Inequivalent bundle representations for the Noncommutative Torus

Transcription

1 Inequivalent bundle representations for the Noncommutative Torus Chern numbers: from abstract to concrete Giuseppe De Nittis Mathematical Physics Sector of: SISSA International School for Advanced Studies, Trieste Noncommutative Geometry and Quantum Physics Vietri sul Mare, August 31-September 5, 2009 inspired by discussions with: G. Landi (Università di Trieste) & G. Panati (La Sapienza, Roma)

2 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

3 The seminal paper [Thouless, Kohmoto, Nightingale & Niji 82] paved the way for the explanation of the Quantum Hall Effect (QHE) in terms of geometric quantities. The model is a 2-dimensional magnetic-bloch-electron (2DMBE), i.e. an electron in a lattice potential plus a uniform magnetic field. In the limit B 0, assuming a rational flux M/N and via the Kubo formula the quantization of the Hall conductance is related with the (first) Chern numbers of certain vector bundles related to the energy spectrum of the model. A duality between the limit cases B 0 and B is proposed: N C B + M C B 0 = 1 (TKNN-formula).

4 The seminal paper [Thouless, Kohmoto, Nightingale & Niji 82] paved the way for the explanation of the Quantum Hall Effect (QHE) in terms of geometric quantities. The model is a 2-dimensional magnetic-bloch-electron (2DMBE), i.e. an electron in a lattice potential plus a uniform magnetic field. In the limit B 0, assuming a rational flux M/N and via the Kubo formula the quantization of the Hall conductance is related with the (first) Chern numbers of certain vector bundles related to the energy spectrum of the model. A duality between the limit cases B 0 and B is proposed: N C B + M C B 0 = 1 (TKNN-formula). In the papers [Bellissard 87], [Helffer & Sjőstrand 89], [D. & Panati t.b.p.] is rigorously proved that the effective models for the 2DMBE in the limits B 0, are elements of two different representations of the (rational) Noncommutative Torus (NCT).

5 In the papers [Bellissard 87], [Helffer & Sjőstrand 89], [D. & Panati t.b.p.] is rigorously proved that the effective models for the 2DMBE in the limits B 0, are elements of two different representations of the (rational) Noncommutative Torus (NCT). Rigorous proof and generalization of the TKNN-formula. The seminal paper [Thouless, Kohmoto, Nightingale & Niji 82] paved the way for the explanation of the Quantum Hall Effect (QHE) in terms of geometric quantities. The model is a 2-dimensional magnetic-bloch-electron (2DMBE), i.e. an electron in a lattice potential plus a uniform magnetic field. In the limit B 0, assuming a rational flux M/N and via the Kubo formula the quantization of the Hall conductance is related with the (first) Chern numbers of certain vector bundles related to the energy spectrum of the model. A duality between the limit cases B 0 and B is proposed: N C B + M C B 0 = 1 (TKNN-formula).

6 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

7 The NCT with deformation parameter θ R is the abstract C -algebra A θ generated by: u = u 1, v = v 1, uv = e i2πθ vu and closed in the (universal) norm: a := sup{ π(a) H π representation of A θ on H }.

8 The NCT with deformation parameter θ R is the abstract C -algebra A θ generated by: u = u 1, v = v 1, uv = e i2πθ vu and closed in the (universal) norm: a := sup{ π(a) H π representation of A θ on H }. The canonical trace : A θ C (unique if θ / Q) is defined by: (u n v m ) = δ n,0 δ m,0. It is a state (linear, positive, normalized), faithful (a a) = 0 a = 0 with the tracial property (ab) = (ba).

9 The NCT with deformation parameter θ R is the abstract C -algebra A θ generated by: u = u 1, v = v 1, uv = e i2πθ vu and closed in the (universal) norm: a := sup{ π(a) H π representation of A θ on H }. The canonical trace : A θ C (unique if θ / Q) is defined by: (u n v m ) = δ n,0 δ m,0. It is a state (linear, positive, normalized), faithful (a a) = 0 a = 0 with the tracial property (ab) = (ba). The canonical derivations j : A θ A θ, j = 1,2 are defined by 1 (u n v m ) = in(u n v m ), 2 (u n v m ) = im(u n v m ). Symmetric j (a ) = j (a), commuting 1 2 = 2 1 and j = 0.

10 A p A θ is a projection if p = p = p 2. Let Proj(A θ ) the collection of the projections of A θ. If θ = M N with M Z, N N and g.c.d(m,n) = 1, then : Proj(AM/N ) { 0, 1 N 1 N,..., N,1}.

11 A p A θ is a projection if p = p = p 2. Let Proj(A θ ) the collection of the projections of A θ. If θ = M N with M Z, N N and g.c.d(m,n) = 1, then : Proj(AM/N ) { 0, 1 N 1 N,..., N,1}. The abstract (first) Chern number of p Proj(A θ ) is defined by: Ch(p) := i 2π (p[ 1 (p); 2 (p)]).

12 A selfadjoint h A θ has a band spectrum if it is a locally finite union of closed intervals in R, i.e. σ(h) = j Z I j. The open interval which separates two adjacent bands is called gap. Let χ Ij the characteristic functions for the spectral band I j, then χ Ij C(σ(h)) C (h) A θ. One to one correspondence between I j and band projection p j Proj(A θ ). Gap projection P j := jk=1 p j. If θ = M/N then 1 j N.

13 An important example of selfadjoint element in Aθ is the Harper Hamiltonian hhar := u + u 1 + v + v 1. Spectrum of hhar for θ Q [Hofstadter 76]:

14 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

15 { Π0,1 (u) =: U 0,1 : ψ n,m e iπθn ψ n,m+1 Π 0,1 : A θ B(H 0,1 ) with H 0,1 := L 2 (T 2 ) defined by: Π 0,1 (v) =: V 0,1 : ψ n,m e iπθm ψ n+1,m ψ n,m (k 1,k 2 ) := (2π) 1 e i(nk 1+mk 2 ) Fourier basis of H 0,1.

16 Π 0,1 : A θ B(H 0,1 ) with H 0,1 := L 2 (T 2 ) defined by: { Π0,1 (u) =: U 0,1 : ψ n,m e iπθn ψ n,m+1 Π 0,1 (v) =: V 0,1 : ψ n,m e iπθm ψ n+1,m ψ n,m (k 1,k 2 ) := (2π) 1 e i(nk 1+mk 2 ) Fourier basis of H 0,1. It is the GNS representation related to, indeed ψ n,m e iπθnm v n u m, cyclic vector ψ 0,0 = (2π) 1, (u a v b ) = δ a,0 δ b,0 = (ψ 0,0 ;Π 0,1 (u a v b )ψ 0,0 ) = e iπθb T 2 ψ b,a (k) dk

17 Π 0,1 : A θ B(H 0,1 ) with H 0,1 := L 2 (T 2 ) defined by: { Π0,1 (u) =: U 0,1 : ψ n,m e iπθn ψ n,m+1 Π 0,1 (v) =: V 0,1 : ψ n,m e iπθm ψ n+1,m ψ n,m (k 1,k 2 ) := (2π) 1 e i(nk 1+mk 2 ) Fourier basis of H 0,1. It is the GNS representation related to, indeed ψ n,m e iπθnm v n u m, cyclic vector ψ 0,0 = (2π) 1, (u a v b ) = δ a,0 δ b,0 = (ψ 0,0 ;Π 0,1 (u a v b )ψ 0,0 ) = e iπθb Π 0,1 is injective since is faithful. T 2 ψ b,a (k) dk

18 Π 0,1 : A θ B(H 0,1 ) with H 0,1 := L 2 (T 2 ) defined by: { Π0,1 (u) =: U 0,1 : ψ n,m e iπθn ψ n,m+1 Π 0,1 (v) =: V 0,1 : ψ n,m e iπθm ψ n+1,m ψ n,m (k 1,k 2 ) := (2π) 1 e i(nk 1+mk 2 ) Fourier basis of H 0,1. It is the GNS representation related to, indeed ψ n,m e iπθnm v n u m, cyclic vector ψ 0,0 = (2π) 1, (u a v b ) = δ a,0 δ b,0 = (ψ 0,0 ;Π 0,1 (u a v b )ψ 0,0 ) = e iπθb T 2 ψ b,a (k) dk Π 0,1 is injective since is faithful. A 0,1 θ := Π 0,1 (A θ ) describes the effective models for the 2DMBE in the limits B 0 (θ f B := flux trough the unit cell).

19 The commutant A 0,1 θ is generated by: { F1 : ψ n,m e iπθn ψ n,m 1 F 2 : ψ n,m e iπθm ψ n+1,m.

20 The commutant A 0,1 θ is generated by: { F1 : ψ n,m e iπθn ψ n,m 1 If θ = M/N then F 2 : ψ n,m e iπθm ψ n+1,m. S 0,1 := C (F 1,F 2 := F N 2 ) is a maximal commutative C -subalgebra of A 0,1 θ.

21 The commutant A 0,1 θ is generated by: { F1 : ψ n,m e iπθn ψ n,m 1 If θ = M/N then F 2 : ψ n,m e iπθm ψ n+1,m. S 0,1 := C (F 1,F 2 := F N 2 ) is a maximal commutative C -subalgebra of A 0,1 θ S 0,1 is a unitary representation of Z 2 on H 0,1, moreover {φ j } N 1 j=0 H 0,1 with φ j := ψ j,0 is wandering for S 0,1, i.e. [ ] S 0,1 {φ j } N 1 = H 0,1, (φ i ;F1 a F 2 b φ j) = δ i,j δ a,0 δ b,0. j=0.

22 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

23 Π q,r : A θ B(H q ) with H q := L 2 (R) C q L 2 (R;Z q ) defined by: { Πq,r (u) =: U q,r = T 1 U Π q,r (v) =: V q,r = T 2 θ r q V r r < q N g.c.d.(r,q) = 1 where T 1 and T 2 are the Weyl operators on L 2 (R), i.e. T 1 := e i2πq, T 2 := e i2πp, Q := multiplication by x, P := i 2π x while U and V act on C q as: 1 ϖ q U :=... ϖ q 1 q where ϖ q := e i2π 1 q, UV = ϖ q VU. 0 1, V :=

24 Π q,r : A θ B(H q ) with H q := L 2 (R) C q L 2 (R;Z q ) defined by: { Πq,r (u) =: U q,r = T 1 U Π q,r (v) =: V q,r = T 2 θ r q V r r < q N g.c.d.(r,q) = 1 where T 1 and T 2 are the Weyl operators on L 2 (R), i.e. T 1 := e i2πq, T 2 := e i2πp, Q := multiplication by x, P := i 2π x while U and V act on C q as: 1 ϖ q U :=... ϖ q 1 q 0 1, V := where ϖ q := e i2π q 1, UV = ϖ q VU. Π q,r is injective and A q,r θ := Π q,r (A θ ) describes the 2DMBE in the limits B (q = 1, r = 0, θ f 1 B ).

25 The commutant A q,r θ is generated by [Takesaki 69]: 1 G 1 := T qθ r 1 U a bq ar = 1. 1 G 2 := T q 2 V 1.

26 The commutant A q,r θ is generated by [Takesaki 69]: 1 G 1 := T qθ r 1 U a bq ar = 1. 1 G 2 := T q 2 V 1. If θ = M/N (with g.c.d.(n,q)=1) then S q,r := C (G 1,G 2 := G N 0 2 ), N N 0 0 := qm rn, G 2 := T q 2 V Nr is a maximal commutative C -subalgebra of A q,r θ.

27 The commutant A q,r θ is generated by [Takesaki 69]: 1 G 1 := T qθ r 1 U a bq ar = 1. 1 G 2 := T q 2 V 1. If θ = M/N (with g.c.d.(n,q)=1) then S q,r := C (G 1,G 2 := G N 0 2 ), N N 0 0 := qm rn, G 2 := T q 2 V Nr is a maximal commutative C -subalgebra of A q,r θ S q,r is a unitary representation of Z 2 on H q L 2 (R;Z q ), moreover {φ j } N 1 j=0 H q with χ j 0 1 if j N 0 φ j := χ j (x) := N x (j + 1)N 0 N. 0 otherwise. 0 is wandering for S q,r, i.e. [ ] S q,r {φ j } N 1 = H q, (φ i ;G1 a Gb 2 φ j) = δ i,j δ a,0 δ b,0. j=0.

28 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

29 Ingredients: 1 separable Hilbert space H (space of physical states); 2 a C -algebra A of bounded operators on H (physical observables); 3 a (maximal) commutative C -subalgebra S of the commutant A (simultaneous implementable symmetries).

30 Ingredients: 1 separable Hilbert space H (space of physical states); 2 a C -algebra A of bounded operators on H (physical observables); 3 a (maximal) commutative C -subalgebra S of the commutant A (simultaneous implementable symmetries). Technical assumptions: 1 S is generated by a unitary representation of Z d, i.e. S := C (U 1,...,U d ) with U j = U 1 j ; 2 S has the wandering property, i.e. it exists a (countable) subset {φ j } H of orthonormal vectors such that S [ {φ j } ] = H, (φ i ;U n Un d d φ j) = δ i,j δ n1,0...δ nd,0.

31 Ingredients: 1 separable Hilbert space H (space of physical states); 2 a C -algebra A of bounded operators on H (physical observables); 3 a (maximal) commutative C -subalgebra S of the commutant A (simultaneous implementable symmetries). Technical assumptions: 1 S is generated by a unitary representation of Z d, i.e. S := C (U 1,...,U d ) with U j = U 1 j ; 2 S has the wandering property, i.e. it exists a (countable) subset {φ j } H of orthonormal vectors such that Consequences: S [ {φ j } ] = H, (φ i ;U n Un d d φ j) = δ i,j δ n1,0...δ nd,0. 1 S is algebraically compatible, i.e. a n1,...,n d U n Un d d = 0 iff a n1,...,n d = 0; 2 the Gel fand spectrum of S is T d ; 3 the Haar measure dz := dtd on T d is basic for S (abs. (2π) d cont. with respect the spectral measures).

32 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

33 Theorem [D. & Panati t.b.p.] i) Let Φ the nuclear space obtained as the inductive limit of vector spaces spanned by finite collections of vectors in S [ {φ j } ]. The map Φ ϕ (U ϕ)(t) := e in 1t 1...e in d t d U n Un d d ϕ Φ is well defined and (U ϕ)(t) is a generalized eigenvector of U j with eigenvalue e it j. ii) Let K (t) Φ the space spanned by { ξ j (t) := (U φ j )(t) }, then H U K (t) dz(t) (L 2 sections) T d is a unitary map between Hilbert spaces. iii) Φ is a pre-c -module over C(T d ) and is mapped by U in a dense set of continuous sections of the vector bundle E S T d with fiber K (t) and frame of sections {ξ j }. iv) A is mapped by U in the continuous sections of End(E S ) T d.

34 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

35 Vector bundle representation of A (0,1) M/N (GNS) H 0,1 := L 2 (T 2 ) U Γ L 2(E 0,1 ) C N dz T 2 E 0,1 T 2 is a rank-n vector bundle with typical fiber K (t) C N, N is the cardinality of the wandering system {φ j } N 1 j=0. The vector bundle is trivial, indeed the frame of sections {ξ j := U φ j } N 1 j=0 satisfies: ξ j (t 1,t 2 ) = ξ j (t 1 +2π,t 2 ) = ξ j (t 1,t 2 +2π) j = 0,...,N 1 (t 1,t 2 ) T 2, then C(E 0,1 ) = 0, where C is the (first) Chern number.

36 Vector bundle representation of A (0,1) M/N (GNS) H 0,1 := L 2 (T 2 ) U Γ L 2(E 0,1 ) C N dz T 2 E 0,1 T 2 is a rank-n vector bundle with typical fiber K (t) C N, N is the cardinality of the wandering system {φ j } N 1 j=0. The vector bundle is trivial, indeed the frame of sections {ξ j := U φ j } N 1 j=0 satisfies: ξ j (t 1,t 2 ) = ξ j (t 1 +2π,t 2 ) = ξ j (t 1,t 2 +2π) j = 0,...,N 1 (t 1,t 2 ) T 2, then C(E 0,1 ) = 0, where C is the (first) Chern number. A (0,1) U...U 1 M/N End C(T 2 ) (Γ(E 0,1)) Γ(End(E 0,1 )) generated by: 1 0 e it 2 U 0,1 (t) := e it ρ 1... V 0,1(t) := ρ N with ρ := e i2π M N.

37 Vector bundle representation of A (q,r) M/N (Weyl) H q,r := L 2 (R) C q U ΓL 2(E q,r ) T 2 C N dz E q,r T 2 is a rank-n vector bundle, N is the cardinality of the wandering system. The vector bundle is non-trivial, indeed 0 e iqt 2 ξ j (t 1,t 2 ) = g(t 2 ) 1 ξ j (t 1 + 2π,t 2 ) g(t 2 ) := 1... ξ j (t 1,t 2 ) = ξ j (t 1,t 2 + 2π) then C(E r,q ) = q.

38 Vector bundle representation of A (q,r) M/N (Weyl) H q,r := L 2 (R) C q U ΓL 2(E q,r ) T 2 C N dz E q,r T 2 is a rank-n vector bundle, N is the cardinality of the wandering system. The vector bundle is non-trivial, indeed 0 e iqt 2 ξ j (t 1,t 2 ) = g(t 2 ) 1 ξ j (t 1 + 2π,t 2 ) g(t 2 ) := 1... ξ j (t 1,t 2 ) = ξ j (t 1,t 2 + 2π) then C(E r,q ) = q. A (q,r) U...U 1 M/N End C(T 2 ) (Γ(E q,r )) Γ(End(E q,r )) generated by: 1 ( U q,r (t) = e i N 0 N t 1... V q,r (t) = e ist 2 0 e iqt 2 1 l 1 ρ q(n 1) N l 0 ) where: lq sn = 1 (g.c.d.(n,q)=1).

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40 Outline 1 Introduction Overview and physical motivations 2 The NCT and its representations Abstract geometry and gap projections The Π 0,1 representation (GNS) The Π q,r representation (Weyl) 3 Generalized Bloch-Floquet transform The general framework The main theorem 4 Vector bundle representations and TKNN formulæ Vector bundles Duality and TKNN formulæ

41 P AdU 0,1 P 0,1 ( ) L 0,1 [p] C 0,1 (p) Π 0,1 Proj(A θ ) p B(H ) Γ(End(E)) E T 2 Z Ran C Π q,r P q,r AdU P q,r ( ) Ran L q,r [p] C C q,r (p) C := first Chern number.

42 P AdU 0,1 P 0,1 ( ) L 0,1 [p] C 0,1 (p) Π 0,1 Proj(A θ ) p B(H ) Γ(End(E)) E T 2 Z Ran C Π q,r P q,r AdU P q,r ( ) Ran L q,r [p] C C q,r (p) What is the relation between C 0,1 (p) and C q,r (p) for a given p Proj(A θ )?

43 P AdU 0,1 P 0,1 ( ) L 0,1 [p] C 0,1 (p) Π 0,1 Proj(A θ ) p B(H ) Γ(End(E)) E T 2 Z Ran C Π q,r P q,r AdU P q,r ( ) Ran L q,r [p] C C q,r (p) Transforms of the torus: f T 2 (t 1,t 2 ) (qnt 1,t 2 ) T 2 g T 2 (t 1,t 2 ) ( N 0 t 1,qt 2 ) T 2. Pullback of vector bundles: f (L q,r [p]) g (L 0,1 [p]) l q 2 l q 2 := line bundle with Chern number q 2 (global twist of E q,r ).

44 P AdU 0,1 P 0,1 ( ) L 0,1 [p] C 0,1 (p) Π 0,1 Proj(A θ ) p B(H ) Γ(End(E)) E T 2 Z Ran C Π q,r P q,r AdU P q,r ( ) Ran L q,r [p] C C q,r (p) Functoriality of C: C q,r (p) = q R(p) N R(p) := Rk(g (L 0,1 [p]) = Rk(L 0,1 [p]). (q MN ) r C 0,1 (p)

45 P AdU 0,1 P 0,1 ( ) L 0,1 [p] C 0,1 (p) Π 0,1 Proj(A θ ) p B(H ) Γ(End(E)) E T 2 Z Ran C Π q,r P q,r AdU P q,r ( ) Ran L q,r [p] C C q,r (p) Abstract version: R(p) N = (p), C 0,1 (p) = Ch(p) C q,r (p) = q (p) + (r qθ)ch(p)

46 Duality between gap projections of hhar Courtesy of J. Avron

47 Thank you for your attention