1mm geometric aspects of quantum condensed matter · 2013-12-03 · geometric aspects of quantum...

Geometric Aspects ofQuantum Condensed Matter

November 27 - December 3, 2013

——————————————————————————–Lecture VI & VII

y An Introduction to Bloch-Floquet Theory (part. II)

——————————————————————————–

Giuseppe De Nittis

Department Mathematik (room 02.317)

� +49 (0)9131 85 67071@ [email protected] gdenittis.wordpress.com/courses/

Outline

1 An Introduction to Bloch-Floquet Theory IIRecommended BibliographyDirect Integral of Hilbert SpacesThe Complete Spectral Theorem (von Neumann)The Nuclear Spectral Theorem (Maurin)The Wandering Property

“Classics” for the Bloch-Floquet theory:

[Ku] Kuchment, P. A.: Floquet Theory for Partial Differential Equations.Birkhäuser, 1993

[RS4] Reed, M. & Simon, B.: Methods of Mathematical Physics IV. AcademicPress, 1978

[Wi] Wilcox, C. H.: J. Anal. Math. 33, 146-167 (1978)

Algebraic approach to Bloch-Floquet theory:

[Dix] Dixmier, J.: von Neumann Algebras. North-Holland, 1981

[DP] De Nittis, G. & Panati, G.: Operator Theory: Advances andApplications, vol. 224, 67-105, Birkhäuser, 2012

[Mau] Maurin, K.: General Eigenfunction Expansions and UnitaryRepresentations of Topological Groups. PWN, 1968

Bloch-bundle:

[DL] De Nittis, G. & Lein, M.: J. Math. Phys. 52, 112103 (2011)

[Pa] Panati, G.: Ann. Henri Poincaré 8, 995-1011 (2007)

Outline


The direct integral of Hilbert spaces is a generalization of the concept ofdirect sum. This concept was introduced in 1949 by J. von Neumann.

Let {X ,B} be a standard Borel space and µ a (regular) Borel measureon X (in many examples X is compact and Polish).

For every x ∈ X let Hx be a (separable) Hilbert space with scalarproduct 〈 · ; · 〉x .

The set F := ∏x∈X Hx (Cartesian product) is called a field of Hilbertspaces over X .

A vector field ϕ is an element of F, namely a map X 3 x 7→ ϕ(x) ∈Hx .

A fundamental family of measurable vector fields is a countable subset{ξj}j∈N ⊂ F such that:

(i) the functions X 3 x 7→ 〈ξi (x);ξj (x)〉x ∈ C are measurable forall i , j ∈N;

(ii) for each x ∈ X the set {ξj (x)}j∈N spans the space Hx .By the Gram-Schmidt orthonormalization one can always build afundamental family of orthonormal measurable fields.

The field F has a measurable structure if it has a fundamental family of(orthonormal) measurable vector fields.

A vector field ϕ ∈ F is called measurable if all the functionsX 3 x 7→ 〈ξj ;ϕ(x)〉x ∈C are measurable for all j ∈N.

Two vector fields ϕ1,ϕ2 ∈ F are µ-equivalent if and only if ϕ1−ϕ2 = 0µ-almost everywhere on X .

Definition (Direct integral)Let F be a field of Hilbert spaces over X with a measurable structure given bythe family {ξj}j∈N. The direct integral H of the Hilbert spaces Hx is the set ofthe equivalence classes, with respect to µ, of measurable vector fields ϕ ∈ Fsatisfying

‖ϕ‖2H :=∫

Xdµ(x) ‖ϕ(x)‖2Hx

< +∞ .

The direct integral H is a Hilbert space with respect to the scalar product

〈ϕ1;ϕ2〉H :=∫

Xdµ(x) 〈ϕ1(x);ϕ2(x)〉x .

The usual notation is H =∫⊕X dµ(x) Hx .

The structure of H depends on µ but not in a dramatica way. Let µ ′ be apositive measure equivalent to µ. The Radon-Nikodym theoremensures the existence of a positive ρ ∈ L1(X ,µ) with ρ−1 ∈ L1(X ,µ)such that µ ′ = ρ µ. Let H be the direct integral with respect to µ and H′

the direct integral with respect to µ ′. The map Uρ : H→ H′, defined byUρ ϕ = ρ

− 12 ϕ, is a unitary equivalence.

Definition (Decomposable operators)A bounded operator A on H is called decomposable if and only if there is afunction X 3 x 7→ A(x) ∈B(Hx ) with bounded essential norm-supremum (i.e.L∞) such that for all ϕ ∈ H(

Aϕ)(x) = A(x) ϕ(x) (µ-almost everywhere) .

If A is decomposable we write

A =∫ ⊕

Xdµ(x) A(x)

and we call A(x) the fiber of A over x .

[RS4, th. XIII.83] If x 7→ A(x) is a L∞ operator-valued function then thereis a unique associated decomposable operator A and

‖A‖B(H) = supx∈X‖A(x)‖B(Hx )

We denote with L∞(H)⊂B(H) the set of the decomposable operator onthe direct integral H. This is an algebra. Moreover L∞(X ,µ) identifies ina natural way with a sub-algbera of the bounded operator as follows: iff ∈ L∞(X ,µ), then the decomposable operator Mf is given by

Mf (x) := f (x) 1x , ∀ x ∈ X .

We refer to L∞(X ,µ) as the algebra of diagonal operators.

Let X be a compact (resp. locally compact) space. Then, in the sameway as above, C(X ) (resp C0(X )) define a sub-algebra of B(H) calledthe algebra of continuously diagonal operators.

[RS4, th. XIII.84] Assume that µ is σ -finite. Then an operator A ∈B(H)is decomposable if and only if [A;Mf ] = 0 for all f ∈ L∞(X ,µ).

Let x 7→ A(x) be a map from X to the self-adjoint unbounded operatorswith domain Dx ⊂Hx . If x 7→ (A(x)± i1x )−1 is a decomposableoperator over H we can define an unbounded operator by(

Aϕ)(x) = A(x) ϕ(x) (µ-almost everywhere) .

with domain DA := {ϕ ∈ H | ϕ(x) ∈ Dx , ‖Aϕ‖H < +∞}.

Theorem ([RS4, th. XIII.85])Let x 7→ A(x) be a measurable family on H =

∫⊕X dµ(x) Hx . Assume that

A(x) is self-adjoint (bounded or unbounded) on Hx for all x ∈ X. Then:

(i) The operator A :=∫⊕X dµ(x) A(x) is self-adjoint on H. Moreover, a

self-adjoint (unbounded) operator A on H decomposes if and only if(A± i1)−1 is a bounded decomposable operator.

(ii) For any founded Borel function g :R→C

g(A) =∫ ⊕

Xdµ(x) g

(A(x)

).

(iii) λ ∈ σ(A) if and only if

µ

({x ∈ X | σ(A(x)) ∩ (λ − ε,λ + ε) 6= /0

})> 0 .

(iv) λ is an eigenvalue of A if and only if

µ

({x ∈ X | λ ist an eigenvalue of A(x)

})> 0 .

(v) If A(x) has pure absolutely continuous spectrum for all x ∈ X, then sodoes A. (This condition is sufficient but not at all necessary!!)

Outline


C∗-algebras: basic definitions

A C∗-algebra A is a complex algebra with an involution (∗-algebra) anda norm (Banach-algebra) such that ‖aa∗‖= ‖a‖2 (C∗-idendity) for alla ∈A . It is unital if 1 ∈A .

C(X ) (Gel’fand isomorphism)

[commutative case]

A

'-

[non-commutative case]

subset of B(H )

'-

(GNS-construction)

In the commutative case the compact Hausdorff space X is called(Gel’fand) spectrum of the algebra.

X :={

x : A → C | continuous ∗−homomorphism}.

The Gel’fand isomorphism A 3 A 7→ fA ∈ C(X ) is defined by theevaluation procedure:

fA(x) := x(A) .

One can use continuous functions on f ∈ C(X ) to “label” elements of acommutative C∗-algebra Af ∈A .

Let A be a unital commutative C∗-algebra of baunded operators on theseparable Hilbert space H . Let X be the Gel’fand spectrum. For allpairs ϕ,ψ ∈H the mapping

C(X ) 3 f 7−→ 〈ϕ ; Af ψ〉H ∈ C

is a continuous linear functional on C(X ). The Riesz- Markov Theoremimplies the existence of a unique regular (complex) Borel measure µϕ,ψ

with finite total variation, such that

〈ϕ ; Af ψ〉H =∫

Xf (x) dµϕ,ψ (x) ∀ f ∈ C(X ) .

µϕ,ψ is called spectral measure.

For every non-void open set O ⊂ X there exists a ϕ ∈H such thatµϕ,ϕ (O) > 0 (check!!) A positive measure µ on X is called basic for A iffor every Y ⊂ X , µ(Y ) = 0 if and only if µϕ,ϕ (Y ) = 0 for every ϕ ∈H .

[Dix, Prop. 4, (Part I, Chap. 7)] Any commutative unital C∗-algebra ofoperators on a separable Hilbert space has a basic measure carried onits Gel’fand spectrum.

Some properties of basic measures [Dix, (Part I, Chap. 7)]:

(i) Let µ be a basic measure on X . Then every other basicmeasure is equivalent (has the same null sets) to µ;

(ii) For all ϕ,ψ ∈H the spectral measure µϕ,ψ is absolutelycontinuous with respect to µ, and there exists a uniquefunction hϕ,ψ ∈ L1(X ) (the Radon- Nikodym derivative) suchthat µϕ,ψ = hϕ,ψ µ;

(iii) The union of the supports of the spectral measures µϕ,ϕ isdense in X , then the support of a basic measure µ is thewhole X .

von Neumann’s complete spectral theorem

Theorem ([Dix, Theo. 1, (Part II, Chap. 6)])Let H be a separable Hilbert space, A a commutative and unital C∗-algebraof operators on H with Gel’fand spectrum X and basic measure µ. Thenthere exist

a) a direct integral H :=∫⊕X dµ(x) Hx with Hx 6= {0} for all x ∈ X,

b) a unitary map F : H → H, called A -Fourier transform,

such that

(i)f ∈ C(X )

[Gel’fand iso.] [canonical iden.]

A 3 AfF . . .F−1

-�

Mf ∈ C(X )

-

(ii) If the operator H commute with A then

F H F−1 =∫ ⊕

Xdµ(x) H(x) .

Some comments about the theorem

It is interesting to see how the fiber Hilbert spaces H x of the directintegral are constructed. For ϕ,ψ ∈H let µϕ,ψ = hϕ,ψ µ be the relationbetween the spectral measure µϕ,ψ and the basic measure µ. Forµ-almost every x ∈ X the value of all the Radon-Nikodym derivativeshϕ,ψ in x defines a semi-definite sesquilinear form on H , i.e.

〈ϕ ; ψ〉x := hϕ,ψ (x) .

Let I x :={

ϕ ∈H | hϕ,ϕ (x) = 0}

. The quotient space H /I x is apre-Hilbert space and H x is defined to be its completion. Byconstruction H x 6= {0} for µ-almost every x ∈ X . Let N ⊂ X be theµ-negligible set on which H x is trivial or not well defined. Then onecan realize H gluing H x for all x ∈ X \N and H x = H ′ for all x ∈ Nwhere H ′ is an arbitrary non-trivial Hilbert space.

The von Neumann’s complete spectral theorem is too general !!

- the construction of the fiber spaces H x is too abstract andnot at all algorithmic !!

- the identification of the measurable structure (X ,µ) isgenerally out of reach !!

Outline


Nuclear Gel’fand triple

A nuclear space is a topological vector space with “many of the goodproperties” of finite-dimensional vector spaces. The nuclear topologycan be defined by a family of seminorms (whose unit balls decreaserapidly in size). Vector spaces whose elements are “smooth” in somesense tend to be nuclear spaces (a typical example is the Schwartzspace)

- Finite-dimensional vector spaces are nuclear.- There are no infinite-dimensional Banach spaces that are

nuclear.But, if a “naturally occurring” topological vector space is not a Banachspace, then there is a “good chance” that it is nuclear !!

Let H be a separable Hilbert space, Φ⊂H a norm-dense subspacesuch that Φ can be endowed with a nuclear topology such that theinclusion map ı : Φ ↪→H is continuous. Let Φ∗ be the topological dualof Φ. The identification H 'H ∗ (self-duality) induces an antilinearinjection ı∗ : H ↪→Φ∗. The sequence

Φı↪→ H

ı∗↪→ Φ∗

is called (nuclear) Gel’fand triple.

The duality pairing between Φ∗ and Φ is compatible with the scalarproduct on H in the sense that if ψ ∈H , ϕ ∈Φ and ı∗(ψ) ∈Φ∗ then

ı∗(ψ)(ϕ)

= 〈ψ;ϕ〉H .

For this reason we will use 〈·; ·〉 : Φ∗×Φ→C for the duality pairing.

Let A be a bounded operator on H such that the adjoint A∗ leavesinvariant the subset Φ and A∗ : Φ→Φ is continuous in the nucleartopology. Then one define a linear operator A : Φ∗→Φ∗ by posing

〈Aη;ϕ〉 := 〈η;A∗ϕ〉 ∀ η ∈Φ∗ ϕ ∈Φ.

The operator A is continuous (in the topology of Φ∗) and is an extensionof A (when restricted to H ).

Relation with the direct integral

Assume that we are in the situation of the von Neumann’s theorem, i.e.:- H an Hilbert space with A a commutative C∗-algebra of operators onH with spectrum X and basic measure µ;- The A -Fourier transform F : H → H =

∫⊕X dµ(x) Hx .

For any ϕ ∈H let ϕ( · ) := Fϕ ∈ H be the related square integrablevector field. Let {ξj}j∈N be a fundamental family of orthonormal vectorfields which define the structure of the direct integral H. Then

ϕ(x) = ∑j∈N

ϕj (x) ξj (x) , x ∈ X

where ϕj ∈ L2(X ,µ). The unitarity of F implies

〈ϕ;ψ〉H = 〈ϕ( · );ψ( · )〉H =∫

Xdµ(x) ∑

j∈Nϕj (x) ψj (x) .

Let Af ∈A be the operator associated with f ∈ C(X ) through theGel’fand isomorphism. Then

(FAf ϕ)j (x) := 〈ξj (x); f (x)ϕ(x)〉Hx =: f (x) ϕj (x) .

� Remark: It looks like an eigenvalue equation !!

Suppose that Φı↪→ H

ı∗↪→ Φ∗ is a Gel’fand triple for H . If ϕ ∈Φ then

the mapΦ 3 ϕ −→ ϕj (x) := 〈ξj (x);ϕ(x)〉Hx ∈ C

is linear and continuous with respect to the nuclear topology of Φ. Then,there exists a ηj (x) ∈Φ∗ such that

〈ηj (x);ϕ〉 := ϕj (x) .

Suppose that Af : Φ→Φ is continuous with respect to the nucleartopology for every f ∈ C(X ). Then we can define the extended operatorAf : Φ∗→Φ∗ by

〈Af η;ϕ〉 = 〈η;Af ϕ〉 ∀ η ∈Φ∗, ϕ ∈Φ .

Then, for all ϕ ∈Φ one has

〈Af ηj (x);ϕ〉 = 〈ηj (x);Af ϕ〉 = f (x) ϕj (x) = 〈f (x) ηj (x);ϕ〉 ,

namely

Af ηj (x) = f (x) ηj (x) ∀ j ∈N .

� Remark: The ηj (x) are a basis of generalized eigenvectors for theall the extended operators Af . We proved a generalized simultaneousdiagonalization for the C∗-algebra A .

Maurin’s nuclear spectral theorem

Theorem ([Mau, Chap. II])Assume the assumptions of the von Neumann’s theorem and the existence of

a Gel’fand triple Φı↪→ H

ı∗↪→ Φ∗ such that A : Φ→Φ is nuclear continuous

for all A ∈A . Then:

(i) for all x ∈ X the map F |x : Φ→Hx given by ϕ → ϕ(x) is continuouswith respect to the nuclear topology .

(ii) there is a family of functionals ηj (x) ∈Φ∗, j ∈N such that equations

〈ηj (x);ϕ〉 = 〈ξj (x);ϕ(x)〉Hx Af ηj (x) = f (x) ηj (x)

hold true for all ϕ ∈Φ and f ∈ C(X ) (µ-a. e.). Moreover, with theidentification ηj (x)≡ ξj (x) the fiber Hilbert space H is (isomorphic to)a vector subspace of Φ∗. With this identification F acts on the denseset Φ as

Φ 3 ϕF−→ ∑

j∈N〈ηj (x);ϕ〉 ηj (x) ∈ Φ∗ .

(iii) Under this identification Hx become the common generalizedeigenspaces of all the operators in A since Af ηj (x) = f (x) ηj (x).

Some comments about the theorem

Up to a canonical identification of isomorphic Gel’fand triples therealization of the fiber spaces Hx as common generalized eigenspacesis canonical in the sense that it does not depend on the choice of afundamental family of orthonormal measurable fields.

The Existence of a good nuclear space Φ⊂H is guaranteed under thefollowing assumptions:

Theorem ([Mau, Chap. II])

Let {A1,A2, . . .} be a countable family of commuting bounded normaloperators on the separable Hilbert space H , generating the commutativeC∗-algebra A . Then there exists a countable A -cyclic system {ψ1,ψ2, . . .}which generates a nuclear space Φ⊂H such that: (a) Φ is dense in H ; (b)the embedding ı : Φ ↪→H is continuous; (c) the maps Aj

m : Φ→Φ arecontinuous for all j ,m ∈N.

The Maurin’s nuclear spectral theorem adds important information.However, there is still a missing point !!

- There is an algorithmic way to construct the nuclear spaceΦ and the generalized eigenvectors ηj (x) ?

Outline


DEFINITION (Wandering property)Let A := C∗(A1,A2, . . .) be a commutative unital C∗-algebra generated by acountable family of commuting bounded normal operators in a separableHilbert space H . A has the wandering property if there exists a (at most)countable family {ψ1,ψ2, . . .} ⊂H of orthonormal vectors such that:

(a) it is A -cyclic in the sense that

Span{

Aψj |A ∈A , j = 1,2, . . .}

is dense in H ;

(b) it generates an orthogonal family, namely

〈ψi ; (A∗)bAaψj 〉H = ‖Aa

ψi‖2H δij δab

for all i , j = 1,2, . . . and a,b ∈N∞fin where Aa = Aa1

1 Aa22 . . .

Let Hj := A [ψj ] be the subspace generated by the action of A on the vectorψj . If A has the wandering property then H decomposes as

H =⊕j∈N

Hj

and each Hj is an A -invariant subspace called a wandering subspace. Thefamily {ψ1,ψ2, . . .} is called a wandering system.

PROPOSITION

Let A be a commutative unital C∗-algebra generated by the (at most)countable family {A1,A2, . . .} of commuting bounded normal operators on aseparable Hilbert space H . Suppose that A has a wandering system ofvectors {ψ1,ψ2, . . .}, then:

(i) The generators are not selfadjoint, and Anj 6= 1 for all n 6= 0;

(ii) Every generator which is unitary has no eigenvectors;

(iii) If A is generated by d unitary operators then it is a Zd -algebra.

Proof (sketch of).(i) follows observing that Aj = Aj

∗ implies Aj ψi = 0 for all the elements of thewandering system. By A -cyclicity and commutativity this means Aj = 0. Byposing b = 0 and i = j in (b) one has that Aa = 1 iff a = 0.(ii) Let A1 = U be unitary. Each ϕ ∈H can be written as

ϕ = ∑n∈Z

Unχn = ∑

n∈ZUn

(∑

j ,a2,a3,...

αj ,,a2,a3,...Aa22 Aa3

3 . . .ψj

)

then Uϕ = ∑n∈ZUnχn−1. If Uϕ = λϕ then χn = λ−nχ0 but this implies‖ϕ‖2H = ∑n∈Z ‖χn‖2H = ∑n∈Z ‖χ0‖2H = +∞ which is a contradiction. �

PROPOSITION

Let A be a commutative unital C∗-algebra generated by the (at most)countable family {A1,A2, . . .} of commuting bounded normal operators on aseparable Hilbert space H . Suppose that A has a wandering system ofvectors {ψ1,ψ2, . . .}, then:

(i) The generators are not selfadjoint, and Anj 6= 1 for all n 6= 0;

(ii) Every generator which is unitary has no eigenvectors;

(iii) If A is generated by d unitary operators then it is a Zd -algebra.

Proof (sketch of).(iii) if A is generated by {U1, . . . ,Ud} then the map

Zd 3 (a1, . . . ,ad ) 7−→ Ua11 . . .Uad

d ∈ U (H )

is a unitary representation of Zd . To show that the representation isalgebraically compatible, suppose that ∑a∈Zd αaUa = 0, then from (b)

0 = 〈Ubψj ; ∑

a∈Zd

αaUaψj 〉H = αb

for all αb ∈Zd . �

T Example (Periodic (continuous) Systems):Let ej := (0, . . . ,1, . . . ,0) with j = 1, . . . ,d be the canonical basis of Rd . On theHilbert space H = L2(Rd ) one defines the unitary operators T1, . . . ,Td

(Tj ψ)(x) := gj (x) ψ(x − ej ) ψ ∈ L2(Rd )

where the twist functions are normalized |gj |= 1 and Zd -periodicgj (·+ n) = gj (·). The map n 7→ T (n) defined by

T (n1, . . . ,nd ) := (T1)n1 . . . (Td )nd n = (n1, . . . ,nd ) ∈Zd

is a unitary representation of Zd on L2(Rd ).

Let Z(d) := C∗(T1, . . . ,Td ). This C∗-algebra has the wandering property. LetW0 := [0,1]d be the unit cell in Rd and Wm = W0 + m the translation of W0 bym ∈Zd . Consider the identifications L2(Wm) ↪→ L2(Rd ) and the orthogonaldecomposition

L2(Rd ) =⊕

m∈Zd

L2(Wm) .

Then each orthonormal system {ψ1,ψ2, . . .} ⊂W0 defines a wanderingsystem for Z(d) of cardinality ℵ.

This proves that Z(d) is a Zd -algebra.

T Example (Mathieu-like System): Let T :=R/(2πR)' [−π,+π]be the 1-dimensional torus and consider the Hilbert space H = L2(T). Abasis is provided by the Furier system

ξj (k) :=1√2π

e(ik)j , j ∈Z.

Let us define two unitary operators u and v by

(uφ)(k) := e(ik) φ(k) , (vφ)(k) := φ(k −2πα) φ ∈ L2(T).

One verifies that uv = ei2πα vu. If the rational condition α = N/M holds truethen [u,vM ] = 0. In this case the map

(n1,n2) 7−→ T (n1,n2) := (u)n1 (v)Mn2 (n1,n2) ∈Z2

defines a unitary representation of Z2 on L2(T).

Let us consider the commutant Z(1) of C∗(u,vM ). Let a ∈B(L2(T)). One canprove that

a commutes whit C∗(u,vM ) ⇔ a = ∑n∈Z

αnwn (check!!)

where wξj := ξj+M . Then Z(1) := C∗(w) is generated by a unitaryrepresentation of Z.

The commutant Z(1) has the wandering property with respect to the system{ξ0, . . . ,ξM−1} which has cardinality M.

Wandering Properties and Spectral Analysis

Commutative C∗-algebras generated by a finite family of unitary operatorsare automatically Zd -algebras if there exists a wandering system. However,the wandering properties implies more!!. It gives a precise characterizationof the spectral properties of the algebra.

THEOREM ([DP, Proposition 5.5])Let Z(d) be a commutative unital C∗-algebra generated by the finite family{U1, . . . ,Ud} of commuting unitary operators on a separable Hilbert space H .Suppose that Z(d) has a wandering system of vectors {ψ1,ψ2, . . .}, then:

(i) The Gel’fand spectrum of Z(d) is homeomorphic to the d-dimensionaltorus Td ;

(ii) The basic measure of Z(d) is the normalized Haar measure dz on Td .

� Remark: The proof of this theorem requires the notions of: (reduced)group algebra C∗-algebra associated with a group G. If G is discrete andabelian one has

C∗r (G) ' C(G) .

1mm geometric aspects of quantum condensed matter · 2013-12-03 · geometric aspects of quantum...

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