Wannier functions, Bloch bundles and topological degree theory

Transcription

1 Wannier functions, Bloch bundles and topological degree theory Horia D. Cornean Department of Mathematical Sciences, Aalborg University, Denmark Bilbao, June 17, 2016

2 The plan of the talk 1. The Bloch-Floquet-Gelfand transform and Bloch bundles; 2 / 35

3 The plan of the talk 1. The Bloch-Floquet-Gelfand transform and Bloch bundles; 2. Adiabatic parallel transport; 2 / 35

4 The plan of the talk 1. The Bloch-Floquet-Gelfand transform and Bloch bundles; 2. Adiabatic parallel transport; 3. The main result: A TRS Bloch bundle is trivial iff a certain number of explicit maps are null-homotopic. 2 / 35

5 The plan of the talk 1. The Bloch-Floquet-Gelfand transform and Bloch bundles; 2. Adiabatic parallel transport; 3. The main result: A TRS Bloch bundle is trivial iff a certain number of explicit maps are null-homotopic. 4. Physical dimensions d 3 are trivial ; 2 / 35

6 The plan of the talk 1. The Bloch-Floquet-Gelfand transform and Bloch bundles; 2. Adiabatic parallel transport; 3. The main result: A TRS Bloch bundle is trivial iff a certain number of explicit maps are null-homotopic. 4. Physical dimensions d 3 are trivial ; 5. Open problems, comments, bibliography. 2 / 35

7 The configuration space Let B R d be a finite set containing D points and let Λ := B + Z d be the discrete configuration space. We assume that every point y Λ can be uniquely written as: y = y + γ, y B, γ Z d. 3 / 35

8 The configuration space Let B R d be a finite set containing D points and let Λ := B + Z d be the discrete configuration space. We assume that every point y Λ can be uniquely written as: y = y + γ, y B, γ Z d unit cell (0, 3 a) 4 a ( 3a a 5a a ) 3 2, 2 2, 2 ( ) 1 2 a1 1 (0,0) (a,0) (3a,0) / 35

9 The configuration space Consider the Hilbert space l 2 (Λ) with the standard basis y, γ, y B, γ Z d. 4 / 35

10 The configuration space Consider the Hilbert space l 2 (Λ) with the standard basis y, γ, y B, γ Z d. Let H be any bounded self-adjoint operator acting on this space. It is completely characterized by its matrix elements H(y, γ; y, γ ) := y, γ H y, γ. 4 / 35

11 The configuration space Consider the Hilbert space l 2 (Λ) with the standard basis y, γ, y B, γ Z d. Let H be any bounded self-adjoint operator acting on this space. It is completely characterized by its matrix elements H(y, γ; y, γ ) := y, γ H y, γ. We will assume that H commutes with the translations acting on Z d. 4 / 35

12 The configuration space Consider the Hilbert space l 2 (Λ) with the standard basis y, γ, y B, γ Z d. Let H be any bounded self-adjoint operator acting on this space. It is completely characterized by its matrix elements H(y, γ; y, γ ) := y, γ H y, γ. We will assume that H commutes with the translations acting on Z d. Important consequence: H(y, γ; y, γ ) = H(y, γ γ ; y, 0), y, y B, γ, γ Z d. 4 / 35

13 The Bloch-Floquet-Gelfand transform Let Ω := [ 1/2, 1/2] d and consider: l 2 (Λ) Ψ (GΨ)(y; k) := e 2πik γ Ψ(y + γ) L 2 (Ω ; C D ) γ Z d and L 2 (Ω ; C D ) f (y; k) (G f )(y + γ) = e 2πik γ f (y; k)dk l 2 (Λ). Ω 5 / 35

14 The Bloch-Floquet-Gelfand transform We want to show that: GHG = Ω h(k)dk, h(k) L(C D ), h(y, y ; k) := e 2πik γ H(y, γ; y, 0). γ Z d 6 / 35

15 The Bloch-Floquet-Gelfand transform Indeed: (GHG f )(y; k) = e 2πik γ H(y, γ γ ; y, 0) γ γ,y = e 2πik γ H(y, γ; y, 0) e 2πik γ y γ γ = y h(y, y ; k)f (y ; k). Ω e 2πik γ Ω e 2πik γ f (y ; k )dk f (y ; k )dk 7 / 35

16 Spectral consequences The D D matrix h(k) has D eigenvalues (Bloch energies), whose range determine the spectrum of H. 8 / 35

17 Spectral consequences The D D matrix h(k) has D eigenvalues (Bloch energies), whose range determine the spectrum of H. h(k) is Z d -periodic and the decay of H(y, γ; y, 0) as a function of γ translates into k-regularity. 8 / 35

18 Spectral consequences The D D matrix h(k) has D eigenvalues (Bloch energies), whose range determine the spectrum of H. h(k) is Z d -periodic and the decay of H(y, γ; y, 0) as a function of γ translates into k-regularity. If H is a real kernel, then: h(y, y; k) = h(y, y ; k) = h(y, y ; k), called conjugation or time reversal symmetry, TRS. For a self-adjoint matrix, TRS imposes: t h(k) = h( k), k R d. 8 / 35

19 Spectral consequences The D D matrix h(k) has D eigenvalues (Bloch energies), whose range determine the spectrum of H. h(k) is Z d -periodic and the decay of H(y, γ; y, 0) as a function of γ translates into k-regularity. If H is a real kernel, then: h(y, y; k) = h(y, y ; k) = h(y, y ; k), called conjugation or time reversal symmetry, TRS. For a self-adjoint matrix, TRS imposes: Consequence: σ(h(k)) = σ(h( k)). t h(k) = h( k), k R d. 8 / 35

20 Spectral consequences Assume that the range of N < D eigenvalues λ j ( ) of h( ) is separated from the range of the other eigenvalues, i.e. there exists a gap in the spectrum of H. 9 / 35

21 Spectral consequences Assume that the range of N < D eigenvalues λ j ( ) of h( ) is separated from the range of the other eigenvalues, i.e. there exists a gap in the spectrum of H. We define P(k) to be the Riesz projection associated to the N eigenvalue cluster: P(k) := i (h(k) z) 1 dz, P(k) 2 = P(k) = P(k). 2π C 9 / 35

22 Spectral consequences Assume that the range of N < D eigenvalues λ j ( ) of h( ) is separated from the range of the other eigenvalues, i.e. there exists a gap in the spectrum of H. We define P(k) to be the Riesz projection associated to the N eigenvalue cluster: P(k) := i (h(k) z) 1 dz, P(k) 2 = P(k) = P(k). 2π C P( ) is periodic and as smooth as h. Moreover: hence P( ) has TRS. t {(h(k) z) 1 } = (h( k) z) 1, 9 / 35

23 Physical motivation: Wannier states Assume that N vectors Ξ j (k) C D form an orthonormal basis for Ran(P(k)). Define for 1 j N, y B, γ Z d : w j (y) = w j (y + γ) := e 2πik γ Ξ j (y; k)dk, Ω 10 / 35

24 Physical motivation: Wannier states Assume that N vectors Ξ j (k) C D form an orthonormal basis for Ran(P(k)). Define for 1 j N, y B, γ Z d : Then the vectors w j (y) = w j (y + γ) := e 2πik γ Ξ j (y; k)dk, Ω {w j ( γ) : γ Z d, 1 j N} l 2 (Λ) form an orthonormal basis for the range of the spectral projection Π corresponding to the part of the spectrum given by σ := N j=1 Ran(λ j ( )), Π := i (H z) 1 dz. 2π C 10 / 35

25 The main problem Consider a family P(k) of orth. proj. of rank N in a Hilbert space H. It is Z d -periodic and jointly analytic in a complex tubular neighborhood I of R d. 11 / 35

26 The main problem Consider a family P(k) of orth. proj. of rank N in a Hilbert space H. It is Z d -periodic and jointly analytic in a complex tubular neighborhood I of R d. Assume that θ is anti-unitary, θ 2 = Id, and θp(k) = P( k)θ. Main question: When is this vector bundle trivial? B := {(k, v) : k T d, v RanP(k)} 11 / 35

27 The main problem Consider a family P(k) of orth. proj. of rank N in a Hilbert space H. It is Z d -periodic and jointly analytic in a complex tubular neighborhood I of R d. Assume that θ is anti-unitary, θ 2 = Id, and θp(k) = P( k)θ. Main question: When is this vector bundle trivial? B := {(k, v) : k T d, v RanP(k)} More precisely, can one construct N vectors {Ξ j (k)} N j=1 H which are jointly analytic in a tubular complex neighborhood of R d, are Z d -periodic, form an orthonormal basis of Ran(P(k)) for all k R d, and θξ j (k) = Ξ j ( k)? 11 / 35

28 Part two: parallel transport Assume d = 1 and let P(x) = P (x) = P 2 (x) be a C 1 -norm differentiable family of orthogonal projections. Define H(x) := i(id 2P(x))P (x) = i[p (x), P(x)], H(x) = H (x). 12 / 35

29 Part two: parallel transport Assume d = 1 and let P(x) = P (x) = P 2 (x) be a C 1 -norm differentiable family of orthogonal projections. Define H(x) := i(id 2P(x))P (x) = i[p (x), P(x)], H(x) = H (x). Given y R, one can define a family of unitary operators A(x, y) such that i x A(x, y) = H(x)A(x, y), A(y, y) = Id, given by: A(x, y) = Id + n 1 x s1 ( i) n ds 1 ds 2... y y sn 1 y ds n H(s 1 )...H(s n ). 12 / 35

30 Parallel transport Furthermore, we have: i y A(x, y) = A(x, y)h(y), [A(x, y)] 1 = [A(x, y)] = A(y, x). 13 / 35

31 Parallel transport Furthermore, we have: i y A(x, y) = A(x, y)h(y), [A(x, y)] 1 = [A(x, y)] = A(y, x). An intertwining identity: P(x)A(x, y) = A(x, y)p(y). 13 / 35

32 Parallel transport Furthermore, we have: i y A(x, y) = A(x, y)h(y), [A(x, y)] 1 = [A(x, y)] = A(y, x). An intertwining identity: P(x)A(x, y) = A(x, y)p(y). If {ψ n } N n=1 form an ONB in Ran(P(0)), then φ n(x) := A(x, 0)ψ n form a smooth ONB in Ran(P(x)). 13 / 35

33 Parallel transport Furthermore, we have: i y A(x, y) = A(x, y)h(y), [A(x, y)] 1 = [A(x, y)] = A(y, x). An intertwining identity: P(x)A(x, y) = A(x, y)p(y). If {ψ n } N n=1 form an ONB in Ran(P(0)), then φ n(x) := A(x, 0)ψ n form a smooth ONB in Ran(P(x)). If θp(x) = P( x)θ, then θh(x) = H( x)θ and θa(x, 0) = A( x, 0)θ. 13 / 35

34 Parallel transport Furthermore, we have: i y A(x, y) = A(x, y)h(y), [A(x, y)] 1 = [A(x, y)] = A(y, x). An intertwining identity: P(x)A(x, y) = A(x, y)p(y). If {ψ n } N n=1 form an ONB in Ran(P(0)), then φ n(x) := A(x, 0)ψ n form a smooth ONB in Ran(P(x)). If θp(x) = P( x)θ, then θh(x) = H( x)θ and θa(x, 0) = A( x, 0)θ. If d > 1, we use induction. Many things can go wrong if we need to impose periodicity conditions. 13 / 35

35 The problem in d = 1: P(k) = P (k) = P 2 (k) has rank N < D, lives in C D, it is Z-periodic and smooth. Let θ denote the anti-unitary involution given by the complex conjugation. We have θp(k) = P( k)θ, or P mn (k) = P nm ( k). 14 / 35

36 The problem in d = 1: P(k) = P (k) = P 2 (k) has rank N < D, lives in C D, it is Z-periodic and smooth. Let θ denote the anti-unitary involution given by the complex conjugation. We have θp(k) = P( k)θ, or P mn (k) = P nm ( k). Problem: Find an orthonormal basis {Ξ j (k)} N j=1 of the range of P(k) such that each Ξ j ( ) is smooth, Z-periodic and θξ j (k) = Ξ j ( k). 14 / 35

37 The problem in d = 1: P(k) = P (k) = P 2 (k) has rank N < D, lives in C D, it is Z-periodic and smooth. Let θ denote the anti-unitary involution given by the complex conjugation. We have θp(k) = P( k)θ, or P mn (k) = P nm ( k). Problem: Find an orthonormal basis {Ξ j (k)} N j=1 of the range of P(k) such that each Ξ j ( ) is smooth, Z-periodic and θξ j (k) = Ξ j ( k). Due to the time-reversal symmetry we can find a real basis for the range of P(0): if f Ran(P(0)), then θf = θp(0)f = P(0)θf, θf Ran(P(0)). 14 / 35

38 The parallel transported basis Due to the symmetry θp(k)θ = P( k), we can find a real basis for the range of P(0) and also θa(k, 0) = A( k, 0)θ. 15 / 35

39 The parallel transported basis Due to the symmetry θp(k)θ = P( k), we can find a real basis for the range of P(0) and also θa(k, 0) = A( k, 0)θ. Then {Ψ j (k) := A(k, 0)Ξ j (0)} N j=1 is an orthonormal basis for the range of P(k) which is smooth and θψ j (k) = Ψ j ( k). 15 / 35

40 The parallel transported basis Due to the symmetry θp(k)θ = P( k), we can find a real basis for the range of P(0) and also θa(k, 0) = A( k, 0)θ. Then {Ψ j (k) := A(k, 0)Ξ j (0)} N j=1 is an orthonormal basis for the range of P(k) which is smooth and θψ j (k) = Ψ j ( k). No periodicity though. 15 / 35

41 The parallel transported basis Due to the symmetry θp(k)θ = P( k), we can find a real basis for the range of P(0) and also θa(k, 0) = A( k, 0)θ. Then {Ψ j (k) := A(k, 0)Ξ j (0)} N j=1 is an orthonormal basis for the range of P(k) which is smooth and θψ j (k) = Ψ j ( k). No periodicity though. Since P( 1/2) = P(1/2), there must exists an N N unitary matrix β such that Ψ m (1/2) = N n=1 β nmψ n ( 1/2). Using Ψ m (1/2) = Ψ m ( 1/2) we obtain: θ c βθ c = β 1, β mn = β nm. 15 / 35

42 The parallel transported basis Due to the symmetry θp(k)θ = P( k), we can find a real basis for the range of P(0) and also θa(k, 0) = A( k, 0)θ. Then {Ψ j (k) := A(k, 0)Ξ j (0)} N j=1 is an orthonormal basis for the range of P(k) which is smooth and θψ j (k) = Ψ j ( k). No periodicity though. Since P( 1/2) = P(1/2), there must exists an N N unitary matrix β such that Ψ m (1/2) = N n=1 β nmψ n ( 1/2). Using Ψ m (1/2) = Ψ m ( 1/2) we obtain: θ c βθ c = β 1, β mn = β nm. The spectral projectors of β have the TRS property θ c Π = Πθ c because: t Π = i t (β z) 1 dz = i (β z) 1 dz = Π. 2π 2π C C 15 / 35

43 And the solution in dimension one is: There exists a unique selfadjoint matrix M with σ(m) ( π, π] and with the TRS property θ c M = Mθ c such that β = e im. 16 / 35

44 And the solution in dimension one is: There exists a unique selfadjoint matrix M with σ(m) ( π, π] and with the TRS property θ c M = Mθ c such that β = e im. Define Ξ m (x) := N n=1 [e ixm ] nm Ψ n (x); these vectors are still smooth, θξ m (x) = Ξ( x), and are also periodic, because: Ξ m (1/2) = N [e im/2 βe im/2 ] nm Ξ n ( 1/2) = Ξ m ( 1/2). n=1 16 / 35

45 Iterating from one dimension Assume that the family of projectors P(k 1, k 2 ) is smooth, Z 2 -periodic and θp(k 1, k 2 )θ = P( k 1, k 2 ). 17 / 35

46 Iterating from one dimension Assume that the family of projectors P(k 1, k 2 ) is smooth, Z 2 -periodic and θp(k 1, k 2 )θ = P( k 1, k 2 ). The family P(0, k 2 ) admits a good basis, i.e. there exist N vectors Ξ j (0, k 2 ) with all the right properties and which form a basis for the range of P(0, k 2 ). 17 / 35

47 Iterating from one dimension Assume that the family of projectors P(k 1, k 2 ) is smooth, Z 2 -periodic and θp(k 1, k 2 )θ = P( k 1, k 2 ). The family P(0, k 2 ) admits a good basis, i.e. there exist N vectors Ξ j (0, k 2 ) with all the right properties and which form a basis for the range of P(0, k 2 ). Keeping k 2 fixed, we can repeat the previous construction of a unitary A k2 (k 1, 0) which is periodic in k 2, obeys θa k2 (k 1, 0)θ = A k2 ( k 1, 0) and P(k 1, k 2 )A k2 (k 1, 0) = A k2 (k 1, 0)P(0, k 2 ). 17 / 35

48 Iterating from one dimension Assume that the family of projectors P(k 1, k 2 ) is smooth, Z 2 -periodic and θp(k 1, k 2 )θ = P( k 1, k 2 ). The family P(0, k 2 ) admits a good basis, i.e. there exist N vectors Ξ j (0, k 2 ) with all the right properties and which form a basis for the range of P(0, k 2 ). Keeping k 2 fixed, we can repeat the previous construction of a unitary A k2 (k 1, 0) which is periodic in k 2, obeys θa k2 (k 1, 0)θ = A k2 ( k 1, 0) and P(k 1, k 2 )A k2 (k 1, 0) = A k2 (k 1, 0)P(0, k 2 ). Define Ψ m (k 1, k 2 ) := A k2 (k 1, 0)Ξ m (0, k 2 ). 17 / 35

49 The vectors Ψ m (k 1, k 2 ) = A k2 (k 1, 0)Ξ m (0, k 2 ) are smooth, periodic in k 2, obey Ψ m (k 1, k 2 ) = Ψ m ( k 1, k 2 ), but are not periodic in k / 35

50 The vectors Ψ m (k 1, k 2 ) = A k2 (k 1, 0)Ξ m (0, k 2 ) are smooth, periodic in k 2, obey Ψ m (k 1, k 2 ) = Ψ m ( k 1, k 2 ), but are not periodic in k 1. Since P( 1/2, k 2 ) = P(1/2, k 2 ), there must exist a matching unitary matrix β(k 2 ) such that Ψ m (1/2, k 2 ) = n β nm (k 2 )Ψ n ( 1/2, k 2 ). 18 / 35

51 The vectors Ψ m (k 1, k 2 ) = A k2 (k 1, 0)Ξ m (0, k 2 ) are smooth, periodic in k 2, obey Ψ m (k 1, k 2 ) = Ψ m ( k 1, k 2 ), but are not periodic in k 1. Since P( 1/2, k 2 ) = P(1/2, k 2 ), there must exist a matching unitary matrix β(k 2 ) such that Ψ m (1/2, k 2 ) = n β nm (k 2 )Ψ n ( 1/2, k 2 ). Since β nm (k 2 ) = Ψ m (1/2, k 2 ) Ψ n ( 1/2, k 2 ), we must have β nm (k 2 ) = β mn ( k 2 ), which is equivalent with θ c β(k 2 )θ c = β( k 2 ) / 35

52 The vectors Ψ m (k 1, k 2 ) = A k2 (k 1, 0)Ξ m (0, k 2 ) are smooth, periodic in k 2, obey Ψ m (k 1, k 2 ) = Ψ m ( k 1, k 2 ), but are not periodic in k 1. Since P( 1/2, k 2 ) = P(1/2, k 2 ), there must exist a matching unitary matrix β(k 2 ) such that Ψ m (1/2, k 2 ) = n β nm (k 2 )Ψ n ( 1/2, k 2 ). Since β nm (k 2 ) = Ψ m (1/2, k 2 ) Ψ n ( 1/2, k 2 ), we must have β nm (k 2 ) = β mn ( k 2 ), which is equivalent with θ c β(k 2 )θ c = β( k 2 ) 1. Can we find a smooth AND periodic logarithm of β(k 2 )? 18 / 35

53 The vectors Ψ m (k 1, k 2 ) = A k2 (k 1, 0)Ξ m (0, k 2 ) are smooth, periodic in k 2, obey Ψ m (k 1, k 2 ) = Ψ m ( k 1, k 2 ), but are not periodic in k 1. Since P( 1/2, k 2 ) = P(1/2, k 2 ), there must exist a matching unitary matrix β(k 2 ) such that Ψ m (1/2, k 2 ) = n β nm (k 2 )Ψ n ( 1/2, k 2 ). Since β nm (k 2 ) = Ψ m (1/2, k 2 ) Ψ n ( 1/2, k 2 ), we must have β nm (k 2 ) = β mn ( k 2 ), which is equivalent with θ c β(k 2 )θ c = β( k 2 ) 1. Can we find a smooth AND periodic logarithm of β(k 2 )? Not always! 18 / 35

54 No good logarithm What can go wrong: [ cos(2πt) sin(2πt) β(t) = sin(2πt) cos(2πt) ] = e i2πtσ 2. Eigenvalues: λ 1 (t) = e i2πt, λ 1 (t) = e i2πt. Doubly degenerate at t = ±1/2 and t = 0. Cannot have a periodic, TRS AND continuous logarithm. 19 / 35

55 If a good logarithm exists Assume for the moment that there exists a self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 )θ c = h( k 2 ) such that β(k 2 ) = e ih(k 2). 20 / 35

56 If a good logarithm exists Assume for the moment that there exists a self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 )θ c = h( k 2 ) such that β(k 2 ) = e ih(k 2). Define Ξ m (x, k 2 ) := N n=1 [e ixh(k 2) ] nm Ψ n (x, k 2 ) if x 1/2 and extend them by periodicity; these vectors are also continuous and obey all the necessary properties. 20 / 35

57 The general case Pick up any self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 ) = h( k 2 )θ c. 21 / 35

58 The general case Pick up any self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 ) = h( k 2 )θ c. Define Φ m (x, k 2 ) := N n=1 [e ixh(k 2) ] nm Ψ n (x, k 2 ) if x 1/2; these vectors are also continuous and obey the following matching property: Φ m (1/2, k 2 ) = N [e ih(k2)/2 β(k 2 )e ih(k2)/2 ] nm Φ n ( 1/2, k 2 ). n=1 21 / 35

59 The general case Pick up any self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 ) = h( k 2 )θ c. Define Φ m (x, k 2 ) := N n=1 [e ixh(k 2) ] nm Ψ n (x, k 2 ) if x 1/2; these vectors are also continuous and obey the following matching property: Φ m (1/2, k 2 ) = N [e ih(k2)/2 β(k 2 )e ih(k2)/2 ] nm Φ n ( 1/2, k 2 ). n=1 The new matching matrix is γ(k 2 ) = e ih(k 2)/2 β(k 2 )e ih(k 2)/2. 21 / 35

60 The general case Pick up any self-adjoint matrix h(k 2 ) which is smooth, Z-periodic and θ c h(k 2 ) = h( k 2 )θ c. Define Φ m (x, k 2 ) := N n=1 [e ixh(k 2) ] nm Ψ n (x, k 2 ) if x 1/2; these vectors are also continuous and obey the following matching property: Φ m (1/2, k 2 ) = N [e ih(k2)/2 β(k 2 )e ih(k2)/2 ] nm Φ n ( 1/2, k 2 ). n=1 The new matching matrix is γ(k 2 ) = e ih(k 2)/2 β(k 2 )e ih(k 2)/2. Maybe we can find a good logarithm for γ! 21 / 35

61 The general case Main idea: given β(k 2 ), we can always construct a selfadjoint matrix h ɛ (k 2 ) which is smooth, Z-periodic, θh ɛ (k 2 )θ = h ɛ ( k 2 ) and: β(k 2 ) e ihɛ(k 2) = e ihɛ(k 2)/2 β(k 2 )e ihɛ(k 2)/2 Id 1/2. 22 / 35

62 The general case Main idea: given β(k 2 ), we can always construct a selfadjoint matrix h ɛ (k 2 ) which is smooth, Z-periodic, θh ɛ (k 2 )θ = h ɛ ( k 2 ) and: β(k 2 ) e ihɛ(k 2) = e ihɛ(k 2)/2 β(k 2 )e ihɛ(k 2)/2 Id 1/2. Then γ ɛ (k 2 ) = e ihɛ(k 2)/2 β(k 2 )e ihɛ(k 2)/2 is close to the identity, hence it admits a good logarithm through the Cayley transform A ɛ = A ɛ := i(id γ ɛ )(Id + γ ɛ ) 1, γ ɛ = (Id ia ɛ )(Id + ia ɛ ) 1, γ ɛ = e i h ɛ, h ɛ (k 2 ) = 1 ( ) 1 iz Ln (A ɛ (k 2 ) z) 1 dz. 2π C 1 + iz 22 / 35

63 The general case After a second rotation, the new matching matrix will be: e i h ɛ(k 2 )/2 e ihɛ(k 2)/2 β(k 2 )e ihɛ(k 2)/2 e i h ɛ(k 2 )/2 = Id. 23 / 35

64 An explicit case of rank two Every SU(2) unitary matrix can be written as: α(t) = m(t)id + if(t) σ, m 2 + F 2 = / 35

65 An explicit case of rank two Every SU(2) unitary matrix can be written as: If F(t) is never zero, then α(t) = m(t)id + if(t) σ, m 2 + F 2 = 1. h(t) := arccos(m(t)) F(t) σ, α(t) = e ih(t). F(t) 24 / 35

66 An explicit case of rank two Every SU(2) unitary matrix can be written as: If F(t) is never zero, then α(t) = m(t)id + if(t) σ, m 2 + F 2 = 1. h(t) := arccos(m(t)) F(t) σ, α(t) = e ih(t). F(t) How to deal with: [ cos(2πt) sin(2πt) β(t) = sin(2πt) cos(2πt) ] = cos(2πt)id + i sin(2πt)σ 2, m(t) = cos(2πt), F = [0, sin(2πt), 0]. 24 / 35

67 Rank two Define β ɛ (t) := ɛ 2 {cos(2πt)id + i(ɛσ 1 + sin(2πt)σ 2 )}, m ɛ (t) = cos(2πt) 1 + ɛ 2, F ɛ(t) = 1 [ɛ, sin(2πt), 0]. 1 + ɛ 2 25 / 35

68 Rank two Define β ɛ (t) := ɛ 2 {cos(2πt)id + i(ɛσ 1 + sin(2πt)σ 2 )}, m ɛ (t) = cos(2πt) 1 + ɛ 2, F ɛ(t) = 1 [ɛ, sin(2πt), 0]. 1 + ɛ 2 Then h ɛ (t) = arccos(m ɛ(t)) F ɛ (t) σ, β ɛ (t) = e ihɛ(t). F ɛ (t) 25 / 35

69 Rank two Define β ɛ (t) := ɛ 2 {cos(2πt)id + i(ɛσ 1 + sin(2πt)σ 2 )}, m ɛ (t) = cos(2πt) 1 + ɛ 2, F ɛ(t) = 1 [ɛ, sin(2πt), 0]. 1 + ɛ 2 Then We have h ɛ (t) = arccos(m ɛ(t)) F ɛ (t) σ, β ɛ (t) = e ihɛ(t). F ɛ (t) lim ɛ 0 e ihɛ(t)/2 β(t)e ihɛ(t)/2 = Id. 25 / 35

70 The main idea behind our method Let β(k) U(N) which is Z d 1 -periodic, smooth, θ c β(k)θ c = β( k) / 35

71 The main idea behind our method Let β(k) U(N) which is Z d 1 -periodic, smooth, θ c β(k)θ c = β( k) 1. Can one find M self-adjoint matrices h j (k), Z d 1 -periodic, smooth and with θ c h j (k) = h j ( k)θ c such that e ih M(k)/2...e ih 1(k)/2 β(k)e ih 1(k)/2...e ih M(k)/2 = Id? 26 / 35

72 The main idea behind our method Let β(k) U(N) which is Z d 1 -periodic, smooth, θ c β(k)θ c = β( k) 1. Can one find M self-adjoint matrices h j (k), Z d 1 -periodic, smooth and with θ c h j (k) = h j ( k)θ c such that e ih M(k)/2...e ih 1(k)/2 β(k)e ih 1(k)/2...e ih M(k)/2 = Id? If yes, then we have an important consequence: T d 1 [0, 1] [k, s] β s (k) U(N) β s (k) := e ish M(k)/2...e ish 1(k)/2 β(k)e ish 1(k)/2...e ish M(k)/2 is a homotopy between β( ) and the constant map, preserving all symmetries, periodicity and smoothness properties. 26 / 35

73 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. 27 / 35

74 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. Then there exists a real, even and periodic p(k) = p( k) such that detβ(k) = e ip(k). 27 / 35

75 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. Then there exists a real, even and periodic p(k) = p( k) such that detβ(k) = e ip(k). Thus e ip(k)/(2n) β(k)e ip(k)/(2n) SU(N) and has the same symmetry properties. 27 / 35

76 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. Then there exists a real, even and periodic p(k) = p( k) such that detβ(k) = e ip(k). Thus e ip(k)/(2n) β(k)e ip(k)/(2n) SU(N) and has the same symmetry properties. If N = 1, this actually solves the problem (Nenciu, 1983). 27 / 35

77 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. Then there exists a real, even and periodic p(k) = p( k) such that detβ(k) = e ip(k). Thus e ip(k)/(2n) β(k)e ip(k)/(2n) SU(N) and has the same symmetry properties. If N = 1, this actually solves the problem (Nenciu, 1983). But is this null-homotopy induced by multiple logarithms method also necessary? 27 / 35

78 The main idea behind our method If β(k) U(N) obeys θ c β(k) = β( k) 1 θ c, then detβ(k) = detβ( k), is continuous and Z d 1 periodic. Then there exists a real, even and periodic p(k) = p( k) such that detβ(k) = e ip(k). Thus e ip(k)/(2n) β(k)e ip(k)/(2n) SU(N) and has the same symmetry properties. If N = 1, this actually solves the problem (Nenciu, 1983). But is this null-homotopy induced by multiple logarithms method also necessary? Yes. 27 / 35

79 Null-homotopy is necessary Let d 2 and assume that RanP(k) admits an ONB F m (k) with all the right properties. We also assume that the multiple logarithm method works up to d / 35

80 Null-homotopy is necessary Let d 2 and assume that RanP(k) admits an ONB F m (k) with all the right properties. We also assume that the multiple logarithm method works up to d 1. Then P(0, k ) has a range spanned both by {F n (0, k )} and by {Ξ n (0, k )} constructed inductively as before, starting from P(0). 28 / 35

81 Null-homotopy is necessary Let d 2 and assume that RanP(k) admits an ONB F m (k) with all the right properties. We also assume that the multiple logarithm method works up to d 1. Then P(0, k ) has a range spanned both by {F n (0, k )} and by {Ξ n (0, k )} constructed inductively as before, starting from P(0). Consider the Kato-unitary A k (k 1, 0) and Ψ n (k) := A k (k 1, 0)Ξ n (0, k ). We have θψ n (k) = Ψ n ( k), periodicity in k but no periodicity in k / 35

82 Null-homotopy is necessary Let d 2 and assume that RanP(k) admits an ONB F m (k) with all the right properties. We also assume that the multiple logarithm method works up to d 1. Then P(0, k ) has a range spanned both by {F n (0, k )} and by {Ξ n (0, k )} constructed inductively as before, starting from P(0). Consider the Kato-unitary A k (k 1, 0) and Ψ n (k) := A k (k 1, 0)Ξ n (0, k ). We have θψ n (k) = Ψ n ( k), periodicity in k but no periodicity in k 1. Define the N N unitary matrix: u nm (s, k ) := Ψ n (s, k ), F m (s, k ), 1/2 s 1/2. 28 / 35

83 Null-homotopy is necessary Let d 2 and assume that RanP(k) admits an ONB F m (k) with all the right properties. We also assume that the multiple logarithm method works up to d 1. Then P(0, k ) has a range spanned both by {F n (0, k )} and by {Ξ n (0, k )} constructed inductively as before, starting from P(0). Consider the Kato-unitary A k (k 1, 0) and Ψ n (k) := A k (k 1, 0)Ξ n (0, k ). We have θψ n (k) = Ψ n ( k), periodicity in k but no periodicity in k 1. Define the N N unitary matrix: u nm (s, k ) := Ψ n (s, k ), F m (s, k ), 1/2 s 1/2. Then: u nm (s, k ) = θψ n (s, k ), θf m (s, k ) = u nm ( s, k ) 28 / 35

84 Null-homotopy is necessary or u nm (1/2, k ) = Ψ n (1/2, k ), F m (1/2, k ) N = β jn (k ) Ψ j ( 1/2, k ), F m ( 1/2, k ) j=1 N = [β(k ) 1 ] nj u jm ( 1/2, k ), j=1 β(k ) = u( 1/2, k )[u(1/2, k )] / 35

85 Null-homotopy is necessary Hence if the bundle is trivial, we can write: β(k ) = u( 1/2, k )[u(1/2, k )] 1, θ c u(s, k )θ c = u( s, k ). 30 / 35

86 Null-homotopy is necessary Hence if the bundle is trivial, we can write: β(k ) = u( 1/2, k )[u(1/2, k )] 1, θ c u(s, k )θ c = u( s, k ). Define α t (k ) := u( t/2, k )[u(t/2, k )] 1, 0 t / 35

87 Null-homotopy is necessary Hence if the bundle is trivial, we can write: β(k ) = u( 1/2, k )[u(1/2, k )] 1, θ c u(s, k )θ c = u( s, k ). Define We have: α t (k ) := u( t/2, k )[u(t/2, k )] 1, 0 t 1. α 0 (k ) = Id, α 1 (k ) = β(k ), θ c α t (k )θ c = [α t ( k )] / 35

88 Null-homotopy is necessary Hence if the bundle is trivial, we can write: β(k ) = u( 1/2, k )[u(1/2, k )] 1, θ c u(s, k )θ c = u( s, k ). Define We have: α t (k ) := u( t/2, k )[u(t/2, k )] 1, 0 t 1. α 0 (k ) = Id, α 1 (k ) = β(k ), θ c α t (k )θ c = [α t ( k )] 1. Thus β( ) must be null-homotopic. 30 / 35

89 Null-homotopy gives the logarithms We saw that N good logarithms give us a TRS null-homotopy. Are there more general ones? 31 / 35

90 Null-homotopy gives the logarithms We saw that N good logarithms give us a TRS null-homotopy. Are there more general ones? Assume that α s (k) U(N), 0 s 1, is periodic, continuous and α 0 (k ) = Id, α 1 (k ) = β(k ), θ c α s (k )θ c = [α s ( k )] / 35

91 Null-homotopy gives the logarithms We saw that N good logarithms give us a TRS null-homotopy. Are there more general ones? Assume that α s (k) U(N), 0 s 1, is periodic, continuous and α 0 (k ) = Id, α 1 (k ) = β(k ), θ c α s (k )θ c = [α s ( k )] 1. There exists n N large enough such that: sup α (j+1)/n (k) α j/n (k) 1/2, 0 j n 1. k 31 / 35

92 Null-homotopy gives the logarithms We saw that N good logarithms give us a TRS null-homotopy. Are there more general ones? Assume that α s (k) U(N), 0 s 1, is periodic, continuous and α 0 (k ) = Id, α 1 (k ) = β(k ), θ c α s (k )θ c = [α s ( k )] 1. There exists n N large enough such that: sup α (j+1)/n (k) α j/n (k) 1/2, 0 j n 1. k Through the Cayley transform, we find a good h 1 (k) such that e ih 1(k)/2 α 1/n (k)e ih 1(k)/2 = Id. 31 / 35

93 Null-homotopy gives the logarithms The key idea (0 j < n): e ih j (k)/2...e ih 1(k)/2 α (j+1)/n (k)e ih 1(k)/2...e ih j (k)/2 Id 1/2. 32 / 35

94 Null-homotopy gives the logarithms The key idea (0 j < n): e ih j (k)/2...e ih 1(k)/2 α (j+1)/n (k)e ih 1(k)/2...e ih j (k)/2 Id 1/2. Then e ihn(k)/2...e ih 1(k)/2 β(k)e ih 1(k)/2...e ihn(k)/2 = Id. 32 / 35

95 Null-homotopy gives the logarithms The key idea (0 j < n): e ih j (k)/2...e ih 1(k)/2 α (j+1)/n (k)e ih 1(k)/2...e ih j (k)/2 Id 1/2. Then e ihn(k)/2...e ih 1(k)/2 β(k)e ih 1(k)/2...e ihn(k)/2 = Id. The basis is made periodic after at most n rotations. So if a null-homotopy exists, it can be used to construct some good logarithms. 32 / 35

96 Known and unknown things When N = 2, then SU(2) can be identified with S 3 and: T d 1 k β(k) S / 35

97 Known and unknown things When N = 2, then SU(2) can be identified with S 3 and: T d 1 k β(k) S 3. If d 3, the above map is null-homotopic and can be straightened up in at most two steps (Cornean, Herbst, Nenciu, 2016). If N 3 the problem is more complicated and more steps are needed. 33 / 35

98 Known and unknown things When N = 2, then SU(2) can be identified with S 3 and: T d 1 k β(k) S 3. If d 3, the above map is null-homotopic and can be straightened up in at most two steps (Cornean, Herbst, Nenciu, 2016). If N 3 the problem is more complicated and more steps are needed. If d = 4, the map T 3 k β(k) S 3 is homotopic with a constant iff its degree is zero (based on Hopf s degree theorem). Here TRS does not help: it turns out that the degree is an even number, not necessarily zero. 33 / 35

99 Known and unknown things When N = 2, then SU(2) can be identified with S 3 and: T d 1 k β(k) S 3. If d 3, the above map is null-homotopic and can be straightened up in at most two steps (Cornean, Herbst, Nenciu, 2016). If N 3 the problem is more complicated and more steps are needed. If d = 4, the map T 3 k β(k) S 3 is homotopic with a constant iff its degree is zero (based on Hopf s degree theorem). Here TRS does not help: it turns out that the degree is an even number, not necessarily zero. Open problem: construct a non-trivial Schrödinger TRS projection in d = 4! 33 / 35

100 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) 34 / 35

101 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) 34 / 35

102 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). 34 / 35

103 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) 34 / 35

104 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) 34 / 35

105 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) 34 / 35

106 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) Panati, G., Pisante, A.: Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions. Commun. Math. Phys. 322(3), (2013) 34 / 35

107 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) Panati, G., Pisante, A.: Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions. Commun. Math. Phys. 322(3), (2013) De Nittis, G., Gomi, K.: Classification of Real Bloch-bundles: Topological quantum systems of type AI. J. of Geom. and Phys. 86, (2014) 34 / 35

108 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) Panati, G., Pisante, A.: Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions. Commun. Math. Phys. 322(3), (2013) De Nittis, G., Gomi, K.: Classification of Real Bloch-bundles: Topological quantum systems of type AI. J. of Geom. and Phys. 86, (2014) Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. H. Poincaré 17(1), (2016) 34 / 35

109 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) Panati, G., Pisante, A.: Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions. Commun. Math. Phys. 322(3), (2013) De Nittis, G., Gomi, K.: Classification of Real Bloch-bundles: Topological quantum systems of type AI. J. of Geom. and Phys. 86, (2014) Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. H. Poincaré 17(1), (2016) Cornean, H.D., Herbst, I., Nenciu, G.: On the construction of composite Wannier functions. First Online Ann. H. Poincaré 2016, DOI /s / 35

110 References Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, (1959) des Cloizeaux, J.: Analytical properties of n-dimensional energy bands and Wannier functions. Phys. Rev., A (1964) Nenciu, G.: Existence of exponentially localized Wannier functions. Commun. Math. Phys. 91, (1983). Helffer, B., Sjöstrand, J.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Phys. 345, (1989) Panati, G.: Triviality of Bloch and Bloch-Dirac bundles. Ann. H. Poincaré , (2007) Cornean, H.D., Nenciu, A., Nenciu, G.: Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators. J. Phys. A: Math. Theor , (2008) Panati, G., Pisante, A.: Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions. Commun. Math. Phys. 322(3), (2013) De Nittis, G., Gomi, K.: Classification of Real Bloch-bundles: Topological quantum systems of type AI. J. of Geom. and Phys. 86, (2014) Fiorenza, D., Monaco, D., Panati, G.: Construction of real-valued localized composite Wannier functions for insulators. Ann. H. Poincaré 17(1), (2016) Cornean, H.D., Herbst, I., Nenciu, G.: On the construction of composite Wannier functions. First Online Ann. H. Poincaré 2016, DOI /s Cornean, H.D., Monaco, D., Teufel, S.: Wannier functions and Z 2 invariants in time-reversal symmetric topological insulators / 35

111 Thank you! 35 / 35