL) to the structure factor of the perturbingpotential (an extension to higher-energy Landau levels isstraightforward). This simple expression reproduces theaforementioned results for disordered and periodic cases,but it also allows us to investigate more subtle potentials(see Fig. 1 for a summary of the results). In particular,we find that when the Fourier spectrum of the potentialis dense and made of Bragg peaks (as in quasicrystals),the variance of the LLL increases algebraically with themagnetic field [see Eq. (22)] with an exponent character- V a r i a n ce o f t h e LLL
Magnetic field0
FIG. 1. Sketch of the LLL variance w as a function of themagnetic field B for different perturbing potentials. Red: dis-ordered w ∼ B (see Ref. 3). Green: hyperuniform with dis-crete scale invariance w ∼ B α + log-periodic oscillations,where α is the hyperuniformity exponent (this work). Blue:periodic w ∼ e − /B (see Ref. 8). izing the hyperuniformity of the potential. This notionof hyperuniformity is commonly used to describe sets ofpoints with an unusually large suppression of density fluc-tuations at long wavelengths . We also show that if thepotential has a discrete scale invariance , then thevariance displays log-periodic oscillations together witha power-law envelope. To illustrate these results, weconsider three examples of quasiperiodic potentials, forwhich we compute exactly the hyperuniformity exponentand the period of these oscillations, when it exists. II. LANDAU LEVELS PERTURBED BY APOTENTIAL
To begin with, let us recall a few well-known results.The Hamiltonian describing a particle of mass m andcharge e in a magnetic field B = ∇ × A is given by H = ( p − e A ) m . (1) a r X i v : . [ c ond - m a t . o t h e r] S e p Here, we consider a two-dimensional system with a mag-netic field perpendicular to the plane. Such a field canbe described by the symmetric gauge where the vectorpotential reads A = B ( − y/ , x/ , H consists in equidistant energy levels known as Landaulevels, E n = (cid:126) ω c ( n + 1 / , ∀ n ∈ N , (2)where ω c = | eB | /m is the cyclotron frequency. EachLandau level has a degeneracy proportional to the samplearea A and the magnetic field. In the following, we set (cid:126) = e = 1.Our aim is to study the behavior of the Landau levelsin the presence of a time-independent potential. Thus,we consider the following general Hamiltonian H = H + V ( x, y ) , (3)and we assume that the magnitude of the potential issmall compared to the Landau level spacing ω c (cid:29) | V | . Inthis regime, we can neglect the coupling between differentLandau levels and use degenerate perturbation theory tocompute the degeneracy splitting of a single level. With-out loss of generality, we assume V ( x, y ) (cid:62) B >
III. VARIANCE OF THE LLL
For simplicity, in the following, we focus on the LLLcorresponding to n = 0 for which the nonperturbed wave-functions, in the thermodynamical limit, can be chosenas ϕ l ( z ) = h x, y | l i = 1 p π l B l ! 2 l z l e −| z | / , (4)where z = ( x + iy ) /l B , l = 0 , , ..., N φ − N φ = A / (2 πl B ) (cid:29) l B = 1 / √ B is the magnetic length.To characterize the broadening of the LLL due to thepotential, we consider its variance defined by w = 1 N φ N φ − X p =0 ε p − N φ N φ − X p =0 ε p ! , (5)where ε p ’s are eigenenergies of H projected onto the LLL.This variance can be recast as w = 1 N φ N φ − X l =0 N φ − X l =0 |h l | V | l i| − N φ N φ − X l =0 h l | V | l i ! , (6)so that one does not need to compute explicitly the ε p ’s. Setting r = ( x, y ) = r (cos θ, sin θ ), a matrix el-ement of the perturbation potential in the LLL basis {| l i , l = 0 , ..., N φ − } reads h l | V | l i = Z d q (2 π ) e V ( q )2 π √ l ! l ! 2 l + l (7) × Z ∞ d rl B (cid:18) rl B (cid:19) l + l e − r l B Z π d θ e i q · r e i θ ( l − l ) , where we introduced the Fourier transform of the poten-tial e V ( q ) = Z d r e − i q · r V ( r ) . (8)In the large- N φ (thermodynamical) limit, one then gets: ∞ X l =0 h l | V | l i = e V (0)2 πl B , (9) ∞ X l =0 ∞ X l =0 |h l | V | l i| = 12 πl B Z d q (2 π ) | e V ( q ) | e −| q | l B / . (10)Finally, one obtains the following expression for the vari-ance w = Z d q (2 π ) S ( q )e −| q | l B / , (11)where we introduced the structure factor S ( q ) = | e V ( q ) | A (1 − δ q , ) . (12)Note that the term proportional to δ q , comes from thesecond term of Eq. (5) and is irrelevant only if e V ( ) = 0.The variance is therefore essentially equal to the integralof the structure factor over a disk of radius l − B whichis the main result of this paper. Before discussing themost interesting case of a potential with a dense Fourierspectrum, let us first show that this expression allows oneto recover known results for simple potentials. IV. PERIODIC POTENTIAL
For a periodic potential of strength V with a singlespatial frequency a − V ( x, y ) = V [cos(2 πx/a ) + cos(2 πy/a )] , (13)Eq. (11) leads to w = V e − π Ba , (14)in agreement with the expression found by Rauh (seeApp. A for details). The generalization to Fourier spectrawith a finite set of frequencies is straightforward, even ifthe potential is no longer periodic. In the zero-field limit,the LLL broadening is exponential and controlled by thesmallest frequency. The case of a dense set of frequenciesis more subtle. V. RANDOM POTENTIAL
Landau level broadening due to an uncorrelated ran-dom potential has been widely studied in the literature .For the simple case of a random potential with zero meanand white-noise correlations V ( r ) = 0 , (15) V ( r ) V ( r ) = ( V a ) δ ( r − r ) , (16)where the overline denotes the average over disorder re-alizations, Eq. (11) gives w = V a πl B = V Ba π , (17)in agreement with the result of Ando (see App. B fordetails). This result is very different from the one ob-tained for a potential with a finite number of frequenciesdiscussed above.For stealthy hyperuniform disorder , the structurefactor is identically zero in a disk of radius q > S ( q ) ∝ Θ( | q | − q ) leading to a LLL broadening w ∝ B e − q B , (18)intermediate between that of a periodic potential,Eq. (14), and that of uncorrelated random disorder,Eq. (17). VI. POTENTIAL WITH A DENSE FOURIERSPECTRUM
The most interesting situation comes from potentialswith a dense Fourier spectrum made of Bragg peaks (asfound, e.g, in quasicrystals). To this aim, let us considera general potential V ( r ) = V a N X j =1 δ ( r − r j ) , (19)built on a set of N scattering points located at position r j with a typical density a − . The random potentialdiscussed above belongs to this family.Before proceeding further, let us stress that the ex-ponential term in Eq. (11) acts as a smooth cutoff thateliminates wave vectors | q | (cid:38) l − B . In order to analyzethe behavior of w , we shall instead consider a sharp cut-off regularization by introducing the integrated intensityfunction Z ( k ) = Z | q |
0, the potential ishyperuniform whereas α < ). The special case α = 0 cor-responds to a potential with a constant S , such as therandom potential considered previously [see Eq. (17)].Interestingly, if Z further manifests a discrete scale in-variance, i.e., if there exists λ > Z ( k/λ ) = Z ( k ) /λ α , (23)then one has Z ( k ) = k α F (ln k/ ln λ ) , (24)where F ( x + 1) = F ( x ) (see the examples below andRefs. 14 and 15 for a review). As a result, the LLL vari-ance w displays log-periodic oscillations together with apower-law envelope in the small- B limit. VII. EXAMPLES OF QUASIPERIODICPOTENTIALS
For illustration, let us consider some potentials ofthe form given in Eq. (19) where the points correspondto vertices of two-dimensional quasiperiodic tilings (seeApp. C). For each tiling considered below, we com-puted exactly the structure factor S , the hyperunifor-mity exponent α characterizing the power-law behaviorof Z ( k ) ∼ k → k α , and the discrete scale invariance factor λ defined in Eq. (23) when it exists. Numerical resultsdisplayed in Fig. 2 have been obtained by integratingmore and more Bragg peaks of smaller and smaller in-tensities. In each case, we checked that the results wereconverged in the range considered. Units are taken suchthat V = p A / ( N a ) and a = 1, where a is the edgelength of the hypercubic lattice that is used to build thetiling in the standard cut-and-project method .Let us first consider the twofold-symmetric Rauzytiling . The hyperuniformity exponent is α = 4 but Z l n Z ( k ) − −
90 ln k − − Z ( k ) / k . . k − − l n Z ( k ) − −
40 ln k − − Z ( k ) / k . . k/ ln(1 + √ − − l n Z ( k ) − −
50 ln k − − Z ( k ) / k . . k/ ln τ − − FIG. 2. Integrated density function Z of the Rauzy (left), Ammann-Beenker (center), and Penrose (right) tilings (see insetsfor illustrations). Top: Log-log plot (blue dots) together with the algebraic envelope k α (green line). Bottom: Z ( k ) /k α asa function of ln k/ ln λ (ln k ) showing 1-periodic (nonperiodic) oscillations for tilings with (without) discrete scale invariance. has no discrete scale invariance (see App. D). By contrast,for the eightfold-symmetric Ammann-Beenker tiling ,the hyperuniformity exponent is α = 2 and Z has a dis-crete scale invariance with λ = 1 + √ α = 6, and Z has a discrete scale invari-ance with λ = τ , where τ = √ is the golden ratio(see App. G). VIII. SUBSTITUTION TILINGS ANDDISCRETE SCALE INVARIANCE
As recently conjectured by Oğuz et al. , the behav-ior of Z in one-dimensional substitution tilings is de-termined by the eigenvalues of the substitution matrix.More precisely, for non-periodic binary substitutions as-sociated with a 2 × λ > | λ | >
0, one has Z ( k/λ ) = Z ( k )( λ /λ ) , (25)when k tends to zero, so that Z ( k ) = k α F (ln k/ ln λ ) , α = 1 − | λ | ln λ , (26)with F ( x + 1) = F ( x ). In two dimensions, it is verylikely that the existence of substitution rules with in-flation/deflation also implies the existence of discretescale invariance for Z . This is clearly the case for theAmmann-Beenker and Penrose tilings, which, contraryto the Rauzy tiling, can be built by inflation/deflation.However, we have not found a simple expression for the hyperuniformity exponent [such as the one givenEq. (26)] for two-dimensional binary substitutions. IX. OUTLOOK
In this paper, we obtained a simple relation betweenthe Landau level broadening and the integrated intensityfunction Z of the perturbing potential. For potentialswith a dense Fourier spectrum made of Bragg peaks, thisrelation implies that the variance of the LLL is drivenby the hyperuniformity exponent α [see Eq. (22)]. Inthe absence of a complete classification of the possiblebehavior of Z , a first step to go beyond would consistin analyzing two-dimensional potentials with a singularcontinuous Fourier spectrum for which one expects morecomplex behavior of Z as observed in one dimension .For instance, one may find noninteger exponents α oreven nonalgebraic decay. Another important issue wouldbe to consider the influence of Landau level mixing whichis known to have dramatic effects on the localizationproperties of the eigenstates , and hence, on integerquantum Hall physics (see, e.g., Ref. 29). It would alsobe important to bridge the gap between the perturbedfree-electron results and the one obtained numerically intight-binding models . A possible route would be to de-velop the analog of Wilkinson semi-classical treatment for nonperiodic potential.Finally, let us mention that the magnetic-field depen-dence of the Landau level broadening induced by disorderhas already been measured in graphene . Combiningsuch an experimental device with a nontrivial superlat-tice potential would allow us to measure the behaviorsdiscussed in the present work. ACKNOWLEDGMENTS
We thank M. Duneau, T. Fernique, and J.-M. Luck forfruitful discussions.
Appendix A: Variance for a periodic potential
The Fourier transform of the periodic potential (13) is given by e V ( q ) = V π ) [ δ ( q x − π/a ) δ ( q y ) + δ ( q x + 2 π/a ) δ ( q y ) + δ ( q x ) δ ( q y − π/a ) + δ ( q x ) δ ( q y + 2 π/a )] , (A1)= V A (cid:2) δ q x , π/a δ q y , + δ q x , − π/a δ q y , + δ q x , δ q y , π/a + δ q x , δ q y , − π/a (cid:3) , (A2)where we used the fact that A δ q , = (2 π ) δ ( q ), in the thermodynamic limit. The structure factor (12) becomes S ( q ) = (1 − δ q , ) A (cid:12)(cid:12)(cid:12)(cid:12) V A (cid:2) δ q x , π/a δ q y , + δ q x , − π/a δ q y , + δ q x , δ q y , π/a + δ q x , δ q y , − π/a (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) , (A3)= V A (cid:2) δ q x , π/a δ q y , + δ q x , − π/a δ q y , + δ q x , δ q y , π/a + δ q x , δ q y , − π/a (cid:3) , (A4)= V π ) [ δ ( q x − π/a ) δ ( q y ) + δ ( q x + 2 π/a ) δ ( q y ) + δ ( q x ) δ ( q y − π/a ) + δ ( q x ) δ ( q y + 2 π/a )] , (A5)where we used the Kronecker delta in order to compute the modulus square of the Fourier transform (otherwise, thesquare of a Dirac delta is ill-defined). Then Eq. (11) involves the integral of Dirac-delta functions, which straightfor-wardly leads to Eq. (14). Appendix B: Variance for a random potential
The Fourier transform of the uncorrelated random potential defined by Eqs. (15)-(16) is given by e V ( q ) = Z d r e − i r · q V ( r ) = 0 , and | e V ( q ) | = Z d r e − i r · q Z d r e i r · q V ( r ) V ( r ) = ( V a ) A . The structure factor (12) becomes S ( q ) = ( V a ) (1 − δ q , ). In Eq. (11) the Kronecker delta does not contributeand the Gaussian integral gives Eq. (17): w = ( V a ) (2 π ) Z d q e −| q | l B / = ( V a ) (2 π ) πl B . Appendix C: Fourier transform of a cut-and-projectquasicrystal
The cut-and-project (CP) method consists in se-lecting points of a D -dimensional periodic lattice if theirprojection onto the ( D − d )-dimensional perpendicular space E ⊥ belongs to a so-called acceptance window. Thetiling is then obtained by projecting these selected pointsonto the complementary d -dimensional parallel space E ∥ .Any vector v in hyperspace can be decomposeduniquely in terms of its projection onto parallel and per-pendicular spaces as v = v ∥ + v ⊥ . (C1)As explained in the early papers introducing theCP method , the Fourier transform of quasiperiodictilings can be computed from the higher-dimensionalspace it stems from. The main idea is that since points ofthe tiling are selected from a periodic tiling via an accep-tance window, computing the Fourier transform of thetiling essentially amounts to compute the Fourier trans-form of this acceptance window.For a tiling with N sites (vertices) at position R ∥ j andobtained by the CP method, the microscopic density is n ( r ∥ ) = N X j =1 δ ( r ∥ − R ∥ j ) , (C2)and its Fourier transform is e n ( q ∥ ) = N X j =1 e − i q ∥ . R ∥ j , (C3)where the sum runs over all sites of the d -dimensionaltiling considered. The convention that we use is thatcapital letters (such as R ∥ j ) refer to discrete points andsmall letters (such as r ∥ ) to a continuum of points.Let R be a point of the D -dimensional hypercubic lat-tice and K a vector of its reciprocal lattice such that K · R = 2 π × integer. These vectors can be decomposedonto the parallel and perpendicular spaces such that theirscalar product reads K · R = K ∥ · R ∥ + K ⊥ · R ⊥ = 2 π × integer . (C4)Equation (C3) is non-zero iff q ∥ = K ∥ , in which case itbecomes e n ( K ∥ ) = N X j =1 e i K ⊥ . R ⊥ j . (C5)For a quasicrystal built along an irrational plane (par-allel space), the points in perpendicular space denselyand uniformly fill the acceptance window such that e n ( K ∥ ) = N Z A ⊥ d r ⊥ A ⊥ e i K ⊥ . r ⊥ , (C6)where the integral is over the acceptance window in per-pendicular space and A ⊥ is its ( D − d )-dimensional vol-ume.Now, for any vector q ∥ in parallel space, the Fouriertransform of the density Eq. (C3) reads e n ( q ∥ ) = X K δ q ∥ , K ∥ N Z A ⊥ d r ⊥ A ⊥ e i K ⊥ . r ⊥ , (C7)where the sum is performed over all vectors K of the re-ciprocal lattice of the hypercubic lattice. As we are con-sidering a quasicrystal, for any K ∥ there is a unique K and therefore K ⊥ is well defined. If { a ∗ j ; j = 1 , ..., D } isa basis of vectors in reciprocal space, then K = P j n j a ∗ j ,where n j are integers. Its parallel and perpendicularcomponents are also functions of the same integers: K = K ∥ ( n , ..., n D ) + K ⊥ ( n , ..., n D ) . (C8)Therefore, the sum over K in Eq. (C7) is actually asum over D integers n , ..., n D , clearly showing that theFourier transform is pure point of rank D > d .Let us define the structure factor in the thermodynam-ical limit as S ( q ∥ ) = | e n ( q ∥ ) | N (cid:0) − δ q ∥ , (cid:1) . (C9)For two-dimensional potentials ( d = 2) of the formgiven by Eq. (19), this definition differs from the onegiven in Eq. (12) by a factor V a N/ A , which disappearsupon choosing units such that V = p A / ( N a ).As explained in Ref. 17, for a spectrum made of a denseset of Bragg peaks (discontinuous S ), the integrated in-tensity function Z ( k ) = Z | q ∥ | The two-dimensional (generalized) Rauzy tiling hasbeen introduced in Ref. 24. This is a codimension-one tiling built from the cubic lattice Z (edge length a = 1) with a one-dimensional perpendicular space ori-ented along the direction e ⊥ = ( θ , θ, 1) where θ is thereal (Pisot-Vijayaraghavan) root of the cubic equation x = x + x + 1. Contrary to the Ammann-Beenker andthe Penrose tilings discussed in the next sections, theRauzy tiling cannot be built by substitution rules.For such a codimension-one quasicrystal, the accep-tance window is a segment of length A ⊥ defined as theprojection of h = (1 , , 1) onto the perpendicular space.This acceptance window also corresponds to the projec-tion of the unit cube onto the perpendicular space. Inthis case, Eq. (C6) gives: | e n ( K ∥ ) | = N (cid:12)(cid:12)(cid:12)(cid:12) sinc (cid:18) K ⊥ . h ⊥ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (D1) 2. Structure factor The structure factor is defined in Eq. (C9). Our goal isto analyze the behavior of S ( q ∥ ) in the limit where | q ∥ | tends to zero. By definition, one has S (0) = 0 but itsbehavior for small | q ∥ | is nontrivial since S ( q ∥ ) (cid:44) q ∥ coincides with the parallel component K ∥ of areciprocal-lattice vector K of the cubic lattice. Thus, weare interested in computing the behavior of S when | K ∥ | goes to 0 for K ∥ (cid:44) S ( K ∥ ) = N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) K ⊥ . h ⊥ (cid:17) K ⊥ . h ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (D2)= N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) K ∥ . h ∥ (cid:17) K ⊥ . h ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (D3) ’ | K ∥ |→ N (cid:12)(cid:12)(cid:12)(cid:12) K ∥ . h ∥ K ⊥ . h ⊥ (cid:12)(cid:12)(cid:12)(cid:12) , (D4)where we used the fact that K is a reciprocal-lattice vec-tor and h a direct-lattice vector. The difficulty comesfrom the fact that, when | K ∥ | goes to 0, | K ⊥ | diverges.So, the goal is to find the relation between these two components.One way to investigate this issue is to follow the ap-proach proposed in Ref. 17 for the Fibonacci chain (seeApp. E). In the Z canonical basis, any reciprocal-latticevector K has coordinates 2 π ( l, m, n ) where l, m , and n are integers. To analyze the behavior of K ∥ . h ∥ and K ⊥ . h ⊥ , let us consider the matrix M = , (D5)that satisfies M = M + M + 1. The eigenvalues of M are the Tribonacci constant θ ’ . ± i φ / √ θ with φ ’ . θ corresponds to the perpen-dicular direction e ⊥ . Since θ is a Pisot-Vijayaraghavannumber, the action of M p onto any vector v , such that v . e ⊥ (cid:44) 0, drives this vector towards the direction e ⊥ inthe large- p limit.Hence, to analyze the behavior of S in the limitwhere K ∥ tends to zero [see Eq. (D5)], let us consider K ( p ) = M p K . More precisely, we are interested in com-puting K ( p ) ∥ . h ∥ and K ( p ) ⊥ . h ⊥ . Keeping in mind that h ⊥ = P ⊥ (1 , , 1) and h ∥ = ( − P ⊥ )(1 , , 1) (where P ⊥ is the projector onto the perpendicular space), one caneasily compute these quantities. After some algebra, onegets: K ( p ) ∥ . h ∥ = ( θ − θ − p +12 n sin( p φ ) (cid:2) (1 + θ ) m − θ ( l + n ) (cid:3) + √ θ [( θm − l ) sin((1 + p ) φ ) + ( θn − m ) sin((1 − p ) φ )] o sin( φ )[( θ − θ + 1] , (D6) K ( p ) ⊥ . h ⊥ = 1sin( φ )[( θ − θ + 1] (cid:0) θ / cos( φ ) − θ − (cid:1) × n − θ p (cid:0) θ + θ + 1 (cid:1) sin( φ ) (cid:16) θl − √ θm cos( φ ) + n (cid:17) + θ − p − h θ / sin(( p + 1) φ ) (cid:2) ( θ − l + θ ( n − m ) (cid:3) + ( θ − θ sin( p φ ) (cid:2) l − ( θ + 1) m + θn (cid:3) + √ θ sin(( p − φ )[ − l + m + ( θ − θn ] + θ ( l − θm ) sin(( p + 2) φ ) + ( m − θn ) sin(( p − φ ) io . (D7)Thus, in the large- p limit, one finds that K ( p ) ∥ . h ∥ van-ishes as θ − p/ , K ( p ) ⊥ . h ⊥ diverges as θ p , and S ( K ( p ) ∥ ) be-haves as θ − p . As a result, one finds that S ( K ∥ ) ∼ | K ∥ |→ | K ∥ | , (D8)for all ( l, m, n ). However, we emphasize that, contraryto the Fibonacci chain Ref. 17 (see also App. E), thispower-law behavior is modulated by a bounded oscillat-ing nonperiodic function as can be seen in Eqs. (D6-D7). 3. Integrated intensity function The integrated intensity function is defined inEq. (C11). The sum over all vector K ∥ with a normsmaller than k can be decomposed into a sum over alltriplets ( l, m, n ) and their iterated under M p . As a re-sult, one has: Z ( k ) = 4 π A X ( l,m,n ) ∞ X p = p ( l,m,n ) S ( K ( p ) ∥ ) , (D9)where p ( l,m,n ) is the smallest integer fulfilling the con-straint | K ( p ) ∥ | < k . As already discussed in the previoussection, in the large- p limit, S ( K ( p ) ∥ ) ’ | K ( p ) ∥ | f ( l, m, n, p ) , (D10) | K ( p ) ∥ | ’ θ − p/ g ( l, m, n, p ) , (D11)where, for a given triplet ( l, m, n ), f and g are boundedoscillating function of p [see Eqs. (D6)-(D7)]). Thus, S is bounded both above and below Z − ( k ) (cid:54) Z ( k ) (cid:54) Z + ( k ) , (D12)where Z ± ( k ) = 4 π A X ( l,m,n ) c ± ( l, m, n ) ∞ X p = p ( l,m,n ) θ − p , (D13) c + ( l, m, n ) = max p f ( l, m, n, p ) g ( l, m, n, p ) , (D14) c − ( l, m, n ) = min p f ( l, m, n, p ) g ( l, m, n, p ) . (D15)Interestingly, Z ± ( k/ √ θ ) = Z ± ( k ) /θ , as can be seenfrom Eqs. (D10)-(D11) since dividing k by √ θ simplyamounts to change p ( l,m,n ) into p ( l,m,n ) + 1 in Eq. (D13).Such a relation reflects a discrete scale invariance (seealso next section) for Z ± and implies a power-law enve-lope Z ( k ) ∼ k → k . (D16)Note that, despite the fact that Z is defined as an integralof S , they are both characterized by a power law with thesame exponent. This is a consequence of the fact that S is discontinuous (discrete) and dense. Appendix E: Integrated intensity function of theFibonacci chain The Fibonacci chain is a one-dimensional tiling builtfrom the square lattice Z (edge length a = 1). Theintegrated intensity function Z of the Fibonacci chain hasbeen widely discussed in Ref. 17. However, one importantproperty has been missed. As a codimension-one system,the Fourier transform of the Fibonacci chain can be easilycomputed. The structure factor is S ( K ∥ ) = N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:16) K ∥ . h ∥ (cid:17) K ⊥ . h ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (E1)where h ∥ and h ⊥ are the projections of thevector h = (1 , 1) onto e ∥ = √ τ ( − , τ ) and e ⊥ = √ τ ( τ, τ = √ is the goldenratio. The integrated intensity function is then Z ( k ) = 2 π A X | K ∥ | 24 ln k − − Z ( k ) / k . . k/ ln τ − − − FIG. 3. Integrated density function Z of the Fibonacci chain(see inset). Top: Log-log plot (blue dots) together with thepower-law envelope k (green line). Bottom: Z ( k ) /k as afunction of ln k/ ln τ showing periodic oscillations (with pe-riod 1). where A is the total length of the chain. As for the Rauzytiling, let us consider the matrix M = (cid:18) (cid:19) , (E3)that satisfies M = M + 1. Eigenspaces of M correspondto the perpendicular and parallel directions with eigen-values τ and − /τ , respectively. The small- k behavior of Z is obtained by analyzing sequences K ( p ) = M p K (seeRef. 17). One then gets, in the large- p limit, S ( K ( p ) ∥ ) ’ | K ( p ) ∥ | f ( l, m ) , (E4) | K ( p ) ∥ | ’ τ − p g ( l, m ) . (E5)However, contrary to the Rauzy tiling, f and g do notdepend on p . Thus, following the same line of reasoningas above, one straightforwardly gets the discrete scalingrelation Z ( k/τ ) = Z ( k ) /τ . (E6)The solution of this equation can be written as Z ( k ) = k F (ln k/ ln τ ) , (E7)where F ( x + 1) = F ( x ) (for a review on discrete scaleinvariance, see Ref. 14). As a result, Z has a power-law envelope together with log-periodic oscillations (seeFig. 3 for illustration). This is in stark contrast with theRauzy tiling where only Z + and Z − obey such a discretescale invariance but not Z itself. Practically, to compute Z , we first select a set of K points in the reciprocal latticeof Z inside a given ball of radius K max around the origin.For each of these points, we consider the sequence ofpoints K ( p ) with p = 0 , ..., p max , and we compute S foreach corresponding K ( p ) ∥ (avoiding possible redundancy). Z is then obtained by summing over these Bragg peaksaccording to Eq. (E2). We check the convergence of theresults displayed in Fig. 3 by increasing K max and p max . Appendix F: The octagonal tiling1. Fourier transform The octagonal (Ammann-Beenker) tiling is acodimension-two tiling built from the four-dimensionalhypercubic lattice Z (edge length a = 1). Perpendicu-lar and parallel spaces are spanned by the eigenvectorsof the matrix M = − − − − − − , (F1) associated to eigenvalues λ ± = 1 ± √ 2. This matrixsatisfies M = 2 M + 1 and its eigenvalues are twofolddegenerate. Here, we choose the following orthonormaleigenbasis e ∥ , = (cid:18) − , , , √ (cid:19) , e ∥ , = (cid:18) , , √ , (cid:19) , e ⊥ , = (cid:18) , − , , √ (cid:19) , e ⊥ , = (cid:18) − , − , √ , (cid:19) , (F2)where the perpendicular (parallel) space is associated to λ + ( λ − ). The acceptance window is an octagon corre-sponding to the projection of the four-dimensional unithypercube onto the perpendicular space. In this case,Eq. (C6) gives: | e n ( K ∥ ) | = Nλ + (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:0) λ + K ⊥ , − K ⊥ , (cid:1) K ⊥ , ( K ⊥ , + K ⊥ , ) − cos (cid:0) λ + K ⊥ , K ⊥ , (cid:1) K ⊥ , ( K ⊥ , − K ⊥ , ) + cos (cid:0) λ + K ⊥ , − K ⊥ , (cid:1) K ⊥ , ( K ⊥ , + K ⊥ , ) − cos (cid:0) λ + K ⊥ , K ⊥ , (cid:1) K ⊥ , ( K ⊥ , − K ⊥ , ) (cid:12)(cid:12)(cid:12)(cid:12) , (F3)for all reciprocal-lattice vector K with components K ∥ ,j = K . e ∥ ,j and K ⊥ ,j = K . e ⊥ ,j . These expressionscoincide with the one given in Ref. 31. 2. Structure factor We are interested in computing the behavior of S when | K ∥ | goes to 0 for K ∥ (cid:44) 0. To this aim, we note that fora vector K = 2 π ( s, t, u, v ) [where ( s, t, u, v ) ∈ Z ], onehas: λ + K ⊥ , = K ∥ , λ + + 2 π ( s − t + v ) , (F4) λ + K ⊥ , = K ∥ , λ + + 2 π ( − s − t + u ) , (F5) K ⊥ , + K ∥ , = 2 πv, (F6) K ⊥ , + K ∥ , = 2 πu. (F7)A close inspection of Eq. (F3) shows that one has todistinguish three different cases. a. Symmetry axes: S ( K ∥ ) ∼| K ∥ | As can be seen in Eq. (F3), the denominator vanishes ifone of the components K ⊥ ,i = 0 or when K ⊥ , = ± K ⊥ , .When K ⊥ belongs to these four symmetry axes, theFourier transform can be recast in a simple form. Forsimplicity, let us focus on the case where K ⊥ , = 0 (theother cases being treated similarly) for which | e n ( K ∥ ) | = Nλ + K ⊥ , (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) K ⊥ , (cid:19) − (cid:18) λ + K ⊥ , (cid:19) + K ⊥ , sin (cid:18) λ + K ⊥ , (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) . (F8)Using Eqs. (F4)-(F6), one then obtains S ( K ∥ ) ’ | K ∥ |→ N λ (cid:12)(cid:12)(cid:12)(cid:12) K ∥ , K ⊥ , (cid:12)(cid:12)(cid:12)(cid:12) . (F9)As previously, to analyze the behavior of the structurefactor for small | K ∥ | , we consider K ( p ) = M p K . By con-struction, in the large- p limit, the parallel components0of K ( p ) tend to zero as λ p − and its perpendicular com-ponents diverge as λ p + . As a result, S ( K ( p ) ∥ ) behaves as λ − p + so that, in this case, S ( K ∥ ) ∼ | K ∥ |→ | K ∥ | . (F10)This result actually holds for all K ⊥ belonging to thefour symmetry axes discussed above. b. Generic cases: S ( K ∥ ) ∼| K ∥ | or S ( K ∥ ) ∼| K ∥ | When K ⊥ does not belong to the four symmetry axesdefined as K ⊥ , = 0, K ⊥ , = 0, and K ⊥ , = ± K ⊥ , ,one can again use Eqs. (F4-F7) to express the structurefactor as S ( K ∥ ) ’ | K ∥ |→ N λ (cid:12)(cid:12)(cid:12)(cid:12) ( K ∥ , K ⊥ , + K ∥ , K ⊥ , )( K ∥ , K ⊥ , − K ∥ , K ⊥ , ) K ⊥ , K ⊥ , ( K ⊥ , + K ⊥ , )( K ⊥ , − K ⊥ , ) (cid:12)(cid:12)(cid:12)(cid:12) , (F11)which leads to S ( K ∥ ) ∼ | K ∥ |→ | K ∥ | . (F12)However, as can be seen in Eq. (F11), this leading con-tribution may vanish for some special K . In this case, S is given by the subleading contribution which gives S ( K ∥ ) ∼ | K ∥ |→ | K ∥ | . (F13) 3. Integrated intensity function To compute the integrated intensity function definedin Eq. (C11), we decompose the sum over all vector K ∥ with a norm smaller than k as a sum over all quadruplets( s, t, u, v ) and their iterated vectors K ( p ) ∥ = M p K ∥ . Onecan then write Z ( k ) = 4 π A X ( s,t,u,v ) ∞ X p = p ( s,t,u,v ) S ( K ( p ) ∥ ) , (F14)where p ( s,t,u,v ) is the smallest integer fulfilling the con-straint | K ( p ) ∥ | < k . As discussed above, the behavior of S in the large- p (small- | K ∥ | ) limit, strongly depends on K ∥ [see Eqs. (F10-F13)]. This is in stark contrast with theRauzy tiling and the Fibonacci chain where there is thesame power-law scaling for all K ∥ [see Eqs. (D10)-(E4)].However, since we are interested in the small- k (large- p ) limit, one only keeps the dominant terms in Eq. (F14)that come from the symmetry axes and gives S ( K ( p ) ∥ ) ’ | K ( p ) ∥ | f ( s, t, u, v ) , (F15) | K ( p ) ∥ | ’ λ − p + g ( s, t, u, v ) . (F16)We emphasize that, as for the Fibonacci chain, f and g are functions that do not depend on p , so that onestraightforwardly gets the following discrete scaling rela-tion Z ( k/λ + ) = Z ( k ) /λ . (F17) The solution of this equation can be written as Z ( k ) = k F (ln k/ ln λ + ) , (F18)where F ( x + 1) = F ( x ). As a result, Z has a power-lawenvelope together with log-periodic oscillations. Appendix G: The Penrose tiling The Penrose rhombus tiling can be built by CPfrom the five-dimensional hypercubic lattice Z (edgelength a = 1) along a well-known procedure (see, e.g.,Ref. 33 for details). For our purpose, let us consider thefollowing orthogonal (non normalized) basis e ∥ , = 25 (1 , c , c , c , c ) , e ∥ , = 25 (0 , s , s , − s , − s ) , e ⊥ , = 25 (1 , c , c , c , c ) , e ⊥ , = 25 (0 , s , − s , s , − s ) , e ∆ = 110 (1 , , , , , (G1)that defines the three subspaces E ∥ , E ⊥ and ∆.Here, we introduced the notation c n = cos(2 πn/ 5) ands n = sin(2 πn/ Z is selected wheneverit projects onto the perpendicular space E ⊥ + ∆ in-side a three-dimensional acceptance window which is theprojection of the five-dimensional unit hypercube ontothis subspace. Remarkably, selected points only fill fiveplanes perpendicular to ∆. Thus, the selection step onlyamounts to consider discrete sections of the acceptancewindow. Among all possible choices, the fivefold symmet-ric canonical Penrose tilings (known as star and sun )considered here correspond to the following sections: onepoint which is the symmetry center of the tiling, tworegular pentagons of side 2 p / π/ τ p / π/ τ = √ is the golden ratio. 1. Fourier transform The Fourier transform of the tiling’s vertices is ob-tained as a weighted sum of the Fourier transform of thefour regular pentagons. For any vector R of the five-dimensional hypercubic lattice, the five-dimensional re-ciprocal lattices vectors K satisfy K · R = K ∥ · R ∥ + K ⊥ · R ⊥ + K ∆ · R ∆ = 2 π × integer . (G2)Then, after some algebra, Eq. (C6) leads to: | ˜ n ( K ∥ ) | = N (cid:12)(cid:12)(cid:12)(cid:12) − s )5 K ⊥ , (cid:26) cos(c K ⊥ , +s K ⊥ , − K ∆ )+cos( K ⊥ , / +s ) K ⊥ , + K ∆ ) − cos( K ⊥ , − K ∆ ) − cos(2c K ⊥ , − K ∆ )5 K ⊥ , − (6s +2s ) K ⊥ , − cos(c K ⊥ , − s K ⊥ , − K ∆ )+cos( K ⊥ , / − (s +s ) K ⊥ , + K ∆ ) − cos( K ⊥ , − K ∆ ) − cos(2c K ⊥ , − K ∆ )5 K ⊥ , +(6s +2s ) K ⊥ , + cos(c K ⊥ , − s K ⊥ , − K ∆ )+cos( K ⊥ , / − (s +s ) K ⊥ , + K ∆ ) − cos(c K ⊥ , − s K ⊥ , − K ∆ ) − cos[(1+c ) K ⊥ , +s K ⊥ , − K ∆ ]5 K ⊥ , − (6s − ) K ⊥ , − cos(c K ⊥ , +s K ⊥ , − K ∆ )+cos( K ⊥ , / +s ) K ⊥ , + K ∆ ) − cos(c K ⊥ , +s K ⊥ , − K ∆ ) − cos[(1+c ) K ⊥ , − s K ⊥ , − K ∆ ]5 K ⊥ , +(6s − ) K ⊥ , (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) , (G3)for all reciprocal-lattice vector K with components K ∥ ,j = K . e ∥ ,j , K ⊥ ,j = K . e ⊥ ,j , and K ∆ = K . e ∆ . 2. Structure factor We are interested in computing the behavior of S when | K ∥ | goes to 0 for K ∥ (cid:44) 0. To this aim, we note that fora vector K = 2 π ( s, t, u, v, w ) [where ( s, t, u, v, w ) ∈ Z ],one hasc K ⊥ , + s K ⊥ , − K ∆ = − c K ∥ , + s K ∥ , − K ∆ + 2 πv, (G4)c K ⊥ , − s K ⊥ , − K ∆ = − c K ∥ , − s K ∥ , − K ∆ + 2 πu, (G5) K ⊥ , / + s ) K ⊥ , + K ∆ = − K ∥ , / − s ) K ∥ , + 5 K ∆ − π ( u + w ) , (G6) K ⊥ , / − (s + s ) K ⊥ , + K ∆ = − K ∥ , / − (s − s ) K ∥ , + 5 K ∆ − π ( v + t ) , (G7) K ⊥ , − K ∆ = − K ∥ , − K ∆ + 2 πs, (G8)2c K ⊥ , − K ∆ = − K ∥ , − K ∆ + 2 π ( t + w ) , (G9)c K ⊥ , + s K ⊥ , − K ∆ = − c K ∥ , − s K ∥ , − K ∆ + 2 πt, (G10)c K ⊥ , − s K ⊥ , − K ∆ = − c K ∥ , + s K ∥ , − K ∆ + 2 πw, (G11)(1 + c ) K ⊥ , + s K ⊥ , − K ∆ = − (1 + c ) K ∥ , + s K ⊥ , − K ∆ + 2 π ( s + v ) , (G12)(1 + c ) K ⊥ , − s K ⊥ , − K ∆ = − (1 + c ) K ∥ , − s K ⊥ , − K ∆ + 2 π ( s + u ) . (G13)Keeping in mind that 5 K ∆ = π ( s + t + u + v + w ), onecan finally rewrite Eq. (G3) as a function of K ∥ ,j and K ⊥ ,j only. In the limit where | K ∥ | → S ( K ∥ ) ’ | K ∥ |→ N ( √ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( K ⊥ , + K ⊥ , )[( K ∥ , − K ∥ , ) K ⊥ , − K ∥ , K ∥ , K ⊥ , ] K ⊥ , (5 K ⊥ , − K ⊥ , K ⊥ , + K ⊥ , ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (G14)2However, when one of the denominators in Eq. (G3)vanishes, one gets different expressions that are easilyobtained along the same line.To analyze the behavior of the structure factor forsmall | K ∥ | , we consider the matrix M = , (G15)whose eigenspaces are E ∥ , E ⊥ and E ∆ with eigenvalues λ ∥ = 1 /τ , λ ⊥ = − τ and λ ∆ = 2, respectively. By con-struction, in the large- p limit, the parallel components of K ( p ) = M p K vanishes as λ p ∥ whereas its perpendicularcomponents diverge as λ p ⊥ . As a result, in the large- p limit, S ( K ( p ) ∥ ) behaves as λ p ∥ so that S ( K ∥ ) ∼ | K ∥ |→ | K ∥ | . (G16) 3. Integrated intensity function As for the octagonal tiling, we decompose the sumover K ∥ in the integrated intensity function defined inEq. (C11) as a sum over all quintuplets ( s, t, u, v, w ) andtheir iterated under K ( p ) ∥ = M p K ∥ . One can then write Z ( k ) = 4 π A X ( s,t,u,v,w ) ∞ X p = p ( s,t,u,v,w ) S ( K ( p ) ∥ ) , (G17) where p ( s,t,u,v,w ) is the smallest integer fulfilling the con-straint | K ( p ) ∥ | < k . In the small- k (large- p ) limit , one cancheck that S ( K ( p ) ∥ ) ’ | K ( p ) ∥ | f ( s, t, u, v, w ) , (G18)for any quintuplet ( s, t, u, v, w ) ∈ Z . However, contraryto the octagonal tiling, the scaling of K ( p ) ∥ with p dependson the quintuplet. For quintuplets that do not annihilatethe denominator in Eq. (G3), one gets: | K ( p ) ∥ | ’ τ − p g ( s, t, u, v, w ) , (G19)or, in other words, | K ( p ) ∥ / K ( p +1) ∥ | = τ . Importantly,when one of the denominators in Eq. 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