Semiclassical Analysis and Magnetic Fields

Transcription

1 Semiclassical Analysis and Magnetic Fields The Peierls-Onsager substitution in the framework of magnetic ΨDO Horia Cornean, Viorel Iftimie & Radu Purice Centre Henri Lebesgue, May 19, 2015 Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

2 Introduction The purpose of my talk is to present how the magnetic ΨDO calculus developed in collaboration with Viorel Iftimie and Marius Măntoiu, and its equivalent version of magnetic integral kernels developed by Horia Cornean and Gheorghe Nenciu allow to obtain a large number of general results concerning the so-called Peierls-Onsager substitution procedure for a periodic quantum system in a magnetic field that is not supposed to vanish at infinity. For the moment we consider only the isolated band situation. The main features of our analysis are that: all the formulae and procedures are explicitly gauge covariant; no adiabatic hypothesis is needed; no decay at infinity is needed for the magnetic field; we work only with absolutely convergent expansions and not with asymptotic ones. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

3 Contents 1 The framework 2 The isolated spectral band 3 The isolated spectral band in a magnetic field 4 The main results Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

4 The framework The framework Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

5 The framework The framework The configuration space: X := R n with a fixed algebraic basis {e 1,..., e d }. The lattice Γ := d j=1 Ze j, with {e 1,..., e d } an algebraic basis of X. It is isomorphic to Z d. The quotient group X /Γ that is canonically isomorphic to the d-dimensional torus T d T. The elementary cell E = {y = d } t j e j X (1/2) t j < (1/2), j {1,..., d}. j=1 Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

6 The framework The framework The dual space X with the duality map <.,. >: X X R. The phase space: Ξ := T X = X X with the canonical symplectic form σ ( (x, ξ), (y, η) ) :=< ξ, y > < η, x >. The dual basis {e 1,..., e d } X defined by < e j, e k >= (2π)δ jk. The dual lattice defined as Γ := {γ X < γ, γ > (2π)Z, γ Γ}. The dual cell (the Brillouin region): E = {ξ = d } t j ej X (1/2) t j < (1/2), j {1,..., d}. j=1 Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

7 The Weyl quantization The framework The Weyl system Two strongly continuous unitary representations on H := L 2 (X ): X x U(x) U(H), [U(x)f ](y) := f (y x) X ξ V (ξ) U(H), [V (ξ)f ](y) := e i<ξ,y> f (y) satisfying the Weyl commutation relations: U(x)V (ξ) = e i<ξ,x> V (ξ)u(x), x X, ξ X. We define W (x, ξ) := e (i/2)<ξ,x> U( x)v (ξ). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

8 The framework The Weyl quantization - 2 The Weyl operators Op(φ) := (2π) d [F Ξ Ξ with [F Ξ φ](x ) := (2π) d Ξ eiσ (X,Y ) φ(y ) dy. Explicitly we have [Op(φ)f ](x) = X φ](x ) W (X ) dx B(H) ( x + z dz dζ e i<ζ,(x z)> φ X 2 ), ζ f (z) Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

9 Hypothesis The framework For m R, ρ [0, 1] we denote by S m ρ (X ) Γ the subspace of Γ-periodic symbols in S m ρ (X ). Hypothesis I We fix some classic Hamiltonian h S m 1 (X ) Γ with m > 0, elliptic and real. Theorem The operator Op(h) is essentially self-adjoint and its closure H is lower semi-bounded and has the domain { } H m (X ) := u L 2 (X ) (1l ) m/2 u L 2 (X ). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

10 The framework The Bloch-Floquet representation We consider the linear space F := {ˆF L 2 loc (X X ) τ γ ˆF = σ γ ˆF γ Γ, τγ ˆF = ˆF γ Γ } (σ γ (ξ) := e i<ξ,x>, [τ γ f ](x) := f (x + γ)) with the Hilbertian norm ˆF 2 := E E ˆF (x, ξ) 2 dξ dx and the the Bloch-Floquet unitary map U Γ : L 2 ( (X ) F, UΓ f ) (x, ξ) = σ γ (ξ)f (x + γ) γ Γ with its inverse having the explicit form ( ) U 1 Γ ˆF (x 0 + γ) = T 1 σ γ (θ)ˆf (x 0, θ)dθ. T Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

11 The framework The Bloch-Floquet representation - 2 We define for any θ T F θ := { f L 2 loc (X ) τ γf } with the Hilbertian norm f 2 F θ = E f (x) 2 dx. Then Theorem F = T F θ dθ; U Γ Op(h)U 1 Γ = T Op(h) Fθ dθ. The operator Op(h) Fθ is essentially self-adjoint and its closure Ĥ(θ) is lower semi-bounded, has domain } Fθ m := {f F θ (1l ) m/2 f F θ and a compact resolvent. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

12 The framework The Bloch structure of the spectrum There exist a family of continuous functions T θ λ j (θ) R, j N such that λ j (θ) λ j+1 (θ) for any j N and any θ T and σ ( Ĥ(θ) ) = {λ j(θ)}. j N Each λ j is smooth on any region of constant multiplicity. There exist a family of measurable functions T θ φ j (θ) F θ, j N such that Ĥ(θ)φ j (θ) = λ j (θ)φ j (θ) for any j N and any θ T. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

13 The isolated spectral band The isolated spectral band Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

14 The Hypothesis II The isolated spectral band There exists a compact interval I R such that σ(h) I =: σ I (H), dist(i, σ(h) \ σ I (H)) =: d 0 > 0. Then we denote by σ (H) := σ(h) \ σ I (H) and notice that there exist j 0 N and N N such that σ I (H) = {λ ( ) j 0 +j T }. 1 j N We denote by J I := {j 0 + 1,..., j 0 + N} and by F I θ := Lin{φ j(θ), j J I }, F I := T F I θ dθ, ( ) π j := U 1 Γ φ j (θ) φ j (θ) dθ U Γ, j J I. T Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

15 The isolated spectral band The band dynamics The following equalities are evident: 1 [H, π j ] = 0, j J I, 2 HE I (H) = j J I Op(λ j ) π j where each λ j is considered as a periodic function of the momenta ξ X. Unfortunately, while T θ j J I φ j (θ) φ j (θ) B ( F ) is a smooth rank N projection valued function on the torus, for each j J I the map T θ φ j (θ) φ j (θ) B ( F ) may be rather singular. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

16 The isolated spectral band The Wannier basis Hypothesis III Under Hypothesis II above, there exist N smooth functions T θ ψ j (θ) F I θ, j N that form an orthonormal basis in ( Fθ I. ) For j J I we denote by w j := U Γ T ψ j (θ)dθ L 2 (X ). Then 1 {W γ,j := τ γ w j } (γ,j) Γ JI is an orthonormal basis in H I := U 1 Γ F I, 2 m N, sup < x γ > m W γ,j (x) <, γ Γ. x X Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

17 The isolated spectral band The reduced Hamiltonian Definition Under Hypothesis III above we define the N N matrix valued function T θ µ(θ) M N,N (C), µ jk (θ) := ψ j (θ), Hψ k (θ) Fθ If Hypothesis I, II and III are true, the following statements are evident: 1 W α,j, HW β,k L 2 (X ) = µ jk (α β) := E 1 T e i<θ,α β> µ jk (θ)dθ, 2 the band Hamiltonian HE I (H) is unitary equivalent with the following Γ-translation invariant matrix valued operator acting on l 2 (Γ) C N : ( M(h)α,β ) jk := µ jk(α β). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

18 Conclusion The isolated spectral band [H, π j ] = 0, HE I (H) = Op(λ j ) π j, j J I W α,j, HW β,k L 2 (X ) = µ jk (α β). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

19 The isolated spectral band in a magnetic field The isolated spectral band in a magnetic field Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

20 The isolated spectral band in a magnetic field The magnetic field The magnetic field is described by a closed 2-form B on X : B : X X X, db = 0. To B we may associate in a highly non-unique way a vector potential, i.e. a 1-form A such that B = da. Gauge transformations: A A = A + dφ; so that B = da = da. Using the transversal gauge we can define a vector potential A ɛ such that B ɛ = da ɛ : A ɛ,j (x) := 1 k d 1 x k B ɛ,jk (sx)sds. 0 Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

21 The isolated spectral band in a magnetic field Hypothesis on the magnetic field Hypothesis IV We consider a magnetic field of the form B ɛ := ɛ B ɛ with ɛ [0, ɛ 0 ] for some small enough ɛ 0 > 0 and with a family of magnetic fields { B ɛ } ɛ [0,ɛ0 ] having components in a bounded subset of BC (X ) (we implicitly assume that d B ɛ = 0 for any ɛ [0, ɛ 0 ]). We shall fix a family of vector potentials { A ɛ } ɛ [0,ɛ0 ] having components in a bounded subset of C pol (X ) such that B ɛ = d A ɛ ; then B ɛ = da ɛ for A ɛ := ɛ A ɛ. We emphasize that all our constructions will be explicitly gauge covariant. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

22 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus Definition For any Schwartz test function Φ S (Ξ) the following oscillating integral defines a continuous linear operator in S (X ): ( Op A (Φ)f ) (x) := (2π) n/2 e i<η,x y> Λ A (x, y)φ ( x + y, η ) f (y)dy dη 2 Λ A (x, y) := exp { i [x,y] A}. Ξ Gauge covariance Note that A = A + dϕ Op A (f ) = e iϕ(q) Op A (f )e iϕ(q). 1 Λ A (x, y)λ A (y, z) = Λ A (x, z) exp 2 Ω B (x, y, z) 1 C B (y x) ((z x). { i <x,y,z> B } Λ A (x, z)ω B (x, y, z), Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

23 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus - 2 Proposition. MP04 The map Op A : S (Ξ) B ( S (X ) ) is an isomorphism of linear topological spaces that extends to an isomorphism Op A : S (Ξ) B ( S (X ); S (X ) ). Proposition. IMP07 Under our Hypothesis IV, for any symbol F S 0 0 (X ) Γ we have that Op A (F ) B ( L 2 (X ) ) and there exist two constants C > 0 and p N, depending only on the dimension d 2 of the configuration space, such that:. Op A (F ) B(L 2 (X )) C sup sup (x,ξ) Ξ a p b p sup ( x a ξ b F ) (x, ξ). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

24 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus - 3 Definition. MP04 On S (Ξ) we define the following separately continuous bilinear form, that is gauge independent: S (Ξ) S (Ξ) (f, g) f B g S (Ξ), Op A (f B g) := Op A (f )Op A (g). We have the explicit formula: ( f B g ) (X ) = (2π) 2d e 2iσ (Y,Z) e i T (x,y,z) B f (X Y )g(x Z)dYdZ, Ξ Ξ with T (x, y, z) having the vertices x y z, x + y z, x y + z. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

25 The isolated spectral band in a magnetic field The magnetic Integral Kernels Calculus (Cornean, Nenciu) For any symbol F S (Ξ) the Weyl operator Op(F ) has as distribution kernel K F S (X X ) given by K F (x, y) = ( WF ) (x, y) := (2π) d X e i<ξ,(x y)> F ( (x + y)/2, ξ ) dξ. Proposition The operator Op A (F ) has the distribution kernel: K A F (x, y) = ΛA (x, y) ( WF ) (x, y) = Λ A (x, y)k F (x, y). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

26 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus - 4 Theorem. IMP07 Under Hypothesis I and IV, Op A (h) is essentially self-adjoint and its closure H A is lower semi-bounded and has domain { } H A (X ) := u L 2 (X ) ( i A) a u L 2 (X ), a N d, a m. Remark. MP04 For h 0(x, ξ) := ξ 2 + V (x), we have Op A (h 0) = ( i j A j (x) )2 + V (Q) 2 A + V (Q). 1 j d Theorem. IMP10 Under Hypothesis I and IV, if z σ(h A ), the resolvent in z is given by ( H A z ) 1 = Op A (r B z (h)) with r B z (h) S m 1 (X ) depending on the magnetic field but not on the choice of gauge. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

27 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus - 5 Development of the magnetic product If f S m 1 ρ (X ) and g S m 2 ρ (X ), then for any n N there exist the family of symbols Ck ɛ(f, g) S m 1+m 2 2kρ ρ (X ), for (k, ɛ) N [0, ɛ 0 ] and Rn(f ɛ, g) S m 1+m 2 2nρ ρ (X ), for (n, ɛ) N, uniformly with respect to ɛ [0, ɛ 0 ], such that f ɛ g = f 0 g + 1 k n 1 ɛ k C ɛ k (f, g) + ɛn R ɛ n(f, g). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

28 The isolated spectral band in a magnetic field The magnetic Pseudodifferential Calculus - 6 Development of the magnetic resolvent 1 [IP15, CP12, CP15]There exists ɛ 0 > 0 small enough such that the bounded interval I R in Hypothesis II satisfies the two conditions in the Hypothesis for H ɛ with ɛ [0, ɛ 0]. 2 For any compact set K C \ σ(h) there exists ɛ 0 > 0 such that K C \ σ(h ɛ ) for ɛ [0, ɛ 0] and the function K z r ɛ z (h) S m 1 (X ) is continuous for the Fréchet topology uniformly for ɛ [0, ɛ 0]. Moreover we have the following development r ɛ z (h) = r 0 z (h) + k N ɛ k r k (h; ɛ, z), with r k (h; ɛ, z) S (m+2n) 1 (X ) uniformly for ɛ [0, ɛ 0] and the series converging in the topology induced from B ( L 2 (X ) ) by the map Op ɛ for any ɛ [0, ɛ 0]. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

29 The main results The main results Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

30 Theorem A The main results Under Hypothesis I, II and IV, there exists ɛ 0 > 0 small enough such that for any ɛ [0, ɛ 0 ] and for any n N we have that: 1 H ɛ E I (H ɛ ) = Op ɛ( ) h I + ɛ k Op ɛ( v ɛ ) k + ɛ n Op ɛ( RI ɛ(h; n)) with: 1 k n 1 1 h I (x, ξ) := [ ( ) ] (2π) d E X e i<ξ,y> T λ j (θ)φ j (x + y/2, θ)φ j (x y/2, θ) dθ dy, j JI 2 vk ɛ := (2πi) 1 C zr k(h; ɛ, z)dz, 3 RI ɛ(h; n) := ( ) (2πi) 1 C z ɛ k r k (h; ɛ, z) dz. k n 2 there exists an orthogonal projection P ɛ I,n in L2 (X ) such that 1 E I (H ɛ ) PI ɛ,n C nɛ n, H ɛ [E I (H ɛ ) PI ɛ,n ] C n(h)ɛ n, 2 [H ɛ, PI ɛ,n ] C n(h)ɛ n, 3 for PI ɛ,n we have a development in powers of ɛ [0, ɛ 0] similar to the one of H ɛ E I (H ɛ ) at point 1. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

31 Corollary The main results For a N-fold degenerated isolated spectral band λ : T R we have that H ɛ E I (H ɛ ) = Op ɛ (λ)e I (H ɛ ) + ɛ k Op ɛ( vk) ɛ + ɛ n Op ɛ( RI ɛ (h; n)). 1 k n 1 Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

32 Theorem B The main results Under Hypothesis I, II, III and IV, there exists an orthonormal magnetic Wannier basis {Wγ,j ɛ } (γ,j) Γ J I such that: 1 E I (H ɛ ) = Wγ,j ɛ Wɛ γ,j, γ Γj J I 2 sup < x γ > m Wγ,j ɛ <, (γ, j) Γ J I, x X 3 Wα,j ɛ, Hɛ Wβ,k ɛ L 2 (X ) = Λ ɛ (α, β) µ jk (α β) + Cɛ for any (α, β) Γ Γ, (j, k) J I J I and ɛ [0, ɛ 0 ]. Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

33 The main results The case of constant magnetic field I. We consider satisfied Hypothesis I and II and a constant magnetic field B. We define the unitary mapping R Γ : L 2 (X ) l 2( Γ; L 2 (E) ) = l 2 (Γ) L 2 (E) given by the formula ( R Γ f ) (ˆx) := f (γ + ˆx). γ We denote by τ : Γ B ( l 2 (Γ) ) the restriction of the translations to l 2 (Γ). We consider the unitary transformation in L 2 (X ) defined by ( Υ A f ) (x) := Λ A ([x], x)f (x). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

34 The main results The case of constant magnetic field I. Theorem I Under the above assumptions we have that H A E I (H A ) = ( Υ A ) 1 R Γ M B (h) ( Υ A ) R Γ with M B (h) α,β := Λ A (α, β)h B I (α β) where H B I (γ) B ( L 2 (E) ) is defined by the following integral kernel: kh,i B (γ, ˆx, ŷ) := Φ B γ (ˆx, ŷ)k h,i (γ, ˆx, ŷ) Φ B γ (ˆx, ŷ) = exp { ( i/2) ( B, γ (ˆx + ŷ) + B, ˆx ŷ )}, k h,i (γ, ˆx, ŷ) := λ j (θ)φ j (ˆx + ŷ/2, θ)φ j (ˆx ŷ/2, θ) dθ. j JI T e i<θ,γ> Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

35 The main results The case of constant magnetic field II. Theorem II Under the above assumptions in the transverse gauge representation, we have the following convolution form for the matrix of the band Hamiltonian in the modified magnetic Wannier basis: W ɛ α,l, H ɛ Wβ,m ɛ = L 2 (X ) Λɛ (α, β) Wα β,l, ɛ H ɛ W0,m ɛ Λ ɛ (α, β)ĥ ɛ L 2 (X ) lm(α β). Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36

36 The main results Thank you! Radu Purice (IMAR) Semiclassical Analysis and Magnetic Fields Rennes, May 19, / 36