Geometric derivation of

Transcription

1 Geometric derivation of the TKNN equations Gianluca Panati Dipartimento di Matematica Università di Roma La Sapienza, Roma Trails in a noncommutative land SISSA International School for Advanced Studies - May 18, 2011 based on joint work with: G. De Nittis (Paris 13) & F. Faure (Institut Fourier, Grenoble) G. Panati (La Sapienza) Derivation of TKNN equations 1 / 39

2 The team Giuseppe De Nittis Institut Galilée Université Paris 13 Paris, France Frédéric Faure Insitute Fourier Université Joseph Fourier Grenoble, France G. Panati (La Sapienza) Derivation of TKNN equations 2 / 39

3 In a milestone paper [TKNN82] Thouless et al. contributed to the understanding of the Quantum Hall effect (QHE) with two important discoveries. 1 for rational magnetic flux the Hall conductance, defined as a bulk observable by linear response theory (Kubo formula), has a topological meaning, which explains its integer value. This topological structure emerges since the Hamiltonian operator commutes with the magnetic translations. 2 the integers corresponding to the Hall conductance, in the limit of weak and strong magnetic field respectively, satisfy a Diophantine equation, nowadays known as TKNN equation (an equation for each spectral gap). G. Panati (La Sapienza) Derivation of TKNN equations 3 / 39

4 1 The first intuition has been later dramatically pushed forward [Connes][Bellissard][Bellissard van Elst Schulz-Baldes], by realizing that the notion of topological invariant can be extended to the case of irrational flux, or of a random perturbation, by replacing the algebra generated by magnetic translations with the rotation C*-algebra, alias the algebra of the non-commutative torus. The relation with alternative approaches to the QHE, based either on a quantum pumping mechanism [Laughlin] [Avron Seiler Simon] or on the edge currents [Halperin][Buttiker] [Kellendonc Richter Schulz-Baldes] [Combes Germinet], has been clarified [Schulz-Baldes Kellendonc Richter] [Elgart Graf Schlein]. 2 On the contrary, the status and the meaning of the TKNN equations have not been completely clarified. No one of the existing derivations took into account the fact that the integers are Chern numbers of suitable vector bundles over T 2. G. Panati (La Sapienza) Derivation of TKNN equations 4 / 39

5 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 5 / 39

6 Theoretical model (without randomness) Independent electrons, effectively 2-dimensional sample periodic potential V Γ : R 2 R, Γ Z 2 uniform orthogonal magnetic field (symmetric gauge) The dynamics is described by the Bloch-Landau Hamiltonian acting in L 2 (R 2 ). H BL = 1 ( 2 i x ϕ B 2 y ) ( ) i y + ϕ B 2 x +V Γ (x, y) }{{} Landau Hamiltonian G. Panati (La Sapienza) Derivation of TKNN equations 6 / 39

7 Theoretical model (without randomness) Independent electrons, effectively 2-dimensional sample periodic potential V Γ : R 2 R, Γ Z 2 uniform orthogonal magnetic field (symmetric gauge) The dynamics is described by the Bloch-Landau Hamiltonian acting in L 2 (R 2 ). H BL = 1 ( 2 i x ϕ B 2 y ) ( ) i y + ϕ B 2 x +V Γ (x, y) }{{} Landau Hamiltonian The competition between the periodic and the magnetic length scales makes a direct study of H BL a formidable task. Thus the need of simpler effective models which capture the physics in suitable asymptotic regimes. G. Panati (La Sapienza) Derivation of TKNN equations 6 / 39

8 Multiscale structure: separation of scales A. The Hofstadter regime (ϕ B 0) B. The Landau regime (ϕ B ) The ratio of the energy scales considered in [TKNN] is ω c / V Γ ϕ B. The ratio of the typical time-scales is Crystal period Cyclotron period = ma2 /2π mc/eb = BS Γ hc/e = Φ = ϕ B Φ 0 Here crystal period is estimated as the time needed to travel trough a distance a (a = S Γ = lattice constant) when the momentum is k = 2π /a G. Panati (La Sapienza) Derivation of TKNN equations 7 / 39

9 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 8 / 39

10 Effective models: Hofstadter regime ϕ B 0 In the Hofstadter regime, ϕ B 0, one considers firstly the unperturbed periodic Hamiltonian H per = + V Γ. The periodicity leads to consider the Bloch-Floquet transform U : L 2 (R 2 ) L 2 (B) L 2 (T 2 }{{ Y) = L 2 (T 2, H } f ) H f (Uψ)(k, y) = γ Γ e i(y+γ) k ψ(y + γ). G. Panati (La Sapienza) Derivation of TKNN equations 9 / 39

11 Effective models: Hofstadter regime ϕ B 0 By using the Bloch-Floquet transform U, one obtains a fibered operator U H per U 1 = B H per (k) dk in L 2 (B, H f ), H per (k) = 1 2 ( i y + k) 2 + V Γ (y) acting on D L 2 (T d Y, dy) =: H f. We assume to know the solution of the eigenvalue problem: H per (k) χ n (k, y) = E n (k) χ n (k, y) G. Panati (La Sapienza) Derivation of TKNN equations 9 / 39

12 Effective models: Hofstadter regime ϕb 0 G. Panati (La Sapienza) Derivation of TKNN equations 9 / 39

13 Effective models: Hofstadter regime ϕ B 0 To any isolated Bloch band E (k 1, k 2 ) := h n,m e i(nk 1+mk 2 ) corresponds an effective Hamiltonian acting in L 2 (T 2, dk 1 dk 2 ) as (formally) E (K 1, K 2 ) with K 1 = k 1 + i 2 πϕ B k 2 C (T 2 ) L 2 (T 2 ) K 2 = k 2 i 2 πϕ B k 1 [K 1, K 2 ] = iπϕ B 1 This procedure corresponds to Peierl s substitution [Peierls 33] and it has been mathematically established in [Bellissard 88] [Hellfer Sjöstrand 89] [Nenciu91] [P Spohn Teufel 03] [DeNittis P 10]. G. Panati (La Sapienza) Derivation of TKNN equations 9 / 39

14 Effective models: Hofstadter regime ϕ B 0 To any isolated Bloch band E (k 1, k 2 ) := h n,m e i(nk 1+mk 2 ) corresponds an effective Hamiltonian acting in L 2 (T 2, dk 1 dk 2 ) as E (K 1, K 2 ) = h n,m e i(nk 1+mK 2 ) = h n,m e iπnmϕ B U 0 m V 0 n U 0 = e ik 2, V 0 = e ik 1, U 0 V 0 = e i2πϕ B V 0 U 0 Hofstadter model: E (k 1, k 2 ) = 2[cos(k 1 ) + cos(k 2 )]. ĥ = U 0 + U V 0 + V 0 1 =: H Hof (ϕ B ). G. Panati (La Sapienza) Derivation of TKNN equations 9 / 39

15 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 10 / 39

16 Effective models: Landau regime ϕ B + In the Landau regime, ϕ B, one considers a perturbation of the Landau Hamiltonian [ ( H L = 1 2 i x ϕ ) 2 ( B 2 y + i y + ϕ ) ] 2 B 2 x = 1 2 [ K K 2 2 ]. For V Γ = 0 the Hamiltonian is quadratic (Landau levels). The operators G 1 and G 2, corresponding to the classical coordinates of the center of the cyclotron orbit, commute with the Hamiltonian. Classical Mechanics Quantum Mechanics x 1 = x 1 (t) + 1 v 2 (t) G 1 = X K 2 = 1 ( P 2 + ϕ ) B ϕ B ϕ B ϕ B 2 X 1 x 2 = x 2 (t) 1 v 1 (t) G 2 = X 2 1 K 1 = 1 ( P 1 + ϕ ) B ϕ B ϕ B ϕ B 2 X 2 [ K 1, K 2 ] = iϕ B 1 [G j, K i ] = 0 [G 1, G 2 ] = i 1 ϕ B 1 G. Panati (La Sapienza) Derivation of TKNN equations 11 / 39

17 Effective models: Landau regime ϕ B + For V Γ 0 to each Landau level corresponds an effective Hamiltonian acting in L 2 (R, ds). In the representation which diagonalizes G 1 the effective Hamiltonian is given (assuming V Γ C ) by V Γ (G 1, G 2 ) with { G1 = s G 2 = i 1 ϕ B d ds Here V Γ (G 1, G 2 ) can be interpreted by means of ordinary ɛ-weyl quantization, with ɛ = ɛ = 1 ϕ B. Since the choice V Γ (p, q) = 2 cos(2πp) + 2 cos(2πq) yields the Harper operator, we say that V Γ (G 1, G 2 ) is a Harper-like Hamiltonian. G. Panati (La Sapienza) Derivation of TKNN equations 11 / 39

18 Effective models: Landau regime ϕ B + For V Γ 0 to each Landau level corresponds an effective Hamiltonian acting in L 2 (R, ds). Explicitly, for V Γ (q, p) = n,m Z v n,m e i2π(np+mq), V Γ (G 1, G 2 ) = v n,m e i2π(ng 1+mG 2 ) = v n,m e iπnmε U m V n U := e i2πg 1, V := e i2πg 2, U V := e i2πε V U. The validity of V Γ as effective Hamiltonian for ϕ B is proved in [Bellissard 88] [Hellfer Sjöstrand 89] [DeNittis P 10]. G. Panati (La Sapienza) Derivation of TKNN equations 11 / 39

19 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 12 / 39

20 Conductance in the effective models How can the value of the conductance be recovered? The key idea is that it corresponds to a TQN related to the symmetries of the system [TKNN]. Each of the models has a Z 2 -symmetry, provided ϕ B = M N Q. For any n = (n 1, n 2 ) Z 2 the symmetry is provided by L 2 L 2 (R 2 ) H BL magnetic translations T N 1, T 2 (ϕ B Q) L 2 L 2 (T 2 ) = l 2 (Z 2 ) E (K 1, K 2 ) L 2 L 2 (R) V Γ (G 1, G 2 ) L 2 (T N 1 ψ) n,m = e iπmnϕ B ψ n 1,m L 2 (T 2 ψ) n,m = e iπnϕ B ψ n,m+1 L 2 (T1 M ψ)(x) = ψ(x M) L 2 (T 2 ψ)(x) = e i 2π ϕ B x ψ(x) One can use a generalized Bloch-Floquet transform to decompose the Hilbert space with respect to the Z 2 -symmetry. G. Panati (La Sapienza) Derivation of TKNN equations 13 / 39

21 The generalized Bloch-Floquet transform is defined as U : H T 2 H(k) dk (Uψ)(k) := n Z 2 e in k T 1 n 1 T 2 n 2 ψ, k T 2. where T 1, T 2 B(H) are the generators of the Z 2 -symmetry mentioned above. Hereafter we assume the rationality condition ϕ B = M N. The relevant Hamiltonian H (resp. H BL, H Hof or H Har ), after BF transform, becomes a fibered operator i.e. U H U 1 = T 2 H(k) dk in T 2 H(k) dk. G. Panati (La Sapienza) Derivation of TKNN equations 14 / 39

22 The direct integral defines a relevant collection of vector spaces. Fix µ R (chemical potential) so that µ / Spec H(k) for all k T 2 (physical condition). Denote by P µ (k) := χ (,µ] (H(k)) the spectral projector up to the energy µ and pose F(k) := Ran P µ (k) H(k) G. Panati (La Sapienza) Derivation of TKNN equations 15 / 39

23 The direct integral defines a relevant collection of vector spaces. Fix µ R (chemical potential) so that µ / Spec H(k) for all k T 2 (physical condition). Denote by P µ (k) := χ (,µ] (H(k)) the spectral projector up to the energy µ and pose F(k) := Ran P µ (k) H(k) The collection of such vector spaces is a vector bundle over T 2. The Chern number of this vector bundle is the Topological Quantum Number we are interested in. G. Panati (La Sapienza) Derivation of TKNN equations 15 / 39

24 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 16 / 39

25 Color-coded quantum butterflies E (k 1, k 2 ) = 2(cos k 1 + cos k 2 ) Hofstadter Hamiltonian V Γ (p, x) = 2(cos(2πp) + cos(2πx)) Harper Hamiltonian G. Panati (La Sapienza) Derivation of TKNN equations 17 / 39

26 Color-coded quantum butterflies ϕ B = θ = 1/ϕ B E (k 1, k 2 ) = 2(cos k 1 + cos k 2 ) Hofstadter Hamiltonian V Γ (p, x) = 2(cos(2πp) + cos(2πx)) Harper Hamiltonian G. Panati (La Sapienza) Derivation of TKNN equations 17 / 39

27 Color-coded quantum butterflies E (k 1, k 2 ) = 2(cos k 1 + cos k 2 ) Hofstadter Hamiltonian V Γ (p, x) = 2(cos(2πp) + cos(2πx)) Harper Hamiltonian G. Panati (La Sapienza) Derivation of TKNN equations 17 / 39

28 Color-coded quantum butterflies E (k1, k2 ) = 2(cos k1 + cos k2 ) Hofstadter Hamiltonian G. Panati (La Sapienza) VΓ (p, x) = 2(cos(2πp) + cos(2πx)) Harper Hamiltonian Derivation of TKNN equations 17 / 39

29 Color-coded quantum butterflies Observation: the colors (Chern numbers) in the limiting models are related by the the TKNN-equation NC (g) + MC0 (g) = dg G. Panati (La Sapienza) g {0, 1,..., gmax } Derivation of TKNN equations (1) 17 / 39

30 Here the integer d g is a gap label: it is equal to g for N odd, while for N even is defined by { g for 0 g N/2 1 d g = g + 1 for N/2 g g max. Moreover, one has the constraint 2 C 0 (g) N. For any g {0, 1,..., g max }, the TKNN equation is solved by infinite pairs (t g, s g ) Z 2. If the constraint is imposed, the solution is unique (provided M, N are coprime, M Z, N N \ {0}). G. Panati (La Sapienza) Derivation of TKNN equations 18 / 39

31 The twofold way We obtain the TKNN formula as a corollary of the duality between vector bundles emerging in two inequivalent representations of the same C -algebra. p Proj(A θ ) P 0 Π 0 U 0...U 1 0 P 0 ( ) Ran L 0 p C C 0 (p) B(H) Γ(End(E)) E T 2 Z Π P P ( ) U...U 1 Ran L p C C (p) C := first Chern number. What is the relation between C 0 (p) and C (p) for a given p Proj(A θ )? G. Panati (La Sapienza) Derivation of TKNN equations 19 / 39

32 Hilbert space: H 0 := L 2 (T 2, dk 1 dk 2 ). Hofstadter unitaries: U 0 = e ik 2, V 0 = e ik 1, U 0 V 0 = e i2πε 0 V 0 U 0. where K 1 = k 1 + iπε 0, K 2 = k 2 iπε 0. k 2 k 1 Hofstadter-like Hamiltonian: h(k 1, k 2 ) = h n,m e i(nk 1+mk 2 ) C 1 (T 2 ) (Peierls substitution) ĥ(ε 0 ) := h n,m e i(nk 1+mK 2 ) = h n,m e iπnmε 0 U 0 m V 0 n B(H 0 ) Hofstadter model: h (k 1, k 2 ) = 2 cos(k 1 ) + 2 cos(k 2 ). ĥ = U 0 + U V 0 + V 0 1 =: H Hof (ε 0 ). G. Panati (La Sapienza) Derivation of TKNN equations 20 / 39

33 Hilbert space: H := L 2 (R, dq). Harper unitaries: where U := e i2πq, V := e i2πp, U V = e i2πε V U. Harper-like Hamiltonian: Q = multiplication by q, P = i ε 2π q. h(p, q) = h n,m e i2π(np+mq) C 1 Z 2 (R 2 ) (Weyl quantization) ĥ(ε ) := h n,m e i2π(np+mq) = h n,m e iπnmε U m V n B(H ) Harper model: h (p, q) = 2 cos(2πp) + 2 cos(2πq). ĥ = U + U 1 + V + V 1 =: H Har (ε ). G. Panati (La Sapienza) Derivation of TKNN equations 21 / 39

34 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 22 / 39

35 Non-Commutative Torus (NCT) abstract C -algebra A θ (θ R deformation parameter) generated by: u = u 1, v = v 1, uv = e i2πθ vu with universal norm a := sup { π(a) H π rep. of A θ on H}. G. Panati (La Sapienza) Derivation of TKNN equations 23 / 39

36 Non-Commutative Torus (NCT) abstract C -algebra A θ (θ R deformation parameter) generated by: u = u 1, v = v 1, uv = e i2πθ vu with universal norm a := sup { π(a) H π rep. of A θ on H}. Hofstadter representation: Π 0 : A ε0 B(H 0 ), Π 0 (u) := U 0, Π 0 (v) := V 0. G. Panati (La Sapienza) Derivation of TKNN equations 23 / 39

37 Non-Commutative Torus (NCT) abstract C -algebra A θ (θ R deformation parameter) generated by: u = u 1, v = v 1, uv = e i2πθ vu with universal norm a := sup { π(a) H π rep. of A θ on H}. Hofstadter representation: Π 0 : A ε0 B(H 0 ), Π 0 (u) := U 0, Π 0 (v) := V 0. Harper representation: Π : A ε B(H ), Π (u) := U, Π (v) := V. G. Panati (La Sapienza) Derivation of TKNN equations 23 / 39

38 Non-Commutative Torus (NCT) abstract C -algebra A θ (θ R deformation parameter) generated by: u = u 1, v = v 1, uv = e i2πθ vu with universal norm a := sup { π(a) H π rep. of A θ on H}. Hofstadter representation: Π 0 : A ε0 B(H 0 ), Π 0 (u) := U 0, Π 0 (v) := V 0. Harper representation: Π : A ε B(H ), Π (u) := U, Π (v) := V. PROPOSITION (isospectrality) Π 0 and Π are injective. Let ε 0 = ε =: θ (duality condition). Then for all a A θ σ(π 0 (a)) = σ(a) = σ(π (a)). G. Panati (La Sapienza) Derivation of TKNN equations 23 / 39

39 Let h := u + u 1 + v + v 1 A θ and θ Q, [Hof76]: ε 0 (B 1 ) = ε (B 2 ) implies B 1 B 2 1 with B 1 1 B 2 (duality) σ(h Hof ) = σ(h) = σ(h Har ) G. Panati (La Sapienza) Derivation of TKNN equations 24 / 39

40 Let h := u + u 1 + v + v 1 A θ and θ Q, [Hof76]: ε 0 (B 1 ) = ε (B 2 ) implies B 1 B 2 1 with B 1 1 B 2 (duality) σ(h Hof ) = σ(h) = σ(h Har ) G. Panati (La Sapienza) Derivation of TKNN equations 24 / 39

41 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 25 / 39

42 Smooth algebra: It is a Fréchet algebra. { } A θ := an,m u n v m {a n,m } S(Z 2 ) A θ Canonical derivations (differentiable structure): j : A θ A θ j = 1, 2 1 (u n v m ) = i2πn(u n v m ), 2 (u n v m ) = i2πm(u n v m ). Symmetric ( j (a ) = j (a) ), commuting ( 1 2 = 2 1 ), trace-compatibility ( j = 0). Canonical trace (measure structure): : A θ C (u n v m ) = δ n,0 δ m,0. state (linear, positive, normalized), faithful ( (a a) = 0 a = 0), tracial property ( (ab) = (ba)), unique if θ / Q. G. Panati (La Sapienza) Derivation of TKNN equations 26 / 39

43 Smooth algebra: It is a Fréchet algebra. { } A θ := an,m u n v m {a n,m } S(Z 2 ) A θ Canonical derivations (differentiable structure): j : A θ A θ j = 1, 2 1 (u n v m ) = i2πn(u n v m ), 2 (u n v m ) = i2πm(u n v m ). Symmetric ( j (a ) = j (a) ), commuting ( 1 2 = 2 1 ), trace-compatibility ( j = 0). Canonical trace (measure structure): : A θ C (u n v m ) = δ n,0 δ m,0. state (linear, positive, normalized), faithful ( (a a) = 0 a = 0), tracial property ( (ab) = (ba)), unique if θ / Q. G. Panati (La Sapienza) Derivation of TKNN equations 26 / 39

44 Projections: Proj(A θ ) := {p A θ p = p = p 2 }. If θ = M N (with M Z, N N and g.c.d(m, N) = 1), then { : Proj(A M/N ) 0, 1 N,..., N 1 } N, 1. G. Panati (La Sapienza) Derivation of TKNN equations 27 / 39

45 Projections: Proj(A θ ) := {p A θ p = p = p 2 }. If θ = M N (with M Z, N N and g.c.d(m, N) = 1), then { : Proj(A M/N ) 0, 1 N,..., N 1 } N, 1. Abstract (first) Chern number: Ch : Proj(A θ ) A θ Ch(p) := i 2π Z (p[ 1 (p); 2 (p)]). G. Panati (La Sapienza) Derivation of TKNN equations 27 / 39

46 Projections: Proj(A θ ) := {p A θ p = p = p 2 }. If θ = M N (with M Z, N N and g.c.d(m, N) = 1), then { : Proj(A M/N ) 0, 1 N,..., N 1 } N, 1. Abstract (first) Chern number: Ch : Proj(A θ ) A θ Ch(p) := i 2π Z (p[ 1 (p); 2 (p)]). G. Panati (La Sapienza) Derivation of TKNN equations 27 / 39

47 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 28 / 39

48 Band projections and gap band projections h A M/N selfadjoint (h = h ). Band spectrum σ(h) = j J I j with I j R closed intervals. Gap, open interval between two bands. G. Panati (La Sapienza) Derivation of TKNN equations 29 / 39

49 Band projections and gap band projections h A M/N selfadjoint (h = h ). Band spectrum σ(h) = j J I j with I j R closed intervals. Gap, open interval between two bands. LEMMA (band projection) Any band I j σ(h) defines a p j Proj(A θ ) A θ. G. Panati (La Sapienza) Derivation of TKNN equations 29 / 39

50 Band projections and gap band projections h A M/N selfadjoint (h = h ). Band spectrum σ(h) = j J I j with I j R closed intervals. Gap, open interval between two bands. LEMMA (band projection) Any band I j σ(h) defines a p j Proj(A θ ) A θ. Gap projection P j := j k=1 p k. If θ = M/N then 1 j N. G. Panati (La Sapienza) Derivation of TKNN equations 29 / 39

51 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 30 / 39

52 Let Γ(E ) be the space of continuous sections of the Hermitian vector bundle E T 2. Any A Γ(End(E )) defines a multiplication operator à acting on s L 2 (E ) by (Ãs) z = A z s z, thus Γ(End(E )) can be regarded as a subalgebra of B(L 2 (E )). DEFINITION (Bundle decomposition) Let A B(H) be a C -algebra. We say that A admits a bundle decomposition over T 2 if there exist a Hermitian vector bundle E T 2 and a unitary map F : H L 2 (E ) such that F A F 1 Γ(End(E )). LEMMA Let A B(H) be a C -algebra admitting a bundle decomposition over T 2 with respect to the vector bundle ι : E T 2. Then any projection p A defines a vector subbundle L (p) E. G. Panati (La Sapienza) Derivation of TKNN equations 31 / 39

53 THEOREM (Bundle decomposition in Hofstadter and Harper representations) Let A M/N be the the rational NCT-algebra, π 0 : A M/N B(H 0 ) the Hofstadter representation and π : A M/N B(H ) the Harper representation. Then: (i) The operator algebra π 0 (A M/N ) B(H 0 ) admits a bundle decomposition over T 2 with respect to a (rank N) Hermitian vector bundle E 0 T 2 and a unitary operator F 0 : H 0 L 2 (E 0 ). The vector bundle E 0 is trivial. The bundle representation a F 0 π 0 (a) F 0 1 is generated by the endomorphism sections U 0 ( ) := F 0 π 0 (u) F 0 1 and V 0 ( ) := F 0 π 0 (v) F 0 1, which are explicitly given in the proof. (ii) The operator algebra π (A M/N ) B(H ) admits a bundle decomposition over T 2 with respect to a (rank N) Hermitian vector bundle E T 2 and unitary transform F : H L 2 (E ). The vector bundle E is non trivial with (first) Chern number C 1 (E ) = 1. G. Panati (La Sapienza) Derivation of TKNN equations 32 / 39

54 The twofold way We obtain the TKNN formula as a corollary of the duality between vector bundles emerging in two inequivalent representations of the same C -algebra. p Proj(A θ ) P 0 Π 0 U 0...U 1 0 P 0 ( ) Ran L 0 (p) C C 0 (p) B(H) Γ(End(E)) E T 2 Z Π P P ( ) U...U 1 Ran L (p) C C (p) C := first Chern number. What is the relation between C 0 (p) and C (p) for a given p Proj(A θ ) for θ = M/N? G. Panati (La Sapienza) Derivation of TKNN equations 33 / 39

55 Outline 1 Introduction The model Effective models: Hofstadter regime Effective models: Landau regime Chern numbers in the effective models The TKNN equations 2 Common algebraic structure & NCT Abstract algebraic structure & isospectrality Measure and differentiable structure on the NCT The band spectrum 3 Main results Bundle decomposition Duality of vector bundles G. Panati (La Sapienza) Derivation of TKNN equations 34 / 39

56 Let p Proj(A M/N ) and r {0, }. The orthogonal projector π r (p) B(H r ) defines a vector subbundle L (π r (p)) of E r T 2, which we denote shortly by L r (p). We write C r (p) := C 1 (L r (p)). THEOREM (Geometric duality (De Nittis, Faure, P)) Let A M/N be the rational NCT-algebra and p Proj(A M/N ). Then: (i) The vector bundles L 0 (p) and L (p) have rank equal to Rk(p). (ii) Consider the continuous functions T 2 T 2 defined by α(z 1, z 2 ) = (z N 1, z 2) β(z 1, z 2 ) = (z M 1, z 2 ). (2) Then the following isomorphism of Hermitian vector bundles holds true, where I is the determinant bundle of E. α L (p) β L 0 (p) I (3) G. Panati (La Sapienza) Derivation of TKNN equations 35 / 39

57 Derivation of the TKNN equations α L p β L 0 p I c 1 first Chern class of a product VB c 1 (α L p ) = c 1 (β L 0 p I) = Rk(β L 0 p) c 1 (I) + Rk(I) c 1 (β L 0 p) functoriality of c 1, Rk(I) = 1, Rk(β L 0 p) = Rk(P 0 ) α c 1 (L p ) = Rk(P 0 ) c 1 (I) + β c 1 (L 0 p) C( ) := c 1 ( ) (first) Chern number, C(I) = 1 T 2 NC(L p ) = Rk(P 0 ) MC(L 0 p) G. Panati (La Sapienza) Derivation of TKNN equations 36 / 39

58 C (p) := C(L p), {0, }. COROLLARY (TKNN equations) Let p Proj(A θ ), θ = M N. The Harper and Hofstadter Chern numbers of p are related by the following (diophantine) equation NC (p) + MC 0 (p) = Rk(p). In particular if h has N disjoint bands (as in the case of h = u + u 1 + v + v 1 for N odd), then p 1... p N = 1. NC (p k ) + MC 0 (p k ) = 1 k = 1,..., N. Let P k := p 1... p k (k th gap projection) NC (P k ) + MC 0 (P k ) = k k = 1,..., N. Remark!! if k = N then P N = 1. C (1) = 1 (Chern number of the total Harper bundle), C 0 (1) = 0 (Chern number of the total trivial Hofstadter bundle). G. Panati (La Sapienza) Derivation of TKNN equations 37 / 39

59 Chern numbers: abstract and concrete NC (p) + MC 0 (p) = Rk(p) C (p) = 1 N Rk(p) M N C 0(p) (Ch = C 0, = 1 N Rk) C (p) = (p) θch(p) ( ) The concrete Chern number of an abstract projection p in the Harper representation is a linear combination of the astract operations Ch and. The differential-geometric Chern number depends on the representation considered. G. Panati (La Sapienza) Derivation of TKNN equations 38 / 39

60 Thank you for your attention!! G. Panati (La Sapienza) Derivation of TKNN equations 39 / 39