Between order and disorder: Hamiltonians for Quasicrystals · Spectral properties of H(!) can be...

QuasicrystalHamiltonians

Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Between order and disorder: Hamiltoniansfor Quasicrystals

Peter Stollmann

Chemnitz University of Technology

Kolloquium Regensburg, 27.10.2011


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Outline

I Quasicrystals?I Mathematical models of aperiodic orderI Hamiltonians

I DynamicsI Quantum transport.I Delone dynamical systems and Hamiltonians.

I Spectral properties:I The Wonderland theorem: generic results.I 1-d models with purely singular continuous spectrum.

Based on collaboration with S. Klassert and D. Lenz.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Quasicrystals?

What these creatures really are is not yet negotiated.However there’s enough evidence to speculate about thequestion ...

... as was done in the September 2005 issue of the Notices.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Quasicrystals?

This years Nobel prize in Chemistrygoes to Dany Schechtmanfor his discovery that was published as:


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Quasicrystals?

As a rule of thumb quasicrystals exhibit:

I Sharp diffraction peaks - usually coming with longrange order.

I Forbidden symmetries - excluding translationinvariance.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Mathematical models for aperiodic order

Aperiodic order can mathematically be described by tilings:

Figure: Original und Falschung: A real quasicrystal and thePenrose tiling


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Delone (Delaunay) sets

An alternative to tilings are Delone sets.

ω ⊂ Rd is called a Delone set, if there exist r ,R ∈ R suchthat

I ∀x , y ∈ ω, x 6= y : Ur (x) ∩ Ur (y) = ∅,I

⋃x∈ω BR(x) = Rd .


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Delone sets

By Dr ,R(Rd) = Dr ,R we denote the set of all (r ,R)-sets; itis a compact metric space in the natural topology.D(Rd) =

⋃0<r≤R Dr ,R(Rd) is the set of all Delone sets.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Hamiltonians: continuum models

The basic idea is very simple: at each point of a Delone setω an ion is sitting, whose potential is given by v . This leadsto the Hamiltonian

H(ω) := −∆ +∑x∈ω

v(· − x)

The potential

Vω =∑x∈ω

v(· − x)

is depicted below


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Hamiltonians: continuum models

If the Delone set ω is periodic, then H(ω) describes acrystal. If we choose the point set ω as the points of aPoisson process (typically no Delone set) then H(ω)describes a disordered solid. If ω is aperiodically ordered,then H(ω) can be used to describe electronic properties of aquasicrystal.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Hamiltonians

We are interested in the Schrodinger equation

ψ′(t) = −iH(ω)ψ(t) (SE )

it describes the time evolution of a wave function ψ(t).Spectral properties of H(ω) can be translated intoqualitative properties of solutions of (SE).The specific form of (dis-)order is encoded in H(ω).It will be very useful to consider a whole collection(H(ω), ω ∈ Ω) at the same time, for physical reasons andfor analytical reasons.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

The Schrodinger equation.

ψ(t) = −i(−∆ + V )ψ(t), ψ(0) = ψ0,


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Quantum dynamics.

Easy: the solution of the Schrodinger equation is

ψ(t) = e−iHtψ0.

But: what does it look like?To this end one studies the spectral resolution of H.

I If ψ0 is an eigenvector of H with eigenvalue E0 ⇒

ψ(t) = e−iE0tψ0.

One speaks of a bound state.

I If the spectral measure ρHψ0of ψ0 w.r.t H is continuous,

⇒ limT→∞

1

T

∫ T

−T‖χBψ(t)‖2dt = 0 for compact B.

One speaks of a scattering state.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Between order and disorder.

I Atomic models are described by a negative potential Vthat decays rapidely enough near infinity:one getsbound states for negative energies and scattering statesfor positive energies.

I Solid states with perfect order are described by aperiodic V ⇒ H = −∆ + V has only scattering states.

I P.W. Anderson proposed a new paradigm for disorderedsolids in dimension ≥ 3: dense pure point spectrum.

I The (dis-) order of quasicrystalline models suggests anintermediate spectral behaviour.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Delone dynamical systems

... simply consist of a translation invariant, compact setΩ ⊂ D(Rd), on which the group Tt : Rd → Rd(t ∈ Rd) oftranslations acts; we denote such a system by (Ω,T ).We interpret such a DDS (Ω,T ) as a model for a certaintype of (dis-)order. Ergodic properties of (Ω,T ) reflectcombinatorial properties of the elements ω ∈ Ω and viceversa. Moreover, spectral properties of the H(ω) aresometimes related to ergodic properties of the DDS. E.g.

(Ω,T ) minimal⇓

σ(H(ω)) = σ(H(ω′)) for all ω, ω′ ∈ Ω.

Minimality and unique ergodicity are equivalent to certaincombinatorial properties of the ω’s.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Spectral properties

For a DDS (Ω,T ) that describes aperiodic order one istempted to expect purely singular continuous spectrum andthis has been verified in some classes of discrete examples inone dimension (quasiperiodic Hamiltonians, substitutionpotentials) as well as continuum models in one dimension.However in higher dimensions there are only very fewrigorous results :-( .


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Spectral properties: generic results

A very modest step has been taken in showing thatgenerically in the topological sense singular continuousspectrum occurs.

Theorem (D. Lenz and P. S., Duke Math. J., 2006)

Let r ,R > 0 with 2r < R and v ≥ 0, v 6= 0. Then thereexists an open ∅ 6= U ⊂ R and a dense Gδ-set Ωsc ⊂ Dr ,R

such that for every ω ∈ Ωsc the spectrum of H(ω) containsU and is purely singular continuous in U.

This follows from a variant of Barry Simon’s WonderlandTheorem and uses heavily the spectral properties of periodicoperators.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Back to Wonderland

We use a variant of a result from B. Simon, Operatorswith singular continuous spectrum: I. General operators.Ann. Math. 141, 131 – 145 (1995)Fix S , a locally compact, σ-compact, separable metric space.Consider M+(S), the set of positive, regular Borel measures.µ ∈M+(S) diffusive or continuous:⇐⇒ µ(x) = 0 forevery x ∈ S .µ ⊥ ν, mutually singular:⇐⇒ ∃ C ⊂ S such thatµ(C ) = 0 = ν(S \ C ).

Theorem (D. Lenz and P. S., Duke Math. J., 2006)Let S be as above. Then(1) The set Mc(S) := µ ∈M+(S)| µ is diffusive is a Gδ-set inM+(S).(2) For any λ ∈M+(S), the set µ ∈M+(S)| µ ⊥ λ is a Gδ-setin M+(S).

The silent majority consists of strange creatures :-)


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

Conclusion of the Wonderland intrigue

I Mc(S) and Ms(S) are Gδ-sets, for polish S .

I This implies the Wonderland theorem and the fact thatgeneric measures are singular continuous in “nicespaces”.

I A particular example is given by “geometric disorder”(= Delone Hamiltonians) for which we can prove thatpurely singular continuous components turn up,generically.


Peter Stollmann

Quasicrystals?

Aperiodic order

Hamiltonians

Dynamical

Spectral

1-d models with purely singular continuousspectrum

Recall that Vω is given by translating a fixed potential v toall points of a Delone set ω. Now we are in 1-d, i.e. in R.?????? KlassertLS Theorem??????? Important difference to generic result??????? Even hard to find a single example

Between order and disorder: Hamiltonians for Quasicrystals · Spectral properties of H(!) can be...

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