Periodic Approximantsto 1D

Aperiodic Hamiltonians

Jean BELLISSARDGeorgia Institute of Technology, Atlanta

School of Mathematics & School of Physicse-mail: jeanbel@math.gatech.edu

Sponsoring

CRC 701, Bielefeld,Germany

ContributorsG. De Nittis, Department Mathematik, Friedrich-Alexander Universität, Erlangen-Nürnberg, Germany

S. Beckus, Mathematisches Institut, Friedrich-Schiller-Universität Jena, Germany

V. Milani, Dep. of Mathematics, Shahid Beheshti University Tehran, Iran

Main ReferencesJ. E. Anderson, I. Putnam,Topological invariants for substitution tilings and their associated C∗-algebras,Ergodic Theory Dynam. Systems, 18, (1998), 509-537.

F. Gahler, Talk given at Aperiodic Order, Dynamical Systems, Operator Algebra and TopologyVictoria, BC, August 4-8, 2002, unpublished.

J. Bellissard, R. Benedetti, J. M. Gambaudo,Spaces of Tilings, Finite Telescopic Approximations,Comm. Math. Phys., 261, (2006), 1-41.

J. Bellissard, Wannier Transform for Aperiodic Solids, Talks given atEPFL, Lausanne, June 3rd, 2010KIAS, Seoul, Korea September 27, 2010Georgia Tech, March 16th, 2011Cergy-Pontoise September 5-6, 2011U.C. Irvine, May 15-19, 2013WCOAS, UC Davis, October 26, 2013online at http://people.math.gatech.edu/∼jeanbel/talksjbE.html

Content

Warning This talk is reporting on a work in progress.

1. Motivation

2. One Dimensional Models

3. Gap-graphs

4. Spectral Properties

5. Conclusion

I - Motivations

MotivationSpectrum of the Kohmoto

model

(Hψ)(n) =

ψ(n + 1) + ψ(n − 1)+λ χ(0,α](x − nα) ψ(n)

as a function of α.

Method:transfer matrix calculation

Motivation

Solvable 2D-model, reducible to 1D-calculations

Motivation

A sample of the icosahedral quasicrystal AlPdMn

Motivation

E. Belin, Z. Dankhazi, A. Sadoc, Y. Calvayrac, T. Klein, J.-M. Dubois, J. Phys.:Condens. Matter, 4, (1992), 4459

Motivation

S. Roche, D. Mayou, G. Trambly de Laissardière, J. Math. Phys., 38, (1997), 1794-1822

Methodologies

• For one dimensional Schrödinger equation of the form

Hψ(x) = −d2ψ

dx2 + V(x)ψ(x)

a transfer matrix approach has been used for a long time toanalyze the spectral properties (Bogoliubov ‘36).

• A KAM-type perturbation theory has been used successfully(Dinaburg, Sinai ‘76, JB ‘80’s).

Methodologies

• For discrete one-dimensional models of the form

Hψ(n) = tn+1ψ(n + 1) + tnψ(n − 1) + Vnψ(n)

a transfer matrix approach is the most efficient method, both fornumerical calculation and for mathematical approach:

– the KAM-type perturbation theory also applies (JB ‘80’s).– models defined by substitutions using the trace map

(Khomoto et al., Ostlundt et al. ‘83, JB ‘89, JB, Bovier, Ghez, Damanik... in the nineties)

– theory of cocycle (Avila, Jitomirskaya, Damanik, Krikorian, Gorodestsky...).

Methodologies• In higher dimension almost no rigorous results are available

• Exceptions are for models that are Cartesian products of 1Dmodels (Sire ‘89, Damanik, Gorodestky,Solomyak ‘14)

•Numerical calculations performed on quasi-crystals have shownthat

– Finite cluster calculation lead to a large number of spuriousedge states.

– Periodic approximations are much more efficient– Some periodic approximations exhibit defects giving contri-

butions in the energy spectrum.

II - One Dimensional Models

The Fibonacci SequenceThe Fibonacci sequence is an infinite word generated by the substi-tution

σ : a −→ ab , b −→ a

Iterating gives

a︸︷︷︸a0

→ ab︸︷︷︸a1

→ ab|a︸︷︷︸a2=a1a0

→ aba|ab︸︷︷︸a3=a2a1

→ abaab|aba︸ ︷︷ ︸a4=a3a2

→ abaababa|abaab︸ ︷︷ ︸a5=a4a3

It can be represented by a 1D-tiling if

a→ [0, 1] b→ [0, σ] σ =

√5 − 12

∼ .618

The Fibonacci Sequence

The Fibonacci Sequence

- Collared tiles in the Fibonacci tiling -

The Fibonacci Sequence

- Gähler-Anderson-Putnam graph for 1.1-collared tiles -

The Fibonacci Sequence

- Gähler’s collaring of order 2 -

The Fibonacci Sequence

- Gähler-Anderson-Putnam graph for 2.2-collared tiles -

The Thue-Morse Tiling

• Alphabet: A = {a, b}

• Thue-Morse sequences: generated by the substitution a →ab , b→ ba starting from either a · a or b · a

Thue-Morse G1,1

The Rudin-Shapiro Tiling

• Alphabet: A = {a, b, c, d}

• Rudin-Shapiro sequences: generated by the substitution a→ab , b → ac , c → db , d → dc starting from either b · a , c · a orb · d , c · d

Rudin-Shapiro G1,1

The Full Shift on Two Letters

• Alphabet: A = {a, b} all possible word allowed.

G1,2 G2,2

III - GAP-graphsJ. E. Anderson, I. Putnam,

Topological invariants for substitution tilings and their associated C∗-algebras,Ergodic Theory Dynam. Systems, 18, (1998), 509-537.

F. Gahler, Talk given at Aperiodic Order, Dynamical Systems, Operator Algebra and TopologyVictoria, BC, August 4-8, 2002, unpublished.

One-Dimensional FLC Atomic Sets

• Atoms are labelled by their species (color ck) and bytheir position xk with x0 = 0

• The colored proto-tile is the pair ([0, xk+1 − xk], ck)

• Finite Local Complexity: (FLC)the set A of colored proto-tiles is finite,it plays the role of an alphabet.

• The atomic configuration L is represented by a dotted infiniteword

· · · a−3 a−2 a−1 • a0 a1 a2 · · · • = origin

Collared Proto-points and Proto-tiles

• The set of finite sub-words in the atomic configuration L is de-noted by W and called the dictionary of L

• If u ∈W is a finite word, |u| denotes its length.

• Vl,r is the set of (l, r)-collared proto-point, namely, a dotted wordu · v with

uv ∈W |u| = l |v| = r

• El,r is the set of (l, r)-collared proto-tiles, namely, a dotted wordu · a · v with

a ∈ A uav ∈W |u| = l |v| = r

Restriction and Boundary Maps

• If l′ ≥ l and r′ ≥ r then πv(l,r)←(l′,r′) : Vl′,r′ → Vl,r is the natural

restriction map pruning the l′ − l leftmost letter and the r′ − rrightmost letters⇒ compatibility.

• Similarly πe(l,r)←(l′,r′) : El′,r′ → El,r,⇒ compatibility.

• Boundary Maps: if e = u · a · v ∈ El,r then

∂0e = πv(l,r)←(l,r+1)(u · av) ∂1e = πv

(l,r)←(l+1,r)(ua · v)

GAP-graphs

•GAP: stands for Gahler-Anderson-Putnam

•GAP-graph: Gl,r = (Vl,r ,El,r , ∂0, ∂1) is an oriented graph.

• The restriction map π(l,r)←(l′,r′) = (πv(l,r)←(l′,r′), π

e(l,r)←(l′,r′)) is a

graph map (compatible with the boundary maps)

π(l,r)←(l′,r′) : Gl′,r′ → Gl,r

π(l,r)←(l′,r′) ◦ π(l′,r′)←(l”,r”) = π(l,r)←(l”,r”) (compatibility)

(l, r) ≤ (l′, r′) ≤ (l”, r”) (with (l, r) ≤ (l′r′)⇔ l ≤ l′ , r ≤ r′)

GAP-graphs Properties

• Theorem If n = l + r = l′ + r′ then Gl,r and Gl′,r′ are isomorphicgraphs. They all might be denoted by Gn

• Any GAP-graph is connected without dandling vertex

• Loops are Growing: if L is aperiodic the minimum size of a loopin Gn grows to infinity as n→∞

Complexity Function

• The complexity function of L is p = (p(n))n∈N where p(n) is thenumber of words of length n.

• L is Sturmian if p(n) = n + 1

• L is amenable if

limn→∞

p(n + 1)p(n)

= 1

• The configurational entropy of a sequence is defined as

h = lim supn→∞

ln(p(n))n

• Amenable sequence have zero configurational entropy

Branching Points of a GAP-graph

• Branching points play the role of a boundary.

• A vertex v of Gl,r is a forward branching point if there is morethen one edge starting at v. It is a backward branching point ifthere is more then one edge ending at v.

• The number of forward (backward) branching points is boundedby p(n + 1) − p(n)

Branching Points of a GAP-graph

• Any GAP-graph of a Sturmian sequence has at most one forward andone backward branching points.

• L is amenable if and only if the number of branching points in Gnbecomes eventually negligible as n→∞

• If the configurational entropy h is positive the ratio of the number ofbranching points in Gn to the number of vertices is bounded below byeh− 1 in the limit n→∞

Tiling Space

• The tiling space Ξ of L is the set of all tilings having the samedictionary as L.

• For any FLC tiling, the tiling space is completely disconnected.(Kellendonk ’96)

• If Ξ has no periodic point under the translation group it is aCantor set.

Periodic Approximations

• Result 1: The family of simple non self-intersecting loops in one ofthe GAP-graphs leads to periodic approximations without defects inthe infinite period limit.

• Result 2: The family of all simple non self-intersecting loops in allGAP-graphs can be glued to the tiling space Ξ to make up a compactmetric space X.

IV - Spectral Properties

Pattern Equivariance•Hamiltonian considered are one dimension lattice models of the

form

(Hψ)(n) =∑m∈Z

hm(n)ψ(n −m)

• H is pattern-equivariant (Kellendonk) whenever

– Finite Range: there is M > 0 such that the coefficients hm(n)vanish for m > M,

– Local Pattern there is N > 0 such that each coefficient hm(n) isdefined only by the local environment of the site n at distanceN.

Main Result

Theorem

Let H be a pattern equivariant self-adjoint operator defined on a one-dimensional aperiodic FLC lattice.

Then there is a sequence of periodic approximants, the spectrum of whichconverges in the Hausdorff metric as the period goes to infinity.

In addition the spectral measures of the approximants converges weaklyto the spectral measure of the limit

Expectation: The convergence of the spectrum is exponentiallyfast w.r.t. the period.

Convergence Techniques

• Each triple (r, ℘,u) where r ∈ N and ℘ is a simple non self-intersecting loop in the GAP-graph Gr,r and u ∈ ℘ defines aperiodic approximation Hr,℘,u of H.

• Each point ξ ∈ Ξ defines an atomic configurationLξ and thus anHamiltonian Hξ defined like H on L.

• If (r, ℘,u) converges in X to a point ξ ∈ Ξ, then the family Hr,℘,uconverges to Hξ, in the sense of continuous family of operators.

Convergence Techniques

• Corollary The spectrum edges and the gap edges of the field(Hn,℘

)n,℘

converges to the spectrum edges and the corresponding gapedges of Hξ as n→∞.

• Proposition The spectral measures of the field(Hn,℘

)n,℘

convergesweakly to the corresponding spectral measures of Hξ as n→∞.

Lipshitz Constant

• Let (T, d) be a complete metric space. A function f : T → C iscalled Lipshitz continuous on T if there is a constant K > 0 suchthat

| f (s) − f (t)| ≤ K d(s, t) , s, t ∈ T

• If f : T→ C is Lipshitz continuous it Lipshitz constant is definedby

‖ f ‖Lip = sups,t

| f (s) − f (t)|d(s, t)

Gap Edges Continuity

•Definition Let (T, d) be a complete metric space. A family (At)t∈Tof self-adjoint operators on a Hilbert spaceH is called Lipshitz con-tinuous if the maps t ∈ T 7→ ‖A2

t + aAt + b‖ are uniformly Lipshitzfor a, b in a compact subset of R.

• Theorem If (At)t∈T is a Lipshitz continuous family of self-adjointoperators on the Hilbert spaceH , such that supt ‖At‖ < ∞, then thespectrum edges and the gap edges of the spectrum of At are Lipshitzcontinuous w.r.t. t ∈ T as long as the corresponding gap is open, andHölder continuous of exponent 1/2 otherwise.

Noncommutative Geometry Tools

• C∗-algebras of operator with unit are used as the noncommutativeanalog of compact space.

• Spectral triples (Connes, ‘94) are used as the noncommutative analogof compact metric space.

• This may allow to prove that the family Hr,℘,u together with itslimit points Hξ is Lipshitz continuous.

Conclusion

Interpretation

•Noncommutative Geometry versus Analysis: The previousformalism puts together both the knowledge about the tilingspace developed during the last fifteen years and the C∗-alge-braic approach proposed since the early 80’s to treat the elec-tronic properties of aperiodic solids.

• Finite Volume Approximation: the Anderson-Putnam com-plex, presented here in the version proposed by Franz Gähler,provides a way to express the finite volume approximationwithout creating spurious boundary states.

Defects

•Defects and Branching Points: The main new feature is theappearance of defects expressed combinatorially in terms ofthe branching points.

• Branching: Since branching comes from an ambiguity in grow-ing clusters, it is likely that such defects be systematic in anymaterial which can be described through an FLC tiling.

• Amenability: If the tiling is not amenable, the accumulationof defects makes the present approach inefficient. The use oftechniques developed for disordered systems might be moreappropriate.

Prospect• Continuous case: This formalism can be extended to the case

of the continuous Schrödinger equation with similar conse-quences.

•Higher Dimension: It also extends to higher dimensional col-ored tilings. However, the geometry is much more demanding.

• A Conjecture: The most expected result is the following con-jecturein dimension d ≥ 3 in the perturbative regime, namely if the potentialpart is small compared to the kinetic part, the Schrödinger operatorfor an electron in the field of an FLC configuration of atoms shouldhave a purely absolutely continuous simple spectrum

• Level Repulsion: It is expected also that this a.c. spectrumcorresponds to a Wigner-Dyson statistics of level repulsion.