Geometric Aspects of Quantum Condensed Matter

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1 Geometric Aspects of Quantum Condensed Matter November 13, 2013 Lecture V y An Introduction to Bloch-Floquet Theory (part. I) Giuseppe De Nittis Department Mathematik (room ) +49 (0) denittis.giuseppe@gmail.com W gdenittis.wordpress.com/courses/

2 1 An Introduction to Bloch-Floquet Theory Recommended Bibliography A Simple Prototypical Example Z d -symmetries Outline

3 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. Academic Press, 1978 [Wi] Wilcox, C. H.: J. Anal. Math. 33, (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 and Applications, vol. 224, , Birkhäuser, 2012 [Mau] Maurin, K.: General Eigenfunction Expansions and Unitary Representations of Topological Groups. PWN, 1968 Bloch-bundle: [DL] De Nittis, G. & Lein, M.: J. Math. Phys. 52, (2011) [Pa] Panati, G.: Ann. Henri Poincaré 8, (2007)

4 1 An Introduction to Bloch-Floquet Theory Recommended Bibliography A Simple Prototypical Example Z d -symmetries Outline

5 Let G = {Id,g 1,...,g l } be a finite Abelian group (FAG) of order l. The fundamental representation theorem for FAG states that ) G Z p1... Z pn ( N j=1p j = l where Z pj = {[0],[1],...,[p j 1]} is the cyclic group of order p j N. Let I j := {0,1,...,p j 1} and define the spectrum Ĝ := I 1... I N. For k := (k 1,...,k N ) Ĝ let g k G defined as g k := ([k 1 ],...,[k N ]). We notice that Ĝ = G = l. Let H be a separable Hilbert space and U : G U (H ) a unitary representation of G. Assume that: (a) U is faithful, i.e. U g = 1 if and only if g = Id; (b) U is algebraically compatible, i.e. the operators {U g g G} are linearly independent over H.

6 The full representation is generated by N unitary generators U 1 := U ([1],[0],...,[0]), U 2 := U ([0],[1],...,[0]),... U N := U ([0],[0],...,[1]). Moreover, we can use the spectrum Ĝ to label the unitary operators Ĝ k = (k 1,k 2,...,k N ) U gk = U k 1 1 Uk Uk N N. The order is not important since the generators U j commute. The condition U p j j = U ([0],...,[pj ],...,[0]) = U ([0],...,[0],...,[0]) = U Id = 1 implies that the eigenvalues of U j have the form e i 2π n p j. 6 Problem (Spectral analysis of the representation U): - Compute eigenvalues and eigenspaces for each generator U j ; - Build the simultaneous diagonalization of the full family {U g g G}; - Let A (G) the commutative C -algebra generated by the family {U g g G}. Compute the Gel fand spectrum of A (G).

7 All the answers are contained in the following formula P t := 1 G χ t (g) U g = 1 ( ) g G l e i 2π t 1 k 1 ( ) p1... e i 2π t N k N pn U k Uk N N k Ĝ }{{} =: χ t (g k ) defined for all t = (t 1,...,t N ) Ĝ. The map χ t : G S 1 is called character. PROPOSITION (Bloch-projection) For all t Ĝ the operator P t is an orthogonal projection. Moreover: (i) P t P t = δ t,t P t for all t,t Ĝ; (ii) t Ĝ P t = P (0,...,0) = 1; (iii) U g P t = χ t (g) 1 P t for all t Ĝ and g G. The map Ĝ t P t Proj(H ) is called Bloch-projection. This induces an orthogonal decomposition of the Hilbert space H = t Ĝ H t =: t Ĝ Im(P t ) If ψ = t ψ t is the decomposition of ψ H, then U(g)ψ = t χ t (g) 1 ψ t.

8 P t := 1 G χ t (g) U g = 1 ( ) g G l e i 2π t 1 k 1 ( ) p1... e i 2π t N k N pn U k Uk N N k Ĝ }{{} =: χ t (g k ) Proof (sketch of). The relation P t = P t follows observing that the adjoint change k j p j k j, i.e. it amounts only to a relabeling of k in the definition of P t. (i) implies that P t is a projection and it can be proved from the equation ( ) P t P t = 1 G 2 χ t (g)χ t (g 1 ) U g+g = g,g G G χ t (g) P t g G χ t (g) and the cyclotomic identity p j 1( ) e i 2π r kj p j = p j δ r,0, (why is it true?!) k j =0 (ii) can be proved using linearity and t Ĝ χ t (g) = lδ g,id which follows again from the cyclotomic identity. (iii) U j P t = e i 2π t j p j P t for each generator j = 1,...,N.

9 1 An Introduction to Bloch-Floquet Theory Recommended Bibliography A Simple Prototypical Example Z d -symmetries Outline

10 We want to generalize the Bloch decomposition to the case of finitely generated Abelian groups (FGAG). We recall that an Abelian group G if FGAG if there exist finitely many element {g 1,...,g N } G such that every g G can be written in the form g = n 1 g n N g N, n 1,...,n N Z In this case, we say that the set {g 1,...,g N } is a generating set of G. The fundamental theorem of FGAG says that if G is a FGAG then G }{{} Z d Z p1... Z pl }{{} free part torsion part where the d 0 is the dimension of the free part and the numbers p 1,...,p l are powers of (not necessarily distinct) prime numbers which are uniquely determined by G. The group is finite if and only if d = 0. Since we know the Bloch decomposition for the torsion part (i.e. when d = 0) we will assume in the following that G is purely free, namely G Z d, d N.

11 Z d -algebra Let H be a separable Hilbert space and U : Z d U (H ) a unitary representation, namely - U(n) is an unitary operator for each n Z d ; - U(0) = 1 and U(n) U(m) = U(n + m) Moreover, we say that: (a) U is faithful, i.e. U(n) = 1 if and only if n = 0; (b) U is algebraically compatible, i.e. the collection operators {U(n) n Z d } is linearly independent over H. DEFINITION (Z d -algebra) Let U : Z d U (H ) be a unitary representation faithful and algebraically compatible. The commutative C -algebra Z(d) B(H ) generated by the set {U(n) n Z d } is a Z d -algebra. Remark: Let H be a Hilbert space which carries a Z d -algebra Z(d). Topological insulators are (self-adjoint) operators H defined on H which are invariant with respect Z(d). More precisely H is a topological insulator if [H,A] = 0, A Z(d).

12 T Example (Periodic Systems): Let e j := (0,...,1,...,0) with j = 1,...,d be the canonical basis of R d and Z d - Continuous (twisted) translations: On the Hilbert space H = L 2 (R d ) one defines the unitary operators T 1,...,T d (T j ψ)(x) := g j (x) ψ(x e j ) ψ L 2 (R d ) where the twist functions are normalized g j = 1 and Z d -periodic g j ( + n) = g j ( ). The map n T (n) defined by T (n 1,...,n d ) := (T 1 ) n 1... (T d ) n d n = (n 1,...,n d ) Z d defines a Z d -algebra on L 2 (R d ). - Discrete (twisted) translations: On the Hilbert space H = l 2 (Z d ) one defines the unitary operators T 1,...,T d (T j ψ)(n) := c j ψ(n e j ) ψ l 2 (Z d ) where c j S 1 are called twist constants. The map n T (n) defined by T (n 1,...,n d ) := (T 1 ) n 1... (T d ) n d n = (n 1,...,n d ) Z d defines a Z d -algebra on l 2 (Z d ).

13 T Example (Mathieu-like System): Let T := R/(2πR) [ π, +π] be the 1-dimensional torus and consider the Hilbert space H = L 2 (T). A basis is provided by the Furier system ξ j (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 L 2 (T). One verifies that uv = e i2πα vu. If the rational condition α = N/M holds true then [u,v M ] = 0. In this case the map (n 1,n 2 ) T (n 1,n 2 ) := (u) n 1 (v) Mn 2 (n 1,n 2 ) Z 2 defines a Z 2 -algebra on L 2 (T). In this case the dimension of the group Z 2 is bigger than the spatial dimension of the manifold T.