topologicalpropertiesofquasiperiodictilings...topologicalpropertiesofquasiperiodictilings yaroslav...

T P Q T Y D, D G, E L E A D P, T – I I T, I A Topological properties of finite quasiperiodic tilings are examined. We study two specific physical quantities: (a) the structure factor related to the Fourier transform of the structure; (b) spectral properties (using scattering matrix formalism) of the corresponding quasiperiodic Hamiltonian. We show that both quantities involve a phase, whose windings describe topological numbers. We link these two phases, thus establishing a “Bloch theorem” for specific types of quasiperiodic tilings. C I Email: [email protected] This work was supported by the Israel Science Foundation Grant No. 924/09. D T Substitutions and Atomic Distributions Define a binary substitution rule by σ ()= α β σ ()= γ δ ⇐⇒ → α β → γ δ Associate occurrence matrix: M = αβ γδ Consider only primitive matrices: • Largest eigenvalue λ 1 > 1 (Perron-Frobenius) • Left and right first eigenvectors are strictly positive Distribution of letters underlies distribution of atoms: 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • Define atomic density ρ ( )= δ (- ) with distances for and given by δ = +1 - = Let ¯ be the mean distance and the deviations from the mean. Define = ¯ + δ δ≡ - Let (ξ )= ∑ e - i ξ be the diffraction pattern, and S (ξ )= | (ξ )| 2 the structure factor. Bragg peaks are located at [1] ξ N = 2π ¯ λ N 1 We consider the following families: Pisot. The second eigenvalue |λ 2 |<1. Non-Pisot. The second eigenvalue |λ 2 |≥ 1. Fluctuations are unbounded [2]; there are no Bragg peaks [3]. Examples Fibonacci. → , → . It is Pisot, M = ( 11 10 ) , λ 1 =( √ 5 + 1)/ 2 ≡τ the golden ratio and λ 2 = -τ -1 . Bragg peaks: ξ =( + τ )2π/ ¯ . Thue-Morse. → , → . Here it is Pisot, M = ( 11 11 ) , λ 1 = 2 and λ 2 = 0. Bragg peaks: ξ N = 2 -N (2π/ ¯ ). T P – S P Another way to define a tiling is by using a characteristic function. We consider the following choice [4, 5]: χ ( φ) = sign cos ( 2πλ -1 1 + φ ) - cos ( πλ -1 1 ) with =0 F N - 1 and [0 2π ] φ→φ =2πF -1 N . The phase φ—called a phason—accounts for the freedom to choose the origin. Let 0 ()= χ ( 0). Let [ 0 ()] = 0 ( + 1) be the translation operator. Define Σ 0 = 0 [ 0 ] ··· F N -1 [ 0 ] = ⇒ Σ 0 ( )= [ 0 ()] Consider now a row permuted Σ 1 Σ 1 ( )= ( ) [ 0 ()] ( )= F -1 N-1 (mod F N ) Lemma. For φ =2π/F N with =0 F N - 1 one has χ ( φ )=Σ 1 ( ). Corollary. This defines a discrete phason φ for the structure. The discrete Fourier transform of Σ 1 reads G (ξ ) ≡ F N -1 =0 ω -ξ Σ 1 ( )= ω ( )ξ ς 0 (ξ ) The structure factor S (ξφ)= |ς 0 (ξ )| 2 is φ-independent. The phase of G (ξ ) reads Θ(ξ ) ≡ arg ω ( )ξ = φ ξ/F N-1 (mod 2π ) Corollary. For any ξ = F N-1 one has the (discrete) winding number at ξ , Θ ( ξ ) = 2π F N = ⇒W ξ = 1 2π 2π 0 ∂Θ ( ξ = ξ φ ) ∂φ dφ = S P T Consider a 1D discrete tight-binding equation, - (ψ +1 + ψ -1 )+ V ψ =2Eψ The gaps in the integrated density of states are given by [6] N = 1 λ N 1 (mod 1) N ∈ Z Here, is the gcd of λ 1 and its corresponding eigenvectors in both M and the collared M 2 . S M Spectral properties are also accessible from the continuous wave equation, - d 2 ψ d 2 - 2 0 ( ) ψ ( )= 2 0 ψ ( ) with scattering boundary conditions. The scattering -matrix is defined by ( - → ← - ) = - → ( ) ( ) ( ) ← - ( ) ( - → ← - ) ≡ ( - → ← - ) , with - → = - → R e i - → and ← - = - → R e i ←- . It is unitary and can be diagonalized to → e i 1 0 0 e i 2 so that det =e 2i δ ( ) with δ ( )=( 1 ( )+ 2 ( )) / 2 and α ( )= - → ( ) - ← - ( ). Using the Krein-Schwinger formula [7] allows to relate the change of density of states to the scattering data, ( ) - 0 ( )= 1 2π Im d d ln det ( ) So that the integrated density of states is ( ) - 0 ( )= δ ( ) /π The total phase shift δ ( ) is independent of the phason φ unlike the chiral phase α (φ), whose winding for values of inside the gaps is given by [8], W α = 1 2π 2π 0 ∂α ( ν = ν φ ) ∂φ dφ =2 Both the winding numbers W ξ 0 previously found and W α are topological. As such, they are robust against perturbations. C P S Yet another method to build quasiperiodic tilings is by the Cut & Project. The procedure is as follows [9]. Cut. 1. Start with an -dimensional space R = R . 2. Insert “atoms” on the integer lattice Z = Z . 3. Divide R into the physical space E and the internal space E ⊥ such that E⊕E ⊥ = R and E∩E ⊥ = ∅. 4. To resolve ambiguity for E , choose an initial location ∈R such that E passes through . There is no such requirement for E ⊥ . Project. 1. Inspect the hypercube I =[-05 05) . 2. The window is its projection on the internal space W = π ⊥ (I ). 3. The strip is the product with the physical space S = W⊗E . 4. Choose only the points inside the strip S∩Z , and project them onto the physical space, Y = π (S∩Z ). 5. The atomic density is given by ρ ( ) ≡ρ ( )= ∑ ∈Y δ ( - ) with ∈E . Note the implicit dependency of Y on . For the 1D systems we consider all along, define the phason φ =2π /W ∈E ⊥ where W is the window above. The slope is given by 1/ = 1 + cot α . R P: A “B ” The windings of the structural phase Θ (νφ), where ν = ξ/F N the normalized wavenumber, correspond to the Bragg peak locations given by ν = + λ 1 = ⇒W ν = Note also, that the integrated density of states of the corresponding Hamiltonian has gap locations given by the Gap-labeling theorem [6] expressed by τ * [K 0 (Ω T )] = = + λ 1 (mod 1) Drawing the integrated density of states on top the structural phase shows the relation between the winding Θ (νφ) and the integer in the integrated density of states (red line), Now, consider the spectral (chiral) phase α (νφ). Its winding W α =2 can be directly read by the following graph, which is analogous to the figure above. It directly shows the relation between the two phases Θ (νφ) and α (νφ). Here, we used F N = 233 sites, A = 1 and B =115 for better discernment. We view this result as a Bloch-like theorem for quasiperiodic tilings [10]. U T Unlike the case of periodic structures, for aperiodic tilings topological numbers cannot be simply expressed as Chern numbers, since the notion of Brillouin zone does not exist any longer. We are thus led to use other set of tools. • Tiling space T (dependent on λ 1 ) and its hull Ω T . • Čech cohomology ˇ H 1 (Ω T ), simplicial cohomology H 1 (Γ ) and Bratteli graphs [11, 12]. •K -theory, K 0 (Ω T ) group and the abstract Gap-labeling theorem [6, 13] μ * [K 0 ( (Ω T ))] = τ * [K 0 ( * (Ω T R ))] • Pattern-equivariant functions PE and cohomology H 1 PE (Ω T ) [14]. C • We have defined two types of phases—a structural and spectral one—whose windings unveil topological features of quasiperiodic tilings. • We found a relation between these two phases, which can be interpreted as a Bloch-like theorem. • We have considered here a subset of tilings, which are known as Sturmian (C&P) words. Our results can be extended to a broader families of tilings in one dimension, and to tiles in higher dimensions (D> 1). • All these features have been observed experimentally [4, 5]. R [1] J. M. Luck, C. Godrèche, A. Janner, and T. Janssen, J. Phys. A 26, 1951 (1993). [2] J. M. Dumont, in Springer proceedings in physics, edited by J.-M. Luck, P. Moussa, and M. Waldschmidt, (Springer Berlin Heidelberg, 1990), pp. 185–194. [3] C. Godrèche, and J. M. Luck, Phys. Rev. B 45, 176 (1992). [4] F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, Phys. Rev. B 95, 161114 (2017). [5] A. Dareau, E. Levy, M. B. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, Phys. Rev. Lett. 119, 215304 (2017). [6] J. Bellissard, A. Bovier, and J.-M. Chez, Rev. Math. Phys. 04, 1 (1992). [7] E. Akkermans, G. V. Dunne, and E. Levy, in Optics of aperiodic structures: funda- mentals and device applications, edited by L. Dal Negro, (Pan Stanford Publish- ing, 2014), pp. 407–449. [8] E. Levy, A. Barak, A. Fisher, and E. Akkermans, (2015) https://arxiv.org/abs/ 1509.04028. [9] M. Duneau, and A. Katz, Phys. Rev. Lett. 54, 2688 (1985). [10] Y. Don, D. Gitelman, E. Levy, and E. Akkermans, (in preperation), 2018. [11] L. A. Sadun, Vol. 46, University Lecture Series (American Mathematical Society, 2008). [12] J. E. Anderson, and I. F. Putnam, Ergod. Th. Dynam. Sys. 18, 509 (1998). [13] J. Kellendonk, and I. F. Putnam, in Directions in mathematical quasicrystals, Vol. 13, edited by M. Baake, and R. V. Moody, CRM Monograph Series (American Mathematical Society, 2000), pp. 186–215. [14] J. Kellendonk, J. Phys. A: Math. Gen. 36, 5765 (2003). [15] E. Bombieri, and J. E. Taylor, J. Phys. Colloq. 47, C3–19–C3 (1986).

topologicalpropertiesofquasiperiodictilings...topologicalpropertiesofquasiperiodictilings yaroslav...

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