Wannier Transform for Quasicrystals: preliminary results

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Wannier Transform for

Quasicrystals

preliminary results

Giuseppe De Nittis

(LAGA,Université Paris 13)

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Challenges in Aperiodic Media

University of Lyon 1, France. February 28 - March 2, 2011

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Joint work with:

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Outline

1 Introduction

Bloch’s legacy

Quasicrystals: phenomenology and modeling Motivations

2 Anderson-Putnam Complex

Voronoi tiling Collar (1-st. order)

Translational equivalence and proto-cells Incident number and homology

3 Wannier transform

Periodic case

Aperiodic case (D-set)

4 Schrödinger operators and boundary conditions

Wannier decomposition of the Laplacian

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[1925-1926]Birth of(modern) Quantum Mechanics:

W. Heisenberg(matrix mechanics) andE. Schrödinger(wave mechanics).

[1928]Birth ofmodern (or Quantum) Theory of Solids: F. Bloch(PhD thesis, supervisor Heisenberg).

[1928 - today]Consequence ofBloch’s theory:

- conductivity properties: band-gap structure of the energy spectrum, conduction/valence band, Fermi level, etc.;

- Bethe-Sommerfeld conjecture: proved in many cases;

- thermodynamic properties: density of states, absolute continuity of the spectrum, etc.;

- semiclassical models: tight-binding models (Wannier functions), Peierls substitution, etc.;

- topological quantization: QHE, piezoelectricity, de Hass-van Alphen effect, etc.;

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Outline

1 Introduction Bloch’s legacy

Quasicrystals: phenomenology and modeling

Motivations

3 Wannier transform

Periodic case

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•ACrystalis ad(=3) dimensionalperiodicarrangement of atoms with translational periodicity along its principal axes. - 230Fyodorov groups(rotations, reflections, inversions, translations)

- 32point groups(without translations)

- 11Laüe groups (center of symmetry)

- 5rotational symmetries(2,3,4,6-fold)≡CR-Theorem

Laüe groups≡(X-ray, electron, ect.) diffraction patterns.

•In 1984,Shechtman,Blech,GratiasandCahnshowed a diffraction patterns (of an Al-Mn alloy) with10-fold symmetry (forbidden by CR-Th).

•Many stable and meta-stable quasicrystals have been found: -pointlike diffraction patterns(like in a perfect crystal),

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Modeling of quasicrystals: “D-set”L ⊂_Rd

i)Discreteandaperiodic. L+a=L iff_Rd₃_a₌_0;

ii)Delone set:

- uniformly discrete, i.e. there isr>0such that every open ball of radius r meetsL at most on one point;

- relatively dense, i.e. there isR>0such that every closed ball of radius R meetsL at least on one point.

iii)Repetitive:

given any patch℘⊂L, there is aR>0such that in any ball of radius R there is a patch℘0⊂L which differs byε>0(in Hausdorff sense) from℘up to a translation.

iv)Finite local coplexity(FLC):

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Outline

1 Introduction Bloch’s legacy

Quasicrystals: phenomenology and modeling

Motivations

3 Wannier transform

Periodic case

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“Strange” physical properties of quasicrystals:

•mostlyinsulatorsat low temperature (even though made of good metals !!);

•mechanicallyhardandfragile;

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•Quantum-mechanicaldescription (L =D-set):

Hqc:=− }

2m∆ +Vqc, Vqc:=

N

∑

i=1

∑

a∈_Li

v(i)(· −a)

i=atomic types,Li⊂L positions ofi-atoms,v(i)typical potential of ani-atom.

- Numerical calculations extremely hard;

- no way of treating the aperiodicity as a perturbation of a periodic structure.

•For periodic crystals:

Translation invariance ⇒ Wannier transform ⇒ Bloch theory.

•Need of new tools to studyHqc:

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Outline

1 Introduction

Bloch’s legacy

Voronoi tiling

Collar (1-st. order)

3 Wannier transform

Periodic case

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•Lper⊂Rd, discrete

0∈Lper, and Γper:=Lper−Lper'Zd.

(open)Voronoi cellina∈Lper

Va:= n

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•L ⊂_Rd_,_D-set_with₀_∈_L_.

THEOREM (J. C. Lagarias, 1999)

The subgroup ofRd generated byL −L is afinitelygenerated free group (Lagarias group)

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•Voronoi tiles

Ta:=Va, a∈L. Convex politope with (natural)punctureina.

•Voronoi-tiling{Ta : a∈L} [

a∈L

Ta=Rd, Va∩Vb=/0 if a6=b∈L.

•(Open)n-faceF(,→_Rn₎_:

Fb:=Ta1∩Ta2∩. . .∩Tan+16=/0, codim.(Fb) =d−n. Fbispuncturedinb∈_Rd _{(eg. its}_barycenter_).

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Outline

1 Introduction

Bloch’s legacy

2 Anderson-Putnam Complex Voronoi tiling

3 Wannier transform

Periodic case

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•aandbarenearest-neighbours(n.-n.) if:

codim.(Fa,b) =1, Fa,b:=Ta∩Tb.

•(1-st. order)collarofVa:

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•Any Voronoicellcan be endowed with a collar:

Col(Va) :=

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•Anyn-facecan be punctured (barycenter) and collared:

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Outline

1 Introduction

Bloch’s legacy

Translational equivalence and proto-cells

Incident number and homology

3 Wannier transform

Periodic case

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DEFINITION

LetFcj ∈F(n)_{, with j} ₌₁_,₂_{, be a pair of n-faces with punctures} incj∈Rd. Then

Fc1∼Fc2 (translationary equivalent) iff:

(i)Fc₁−c1=Fc2−c2 (samegeometric support); (ii)Col(Fc1) =Col(Fc2) (samecollar).

•Because of theFLC

Q(d)_:=_F(d)_/_∼₌_{_V

1, . . . ,VNd} (proto-cells)

Q(d−1)_:=_F(d−1)_/_∼₌_{_F

1, . . . ,FNd−1} (proto-faces)

..

. (proto-n-faces)

Q(0)_:=_F(0)_/_∼₌_{_p

1, . . . ,pN0} (proto-vertices)

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Periodic case

1proto-cell Q(d)₌_{_V_}

1proto-vertex Q(0)₌_{_p_}

•Q(d−1)₌_{_F

1, . . .FN}; anyproto-facesis a “double” face forV:

∂V=F+₁ ∪ F−₁

| {z } ∼_F1

∪ F+₂ ∪ F−₂

| {z } ∼_F2

∪ . . . ∪ F_N+ ∪ F_N−

| {z } ∼_FN

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Outline

1 Introduction

Bloch’s legacy

Translational equivalence and proto-cells

Incident number and homology

3 Wannier transform

Periodic case

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Relative position number

N :Q(n)×Q(n−1)−→ {0,1,2}

N(Fj,Gk) :=]{r =ck−bj : Gck ⊂Fbj, ∀Gck ∼Gk andFbj ∼Fj}.

N(Fj,Gk) =



 

 

0, forbidden attachment,

1, unique attachment, 1 rel. positionr_F_j→_Gk,

2, local periodicity, 2 rel. positionsr_F±

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•Any proto-n-face can beorientedbyFj,→_Rn.

•The orientation of a proto-n-facesFj fixes the orientation ofall

the equivalent “real”n-faces in the Voronoi tiling. Arelative orientationbetweenn-faces in the Voronoi tiling is fixed.

Incidence Number (I.N.)

[·; ·]∼:Q(n)×Q(n−1)−→ {−1,0,1}

[Fj;Gk]∼=         

0, ifN(Fj,Gk) =0

±1= [Fb_j;Gck] ifN(Fj,Gk) =1 ck−bj=rFj→Gk

0=

_∑

σ∈{±}

[Fbj;Gcσ

k] ifN(Fj,Gk) =2 c ±

k −bj=r_F±_j→_Gk

Short notation:

[Fj,Gσ

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•Homological relation (S. Eilenberg, 1944):

∑

Fj∈Q(n)

[Ri,Fj]∼[Fj,Gk]∼=0, ∀Ri∈Q(n+1), Gk ∈Q(n−1).

THEOREM

The graded set

Q:=

d ]

j=0 Q(j)

is aCW-complexwith (singular)homologyinduced by the Incidence Number [·; ·]∼ .

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Outline

1 Introduction

Bloch’s legacy

3 Wannier transform

Periodic case

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Wannier decomposition

L2(Rd) I //L2(V)⊗`2(Lper) II //L2(V)⊗L2(B)

ψ //∑a∈_Lper δa⊗(T−aψ)V

_/_/

∑a∈_Lper

ˇ

δa⊗(T−aψ)V

Step I

- Rd3x7→(s,a)∈V×Lper (spatial decomposition), - (Taψ)(·) :=ψ(· −a) (translation operators),

- {δa} ⊂`2(Lper)'`2(Γ) (canonical basis).

Step II

- Id_L2₍_V₎⊗Fˇ ( inverse-Fourier transform),

- _B:=Rd/Γ∗, (Brillouin zone),

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•Wannier transform(periodic setLper):

L2(Rd)3ψ7−→W (Wψ)∈L2(V)⊗L2(B), W := (step II)◦(step I)

(Wψ)(s;k) :=

_∑

a∈Lper

e−ik·aψ(s+a), s∈V, k∈B

s∈Vthe position w.r.t. the puncture0!

•Plancherel’s formula(unitarity):

kψk2_L2₍_Rd₎=

Z

B Z

V|(Wψ)(s;k)|

2_ds

| {z }

k(Wψ)(·;k)k2

L2(V)

dk.

dk=normalized Haar measure.

•Smoothness:

W ∂αψ ∂xα

(s;k) = ∂

α

∂sα(Wψ) (s;k), ψ ∈C

∞₍

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Outline

1 Introduction

Bloch’s legacy

3 Wannier transform Periodic case

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•Wannier transform(D-setL):

L2(Rd)3ψ7−→W



 

(W(1)ψ)

.. .

(W(Nd)ψ)



 ∈L

2₍_Q₎_⊗_L2₍ B)

(W(j)ψ)(s;k) :=

_∑

a∈_Lj

e−ik?aψ(s+a), s∈Vj, k∈B.

- L_j:={a∈L : Va∼Vj} ⊂ΓL whereQ(d)={V1, . . . ,VNd}

L=L₁ ]. . . ]L_N_d;

- L2₍_Q_{) :=}LNd

j=1 L2(Vj);

- _B:=Rg/Γ∗_L, (Brillouin zoneor Pontryagin dual ofΓ_L);

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THEOREM (J. V. Bellissard, G. D., V. Milani)

TheWannier transformassociated toL is aunitarymap

W :L2(Rd)−→Π_LL2(Q)⊗L2(B).

ThePlancherel’s formulaholds true:

kψk2_L2₍_Rd₎=

Z

B

Nd

∑

j=1

k(W(j)ψ)(·;k)k2_L2₍_V

j)

!

dk.

•Hj⊂L2(B)closed subspace generated by

n

ξa(k) :=e−ik?a : a∈Lj

o

.

•TheprojectionΠ_L

Π_L :L2(Q)⊗L2(B)−→

Nd

M

j=1

L2(Vj)⊗Hj

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Outline

1 Introduction

Bloch’s legacy

3 Wannier transform

Periodic case

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•Quasicrystal-Hamiltonianfor the D-setL:

Hqc:=−∆ +

Nd

∑

j=1 a

∑

∈_Lj

v(j)(· −a)

!

=−∆ +V_L

selfadjoint onH2(Rd).

•Under the assumptionsupp(v(j))⊂Vj:

W : (V_Lψ)−→



 

v(1)(W(1)ψ)

.. .

v(Nd)(W(Nd)_ψ₎





, ψ∈ L

2₍

Rd)

i.e.WV_LW−1₌ L

jv(j)

⊗Id_L2₍_B₎(diagonalandk-indep.).

•W∆W−1can be studied by means of the quadratic form

h∇ψ;∇ψi_L2₍_Rd₎=

Z

B

Nd

∑

j=1

h∇s(W(j)ψ)(·;k);∇s(W(j)ψ)(·;k)i_L2₍_V j)

!

| {z }

Q_k0[(Wψ)(·;k)]

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PROBLEM ! Determination of the form domainD(Q_k0)

•EvidentlyD(Q_k0)⊂H1(Q(d))

Hq(Q(d)) =

Nd M

j=1

Hq(Vj), Hq(Q(d−1)) =

Nd−1

M

j=1

Hq(F_j), . . . (Sobolev spaces)

•Thetrace operatoris a linear bounded:

H1(Vj)3 φ7−→τ φ_∂_V_j ∈H

1 2₍_∂V

j). T_j:=V_j is a polytope hence∂Vis aLipschitz boundary.

•W inducesboundary conditionsonH1(Q(d))for anyk∈_B: D(Q_k0) :=nφ∈H1(Q(d)) : φverifies thek−boundary condition.

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The quadratic form Q_k0[φ] =

Nd

∑

j=1

h∇sφj;∇sφjiL2₍_V j)

has domain D(Q_k0) ={φ∈H1(Q(d)) : ðkφ=0}.

Let∆˜k be the selfadjoint operator defined byQ_k0on L2(Q(d)),

then

W ∆W−1= Π_L ∆˜ Π_L, with ∆˜ :=R⊕

B∆˜k dk.

! REMARK !The rôle of the projectionΠ_L isnot innocent.

W HqcW−1= Π_L H˜ Π_L, with H˜:= ˜∆ + LN_j₌₁d v(j)⊗IdL2₍_B₎.

-H˜ has band spectrum (standard facts!) butHqcmay have

Cantor spectrum(eg. in the one-dimensional cases).

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Outline

1 Introduction

Bloch’s legacy

3 Wannier transform

Periodic case

4 Schrödinger operators and boundary conditions Wannier decomposition of the Laplacian

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Boundary operator:

H1(Q(d))3

   φ1 .. . φNd    ðk 7−→   

(ðkφ)1

.. .

(ðkφ)Nd−1



 ∈H

1

2₍Q(d−1)₎

(ðkφ)`:= Nd

∑

j=1 σ

∑

∈Ij,`

[Vj;Fσ `]∼ e

−ik?rσ Vj→F` _φ

jFσ `

!

where Ij,`= (

{0} if N (Vj,F`) =1

{+,−} if N (Vj,F`) =2.

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•N (Vj,F2) =δj,2+δj,4

(ðkφ)2=

∑

j={2,4}

[Vj;F2]∼ e

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•N (Vj,F1) =δj,2+δj,3+2δj,1

(ðkφ)1=

_∑

j={2,3}

[Vj;F1]∼ e−ik?rVj→F1 φjF1 (aperiodic)

+

_∑

σ={+,−}

[V1;F1σ]∼ e −ik?rσ

V1→F1 _φ_j_Fσ

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Proof of the boundary conditions: periodic case

•1 proto-cell ofV.N(V,Fj) =2 for any proto-faceFj.

•Letr_V±_→_F

j be therelative positionsofFj w.r.tV.

∂V=F+₁ ∪ F−₁ ∪ F+₂ ∪ F−₂ ∪ . . . ∪ F+_N ∪ F−_N.

•Ifs∈F_j±thens=r_T±_→_F j+s⊥. ψ_F±

j (s⊥) :=ψ(r

±

V→_Fj+s⊥) s⊥is an “affine coordinate” forF±_j .

•Letψ∈H1(Rd). A simple computation shows

(Wψ)_F−

j (s⊥;k) : = (Wψ)(r −

V→_Fj+s⊥;k)

= (Wψ)(r_V−_→_F

j−r + V→_Fj+r

+

V→_Fj+s⊥;k)

=:eik·(r

−

V→Fj−r

+

V→Fj)₍W_ψ₎

F+

j (s⊥;k) wherer_V−_→_F

j−r +

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•F±_j ,→_Rd−1_{induces a}_{relative orientation}_w.r.t._V_.

Incidence number:

[V;F_j±] =±1.

•k-boundary conditioninduced byW: k ∈_B, j=1,2, . . .

∑

σ∈{+,−}

[V;Fσ j ]e

−ik·rσ

T→Fj ₍W_ψ₎ Fσ

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Conclusions and Remarks

•D-setsL models thelong-range orderof quasicrystals.

•TheLagarias groupΓ_L plays the role ofZg(g>d) in general. The

Brillouin zone_Bis the Pontryagin dual ofΓ_L.

•The classification of theproto-cellsby means of the (1-st deg.) collar leads to theAnderson-Putnam complexesQ. TheWannier transformW identifies vectors inL2(Rd)with a proper subspace ofL2(Q)⊗L2(B).

•The Schrödinger operators are represented viaW as thecompressionof a Bloch-type operators depending onk∈Bby means of the boundary conditions (cohomoogical description).

•Usingn-th deg. collars one obtains a sequence of AP-complexes{Qn}

together with the mapsQ_n+₁→Qn. This gives rise to the notion of an

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Thank you for your attention

&

Joyeux anniversaire

Prof. Bellissard

“Tout l’art réside dans le fait de devenir adulte sans devenir vieux”.

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Classification of the collared vertices

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Classification of the collared edges

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Classification of the collared cells

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The Anderson-Putnam Complex

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