Inequivalent bundle representations for the Noncommutative Torus - PowerPoint PPT Presentation

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)

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

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) .

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 & Sj˝ ostrand ’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).

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 & Sj˝ ostrand ’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.

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

The NCT with deformation parameter θ ∈ R is the “abstract” C ∗ -algebra A θ generated by: u ∗ = u − 1 , v ∗ = v − 1 , uv = e i 2 πθ vu and closed in the (universal) norm : � a � := sup {� π ( a ) � H | π representation of A θ on H } .

The NCT with deformation parameter θ ∈ R is the “abstract” C ∗ -algebra A θ generated by: u ∗ = u − 1 , v ∗ = v − 1 , uv = e i 2 πθ 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 NCT with deformation parameter θ ∈ R is the “abstract” C ∗ -algebra A θ generated by: u ∗ = u − 1 , v ∗ = v − 1 , uv = e i 2 πθ 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 ) . ∂ ∂ − − − − − − − − j ( a ∗ ) = ∂ − − j ( a ) ∗ , commuting ∂ − − − − − − − − Symmetric ∂ 1 ◦ ∂ 2 = ∂ 2 ◦ ∂ 1 and � − − − − ◦ ∂ j = 0. − −

A p ∈ A θ is a projection if p = p ∗ = p 2 . Let Proj ( A θ ) the collection of the projections of A θ . N with M ∈ Z , N ∈ N ∗ and g.c.d ( M , N ) = 1, then If θ = M � � � N ,..., N − 1 0 , 1 − : Proj ( A M / N ) → N , 1 . − −

A p ∈ A θ is a projection if p = p ∗ = p 2 . Let Proj ( A θ ) the collection of the projections of A θ . N with M ∈ Z , N ∈ N ∗ and g.c.d ( M , N ) = 1, then If θ = M � � � N ,..., N − 1 0 , 1 − : Proj ( A M / N ) → N , 1 . − − The “abstract” (first) Chern number of p ∈ Proj ( A θ ) is defined by: Ch ( p ) := i � − − − − − − − ( p [ ∂ 1 ( p ); ∂ 2 ( p )]) . − − 2 π

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 I j the characteristic functions for the spectral band I j , then χ I j ∈ 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 := � j k = 1 p j . If θ = M / N then 1 � j � N .

An important example of selfadjoint element in A θ is the Harper Hamiltonian h Har := u + u − 1 + v + v − 1 . Spectrum of h Har for θ ∈ Q [Hofstadter ’76]:

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

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

Π 0 , 1 : A θ → B ( H 0 , 1 ) with H 0 , 1 := L 2 ( T 2 ) defined by: � Π 0 , 1 ( u ) =: U 0 , 1 : ψ n , m �→ e i πθ n ψ n , m + 1 Π 0 , 1 ( v ) =: V 0 , 1 : ψ n , m �→ e − i πθ m ψ n + 1 , m ψ n , m ( k 1 , k 2 ) := ( 2 π ) − 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 i πθ nm v n u m , cyclic vector ψ 0 , 0 = ( 2 π ) − 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 i πθ b T 2 ψ b , a ( k ) dk − −

Π 0 , 1 : A θ → B ( H 0 , 1 ) with H 0 , 1 := L 2 ( T 2 ) defined by: � Π 0 , 1 ( u ) =: U 0 , 1 : ψ n , m �→ e i πθ n ψ n , m + 1 Π 0 , 1 ( v ) =: V 0 , 1 : ψ n , m �→ e − i πθ m ψ n + 1 , m ψ n , m ( k 1 , k 2 ) := ( 2 π ) − 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 i πθ nm v n u m , cyclic vector ψ 0 , 0 = ( 2 π ) − 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 i πθ b T 2 ψ b , a ( k ) dk − − � − is faithful. Π 0 , 1 is injective since − −