WANNIER TRANSFORM APERIODIC SOLIDS. for. Jean BELLISSARD. Collaboration:

Transcription

1 WANNIER TRANSFORM for APERIODIC SOLIDS Jean BELLISSARD Georgia Institute of Technology, Atlanta School of Mathematics & School of Physics Collaboration: G. DE NITTIS (SISSA, Trieste, Italy) V. MILANI (U. Tehran, Tehran, Iran)

2 Main References J. E. ANDERSON, I. PUTNAM, Topological invariants for substitution tilings and their associated C -algebras, Ergodic Theory Dynam. Systems, 18, (1998), J. C. LAGARIAS, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom., 21, (1999), J. BELLISSARD, R. BENEDETTI, J. M. GAMBAUDO, Spaces of Tilings, Finite Telescopic Approximations, Comm. Math. Phys., 261, (2006), J. BELLISSARD, G. DE NITTIS, V. MILANI, Wannier transform for aperiodic tilings, in preparation, (2010)

3 Motivation

4 Motivation

5 Motivation Figure 1: A sample of the icosahedral quasicrystal AlPdMn

6 Motivation

7 Motivation

8 Motivation For periodic crystals, the Wannier transform leads to band spectrum calculation (Bloch theory) The Wannier transform uses the translation invariance of the crystal Is it possible to extend it to aperiodic solids?

9 Content 1. An example: Fibonacci 2. The Wannier Transform 3. The Schrödinger Operator 4. To conclude

10 I - An example: Fibonacci

11 The Fibonacci Sequence The Fibonacci sequence is an infinite word generated by the substitution Iterating gives ˆσ : a ab, b a }{{} a }{{} ab ab a aba ab abaab aba }{{}}{{}} {{ } a 0 a 1 a 2 =a 1 a 0 a 3 =a 2 a 1 a 4 =a 3 a 2 abaababa abaab } {{ } a 5 =a 4 a 3 It can be represented by a 1D-tiling if a [0, 1] b [0, σ] σ =

12 The Fibonacci Sequence

13 The Fibonacci Sequence

14 The Fibonacci Sequence

15 The Fibonacci Sequence

16 The Fibonacci Sequence

17 The Fibonacci Sequence - Collared tiles in the Fibonacci tiling -

18 The Fibonacci Sequence - The Anderson-Putnam complex for the Fibonacci tiling -

19 The Fibonacci Sequence - The substitution map -

20 The Fibonacci Sequence Let Ξ n X n be the set of tile endpoints (0-cells). The sequence of complexes (X n ) n N together with the maps f n : X n+1 X n gives rise to inverse limits lim (X n, f n ) = Ω lim (Ξ n, f n ) = Ξ The space Ω is compact and is called the Hull. It is endowed with an action of R generated by infinitesimal translation on the X n s The space Ξ is a Cantor set and is called the transversal Ξ parametrizes a the set of all tilings sharing the same words as the Fibonacci sequence with one tile endpoint at the origin. There is a two-to one correspondence between Ξ and the window.

21 The Fibonacci Sequence

22 The Fibonacci Sequence

23 The Fibonacci Sequence

24 The Fibonacci Sequence

25 The Fibonacci Sequence

26 The Fibonacci Sequence

27 The Fibonacci Sequence

28 The Fibonacci Sequence: Groupoid Γ Ξ is the set of pairs (ξ, a) with ξ Ξ and a L ξ. It is a locally compact groupoid when endowed with the following structure Units: Ξ, Range and Source maps: r(ξ, a) = ξ, s(ξ, a) = T a ξ Composition: (ξ, a) (T a ξ, b) = (ξ, a + b) Inverse: (ξ, a) 1 = (T a ξ, a) Topology: induced by Ξ R

29 II - Wannier Transform J. BELLISSARD, G. DE NITTIS, V. MILANI, Wannier transform for aperiodic tilings, in preparation, (2010)

30 Wannier Transform: Periodic Case If Z R is a one dimensional lattice the Wannier transform is defined for a function f C c (R) by g(s; k) = W f (s; k) = f (s + a) e ık a a Z Here k belongs to the dual group of Z, called Brillouin zone B T = R/2πZ Bloch boundary conditions: g(s + a; k) = g(s; k)e ık a whenever a Z. Plancherel s formula:. 1 0 ds T dk 2π g(s; k) 2 = R dx f (x) 2 Unitarity: W : L 2 (R) L 2 ([0, 1]) L 2 (T) is a unitary operator.

31 Wannier Transform: Definition In the case of the Fibonacci sequence: ξ Ξ, L ξ being the corresponding Delone set, v = ˆσ n (w) the n-th substitute of a collared tile. Denote by B T 2 the dual group of Z 2. Then, for s R and k B the Wannier transform of a function f C c (R) is W ξ f (v, s; k) = a L ξ (v) f (s + a)e ık a

32 Wannier Transform: Properties Smoothness: if f is smooth, then d W ξ k f dx k = k W ξ f s k Covariance: if g = W f then g ξ (v, s + b; k) = g T b ξ (v, s; k) eık b b L ξ Inversion: if dk denotes the normalized Haar measure on B f (s + a) = B dkg ξ (v, s; k) e ık a a L ξ, s R

33 Wannier Transform: Momentum Space Let E ξ (v) L 2 (B) be the closed subspace generated by {e a : k B e ık a ; a L ξ } E(v) = ( E ξ (v) ) ξ Ξ is a continuous field of Hilbert spaces. If W v (ξ, a) : E T a ξ (v) E ξ(v) is defined by W v (ξ, a)e b = e a+b then the family ( W v (γ) ) γ Γ Ξ defines a strongly continuous unitary representation of the groupoid Γ Ξ.

34 Wannier Transform: Momentum Space Define H ξ = v L2 (v) E ξ (v) L 2 (X n ) L 2 (B) where v varies among the d-cells of the Anderson-Putnam complex. Let Π ξ : L 2 (X n ) L 2 (B) H ξ be the corresponding orthogonal projection. H = ( H ξ )ξ Ξ is a continuous field of Hilbert spaces. Similarly U(γ) = v 1 v W v (γ) defines a strongly continuous unitary representation of the groupoid Γ Ξ on H.

35 Wannier Transform: Plancherel The Wannier transform is a strongly continuous field of unitary operators defined on the constant field L 2 (R) with values in H R dx f (x) 2 = v v ds The Wannier transform is covariant: B dk W ξ f (v, s; k) 2 U(ξ, a) W T a ξ = W ξ U reg (a) where U reg is the usual action of the translation group R in L 2 (R).

36 III - Schrödinger s Operator J. BELLISSARD, G. DE NITTIS, V. MILANI, Wannier transform for aperiodic tilings, in preparation, (2010)

37 The Schrödinger Operator: Model As an example let an atomic nucleus be placed is each tile, namely sites in L ξ. The atomic species are labeling the tiles. The corresponding atomic potential has compact support small enough to be contained in one tile V ξ (x) = v a L ξ (v) v (v) at (x a) The Schrödinger operator describing the electronic motion is then a covariant family H ξ (x) = + V ξ

38 The Schrödinger Operator: Form If f C 1 c (R) then, like in the Bloch Theory for periodic potentials Q ξ ( f, f ) = f H ξ f L 2 (R) = v v ds B dk ( s W ξ f (v, s; k) 2 + v (v) at (s) W ξ f (v, s; k) 2 ) = B dk ˆQ k ( (Wξ f ) k, (W ξ f ) k ) with ( ) ˆQ k g, g = v v ds ( s g(v, s) 2 + v (v) at (s) g(v, s) 2)

39 The Schrödinger Operator: Form A function g belongs to the form domain of ˆQ k if and only if 1. both g(v, s) and its derivative are in L 2 (v) for all (d = 1)-cell v 2. g satisfies the following cohomological equation: at each ({d 1} = 0)-cell u of the Anderson-Putnam complex v: 0 v=u g(v, u)e ık a ˆv v = w: 1 v=u g(w, u)e ık a ˆv w where a v w is the translation vector sending the initial point of v the initial point of w, and ˆv is one tile touching u.

40 The Schrödinger Operator: Form

41 The Schrödinger Operator: Bands The form ˆQ k generates a selfadjoint operator Ĥ k defined by g Ĥ k g L 2 (X n ) = ˆQ k (g, g) On each d-cell v, Ĥ k = s + v (v) at, with k-dependent boundary conditions. Since a cell is compact it follows that Ĥ k is elliptic, thus it has compact resolvent. In particular its spectrum is discrete with finite multiplicity, namely its eigenvalues are E 0 (k) E 1 (k) E r (k) with each E r (k) a smooth function of k B.

42 The Schrödinger Operator: Bands What is the connection with the original operator? Theorem The Schrödinger operator H ξ is given by H ξ = Π ξ B dk Ĥ k Π ξ if Π ξ is the orthogonal projection from L 2 (X n ) L 2 (B) onto H ξ. The restriction to the subspace H ξ is NOT INNOCENT and reduces the band spectrum to produce a Cantor spectrum in the onedimensional cases.

43 IV - To Conclude

44 1. The Fibonacci sequence can be replaced by aperiodic, repetitive tilings on R d with finite local complexity. The Hull and the transversal are well-defined. 2. The Lagarias group L plays the role of Z 2 in general. It is always free with finite rank. Then B is the group dual to L. 3. The definition of the Wannier transform can be extended to this case 4. The sequence of Anderson-Putnam complexes (X n ) n N can be defined in this general case as well. 5. The Wannier transform identifies wave functions in L 2 (R d ) with a proper subspace of L 2 (X n ) L 2 (B) 6. The Schrödinger operator can be written in terms of this new representation as the compression of a Bloch-type operator exhibiting band spectrum.