Superconductivity in domains with corners

Transcription

1 Superconductivity in domains with corners Virginie BONNAILLIE-NOËL IRMAR, Université Rennes 1 and ENS Cachan Bretagne Each China Normal University, Shanghai

2 Outline 1. Introduction 2. Linear spectral problem Model domains: plane, half-plane, sectors Smooth domains Polygons Curvilinear polygons 3. Applications to the onset of superconductivity Asymptotic of the critical field H C3 localtion of nucleation of superconductivity

3 Introduction Framework Ω R 2 : a bounded simply connected domain with Lipschitz boundary. Ginzburg-Landau functional: E κ,h [ψ,a] = { p κha ψ 2 κ 2 ψ 2 + κ2 2 ψ 4} dx+κ 2 H 2 curla 1 2 dx, R 2 Ω with ψ W 1,2 (Ω; C), A Ḣ1 F,div, and p A = ( i A). We denote Ḣ 1 F,div = F + Ḣ 1 div, Ḣ 1 div = {A Ḣ 1 (R 2, R 2 ) diva = 0}. F(x 1, x 2 ) = 1 2 ( x 2, x 1 )

4 Euler-Lagrange equations Minimizers (ψ,a) W 1,2 (Ω) Ḣ 1 F,div satisfy p 2 κhaψ = κ 2 (1 ψ 2 )ψ in Ω, (1a) curl 2 A = { i 2κH (ψ ψ ψ ψ) ψ 2 A } 1 Ω (x) in R 2, (1b) (p κha ψ) ν = 0 on Ω. (1c) For all κ, H > 0, the functional E κ,h has a minimizer. Giorgi-Phillips: for κ fixed and H sufficiently large, the unique solution is (ψ,a) = (0,F) the superconductivity is destroyed.

5 Critical field(s) H C3 (κ) = inf{h > 0 : (0,F) is a minimizer of E κ,h }. H C3 (κ) = inf{h > 0 : (0,F) is the unique minimizer of E κ,h for all H > H}. References: Bernoff-Sternberg, Fournais-Helffer, Helffer-Morame, Helffer-Pan, Jadallah, Lu-Pan, del Pino-Felmer-Sternberg,... Aim: asymptotics of the critical fields for domains with corners in terms of linear spectral data estimates on the location of nucleation of superconductivity.

6 Linear spectral problem Notation Ω open bounded domain in R 2 A smooth magnetic potential B = curl A magnetic field associated with A h semi-classical parameter (h 0) Assumption: B > 0 P h : Neumann realization on Ω for the Schrödinger operator (h ia) 2 p h : sesquilinear form associated with P h defined on H 1 (Ω) by p h (u, v) = (h ia)u(x) (h ia)v(x) dx Ω Aim: investigate the behavior of the eigenpairs (µ h,n, u h,n ) of P h as h 0

7 Model operators Ω sector Ω Ω half-plane Ω plane F(X) = 1 2 ( X 2, X 1 ) : magnetic potential with constant magnetic field ( if) 2 on the plane, the half-plane and sectors.

8 Plane and half-plane Proposition The smallest eigenvalue of the Neumann realization of ( if) 2 on R 2 is equal to 1 (Landau). 2. The bottom of the spectrum of ( if) 2 on R R + is equal to Θ (Dauge-Helffer, 1993; Bolley-Helffer, 1993).

9 Applications for smooth domains and B > 0 Bernoff-Sternberg (1998), Helffer-Morame (96, 01), Lu-Pan (99, 00), del Pino-Felmer-Sternberg (00), Helffer-Pan (03), Fournais-Helffer (05, 06) µ h,1 h Θ 0 as h 0 Under a simplicity assumption, u h,1 concentrates at scale h around the point of maximal curvature as h 0 For B = 1, asymptotics of µ h,n, estimate of the decay of u h,n, determination of the critical field H C3 (κ), location of nucleation H C3 (κ) = H C3 (κ) = H C3 (κ) = κ Θ 0 + O(1) as κ +.

10 Sector in R 2 X = (X 1, X 2 ) the Cartesian coordinates in R 2 G α = {X R 2, X 1 > 0, 0 < X 2 < X 1 tanα} Let Q α be the Neumann realization of ( if) 2 on G α Q α = + i(x 1 X2 X 2 X1 ) X 2 µ k (α) : k-th smallest eigenvalue of Q α Quarter plane: Jadallah (01), Pan (02)

11 Spectrum of Q α 1. Bottom of the spectrum The infimum of the essential spectrum of Q α is equal to Θ 0. For all α (0, π/2], µ 1 (α) < Θ 0. α (0, 2π), K α := the largest integer such that µ Kα (α) < Θ 0 2. Decay of the eigenfunctions Let α > 0 be such that K α > 0. Let 0 < k K α and Ψ α k be a normalized eigenfunction associated with µ k (α). Then ǫ > 0, C ǫ,α : e Θ0 µ k (α) ǫ X Ψ α k (x) C ǫ,α V(q α ) with u 2 V(q α ) = u 2 L 2 (G α ) + ( if)u 2 L 2 (G α ).

12 Numerical computations Modulus of the first eigenfunction for several angles

13 Numerical computations Estimates for the first eigenvalue according to the opening µ 1 (α) Essential spectrum Upper bound Lower bound Numerical estimates α/π Conjecture: µ 1 increasing from (0, π] onto (0, Θ 0 ], equal to Θ 0 on [π, 2π)

14 Schrödinger operator with constant magnetic field in a polygonal domain (with M. Dauge, IRMAR, France) A = F Construction of quasi-modes Ω convex polygon Σ set of vertices s of Ω α s angle at s G α s sector in R 2 with opening α s Let s Σ and k 1 be such that µ k (α s ) < Θ 0 Ψ α s k a normalized eigenfunction of Qα s on G α s for µ k (α s )

15 Scaling X = x h to link Q α s = ( if) 2 with P h = (h if) 2 on G α s x Ψ α s k ( x h ) eigenfunction of P h on G α s associated with hµ k (α s ) Translation and rotation G s to send G α s on G s Ω which Ω coincides with Ω around s α s G α s β s s 0 ψ h,s,k (x) = 1 ( ) i exp h 2h x s Ψ α s k ( ) Rs (x s) h eigenfunction of P h on G s

16 Cut-off For each vertex s Σ, we denote by ρ s the distance to the other vertices : ρ s = dist(s, Σ \ {s}) 0 if x / B(s, ρ s ) Smooth cut-off function: χ s (x) = 1 if x B(s, ρ ) with ρ < ρ s Quasi-mode defined on Ω x ψ h,s,k (x) = χ s (x) ψ h,s,k (x)

17 Properties of quasi-modes For any ε > 0, there exists C ε such that Norm: ( ) 1 ψ h,s,k 2 C ε exp 2 ρ Θ 0 µ k (α s ) ε h Rayleigh quotient: ( ) p h (ψ h,s,k, ψ h,s,k ) ψ h,s,k 2 hµ k (α s ) C ε exp 2 ρ Θ 0 µ k (α s ) ε h Approximation of the eigenpair equation: ( ) P h ψ h,s,k hµ k (α s )ψ h,s,k C ε exp ρ Θ 0 µ k (α s ) ε h

18 Tubular estimates for eigenvalues µ h,n the n-th eigenvalue of P h repeated according to multiplicity λ n the n-th eigenvalue of Q α s repeated according to multiplicity s Σ and given by the max-min principle K Ω the largest integer such that λ KΩ < Θ 0 (K Ω = s Σ K α s ) Let n K Ω, Σ n = { } s Σ, λ n is an eigenvalue for Q α s r(λ n ) = min s Σn d(s, Σ \ {s}) Theorem 2. For any ε > 0, there exists C ε s. t. ( µ h,1 hλ 1 + C ε exp 2 (r(λ 1 ) ) ) Θ 0 λ 1 ε h µ h,n hλ n C ε exp ( 1 h (r(λ n ) Θ 0 λ n ε) ), n K Ω

19 Estimates for clusters of eigenspaces Repetitions in {λ 1,, λ KΩ } gather of µ h,n into clusters (µ h,n, u h,n ) the n-th eigenpair of P h {Λ 1 <... < Λ M } the set of disctinct values in {λ 1,...,λ KΩ } Let m M m-th cluster of eigenspaces for P h : F h,m = span{u h,n for any n s. t. λ n = Λ m } Corresponding cluster of quasi-modes (χ s = 1 on B(s, r(λ m ) δ)): E h,m = span{ψ h,s,k = χ s ψh,s,k for any s Σ, k 1 s. t. µ k (α s ) = Λ m } Theorem 3. For any ε > 0, there exists C ε s. t. ( d(e h,m ; F h,m ) C ε exp (r(λ m) δ) Θ 0 Λ m ε ), m M h where d(e, F) = Π E Π F Π E H with Π E the orthogonal projection on E

20 Numerical simulations (with M. Dauge, D. Martin and G. Vial, IRMAR, France) Two-scale structure: 1. a corner layer at scale h, 2. an oscillatory term at scale h. high order finite element method (Q 10 ). Compute eigenpairs with the FEM code MELINA On a square Ω = ( 1, 1) ( 1, 1) λ 1 = λ 2 = λ 3 = λ Θ

21 Square Modulus and phasis of first four eigenfunctions, h = 0.1

22 Modulus and phasis of first four eigenfunctions, h = 0.08

23 Modulus and phasis of first four eigenfunctions, h = 0.06

24 Modulus and phasis of first four eigenfunctions, h = 0.04

25 Modulus and phasis of first four eigenfunctions, h = 0.02

26 Modulus and phasis of first four eigenfunctions, h = 0.01

27 Modulus and phasis of eigenfunctions 5-8, h = 0.1

28 Modulus and phasis of eigenfunctions 5-8, h = 0.08

29 Modulus and phasis of eigenfunctions 5-8, h = 0.06

30 Modulus and phasis of eigenfunctions 5-8, h = 0.04

31 Modulus and phasis of eigenfunctions 5-8, h = 0.02

32 Modulus and phasis of eigenfunctions 5-8, h = 0.01

33 Tunneling effect Exponential tube: λ 1 ± C exp ( 2 Θ 0 λ 1 / ) h h 1 µ h,n versus h 1

34 Eigenmodes on polygon domain Rhombus λ 1 = λ 2 < λ 3 = λ 4 < Θ 0 µ 2 (α) = Θ

35 Rhombus Modulus and phasis of first six eigenfunctions, h = 0.1

36 Modulus and phasis of first six eigenfunctions, h = 0.08

37 Modulus and phasis of first six eigenfunctions, h = 0.06

38 Modulus and phasis of first six eigenfunctions, h = 0.04

39 Modulus and phasis of first six eigenfunctions, h = 0.02

40 Modulus and phasis of first six eigenfunctions, h = 0.01

41 Eigenmodes on polygon domain Trapezoid λ 1 < λ 2 = λ 3 < λ 4 < Θ 0 µ 2 (α) = Θ

42 Trapezoid Modulus and phasis of first four eigenfunctions, h = 0.1

43 Modulus and phasis of first four eigenfunctions, h = 0.08

44 Modulus and phasis of first four eigenfunctions, h = 0.06

45 Modulus and phasis of first four eigenfunctions, h = 0.04

46 Modulus and phasis of first four eigenfunctions, h = 0.02

47 Modulus and phasis of first four eigenfunctions, h = 0.01

48 Eigenmodes on non convex polygon domain L-shape λ 1 = λ 2 = λ 3 = λ 4 = λ 5 < Θ 0 µ 1 (3π/2) = Θ 0, µ 2 (π/2) = Θ

49 L-shape Modulus and phasis of eigenfunctions 1, 3, 5, h = 0.02

50 Schrödinger operator in a curvilinear polygon Ω bounded curvilinear polygon with a piecewise smooth boundary p h (u, v) = Ω (h ia)u (h ia)v dx, u, v H 1 (Ω) P h = (h ia) 2 on D(P h ) = {u H 2 (Ω), ν (h ia)u Ω = 0} b = inf B(x) and b = inf B(x) x Ω x Ω µ h,n the n-th eigenvalue of P h counted with multiplicity λ n the n-th eigenvalue of s Σ B(s)Q α s counted with multiplicity

51 Assumptions 1. 0 < α s < π for any s Σ K Ω,B the largest integer s. t. λ KΩ,B < min(θ 0 b, b) 2. For n K Ω,B and s Σ, k K αs s. t. B(s)µ k (α s ) = λ n, µ k (α s ) is a simple eigenvalue of Q α s Construction of quasi-modes 1. Change of variables around a corner to be reduced to a sector 2. Gauge transform: to have a magnetic field close to 1 3. Scaling and Taylor expansion: to introduce a formal series expansion 4. Solutions of the formal series equation at any order

52 Eigenvalue asymptotic Theorem 4. Let L 2, E L (h) be the set of the K Ω,B smallest asymptotics at the order L given by the construction of the formal series equation and arranged in increasing order : E L (h) = {µ [L] h,s,k, s Σ, k K α s such that B(s)µ k (α s ) < min(b Θ 0, b)}. Let n K Ω,B. There exists h 0, s Σ and k K αs such that µ [L] h,s,k n-th element of E L (h) for any h (0, h 0 ). We have is the There holds µ [L] h,s,k = hb(s) L l=0 h l/2 µ l s,k with µ 0 s,k = µ k (α s ). µ h,n µ [L] L+1 h,s,k Ch 2, h (0, h0 ).

53 Eigenspaces {Λ 1 <... < Λ M } the set of distinct values in {λ 1,...,λ N } For any n N, (µ h,n, u h,n ) the n-th eigenpair of P h For any m M, m-th cluster of eigenspaces of P h F h,m = span{u h,n for any n such that λ n = Λ m } and the corresponding cluster of quasi-modes for any L N E [L] h,m = span{φ[l] h,s,k for any s Σ, k 1 such that B(s)µ k(α s ) = Λ m } Theorem 5. For any m M and L 2, there exists C > 0 such that d(f h,m, E [L] L 1 h,m ) Ch 2

54 Tunneling effect on a square and a curved square

55 Modulus and phasis of first four eigenfunctions, h = 0.1

56 Modulus and phasis of first four eigenfunctions, h = 0.08

57 Modulus and phasis of first four eigenfunctions, h = 0.06

58 Modulus and phasis of first four eigenfunctions, h = 0.04

59 Modulus and phasis of first four eigenfunctions, h = 0.02

60 Modulus and phasis of first four eigenfunctions, h = 0.01

61 Applications to the onset of superconductivity (with S. Fournais, Univ. Aarhus, Denmark) Notation Ω bounded, simply connected curvilinear polygon Σ set of vertices, N = Σ > 0 α s H Ω (B) λ n,ω (B) angle at the vertex s (measured towards the interior) the Neumann realization associated with Ω ( i BF)u 2 dx, n th eigenvalue of H Ω (B) counted with multiplicity Assumptions: for all s Σ 1. µ 1 (α s ) < Θ 0 2. α s (0, π) Λ 1 := min s Σ µ 1(α s )

62 Asymptotic for H C3 (κ) Theorem 6. There exists κ 0 > 0 such that if κ κ 0 then the equation has a unique solution H = H lin C 3 (κ). λ 1,Ω (κh) = κ 2, Furthermore there exists a real valued sequence (η j ) j 1 such that H lin C 3 (κ) = κ Λ 1 (1 + in the sense of asymptotics series. η j κ j), If κ 0 is chosen sufficiently large, then for κ κ 0, the critical fields coincide and satisfy j=1 H C3 (κ) = H C3 (κ) = H lin C 3 (κ).

63 Proof. The limits of λ 1,+(B) and λ 1, (B) as B + exist, are equal and we have lim B + λ 1,+(B) = lim B + λ 1, (B) = Λ 1. Therefore, B λ 1 (B) is strictly increasing for large B. isomorphism for κ large enough. asymptotics series We prove that the following statements are equivalent for κ large: 1. There exists a minimizer (ψ,a) of E κ,h with ψ The parameters κ, H satisfy κ 2 λ 1 (κh) > 0. We need to prove some new results about the linear problem

64 Remarks If Σ =, asymptotics given by Fournais-Helffer H C3 (κ) = κ Θ C 1k max Θ0 κ C 3k2 1 2 κ κ 7 4 η j κ j 4. j=0 For a rectangle, leading order term given by Pan H C3 (κ) = κ µ 1 ( π 2 ) + O(1).

65 Location of the onset of superconductivity In smooth domains: H C3 (κ) = κ Θ 0 + O(1) + location of the minimizers around points with maximum curvature In domains with corners: H C3 (κ) = κ Λ 1 + O(1) with Λ 1 < Θ 0 the corners change the leading order term of H C3 (κ). superconductivity is dominated by the corners in the regime parameter κ Θ 0 H H C3 (κ)

66 Theorem 7. Let µ > 0 satisfy min s Σ µ 1 (α s ) < µ < Θ 0 and define Σ := {s Σ µ 1 (α s ) µ}. There exist constants κ 0, M, C, ǫ > 0 such that if κ κ 0, H κ µ 1, and (ψ,a) is a minimizer of E κ,h, then ( e ǫ κhdist(x,σ ) ψ(x) ) Ω κh p κhaψ(x) 2 dx C {x: κhdist(x,σ ) M} ψ(x) 2 dx.

67 Contribution of the corners Let α (0, π) be s.t. µ 1 (α) < Θ 0. Define, for µ 1, µ 2 > 0, Jµ α 1,µ 2 [ψ] = Γ α { ( i F)ψ 2 µ 1 ψ 2 + µ 2 2 ψ 4} dx, with domain {ψ L 2 (Γ α ) ( i F)ψ L 2 (Γ α )}. Define also the corresponding ground state energy E α µ 1,µ 2 := inf J α µ 1,µ 2 [ψ]. Theorem 8. Suppose κ H(κ) µ (0, Θ 0) as κ. Let (ψ,a) = (ψ,a) κ,h(κ) be a minimizer of E κ,h(κ). Then E κ,h(κ) [ψ,a] s Σ E α s µ,µ as κ

68 Agmon estimates near corners for the linear problem Lemma 9. Let δ > 0. Then there exist constants M 0, B 0 > 0 such that if B B 0 then H(B) satisfies the operator inequality H(B) U B, where U B is the potential given by (µ 1 (α s ) δ)b, dist(x, s) M 0 / B, U B (x) := (Θ 0 δ)b, dist(x, Σ) > M 0 / B, dist(x, Ω) M 0 / B, (1 δ)b, dist(x, Ω) > M 0 / B. Theorem 10. Let ψ B be the ground state eigenfunction of H(B). Then there exist constants ǫ, C, B 0 > 0 such that e ǫ Bdist(x,Σ) { ψ B (x) 2 +B 1 p BF ψ B (x) 2} dx C ψ B 2 2, for all B B 0.

69 Basic estimates Let (ψ,a) a minimizer of E κ,h, then ψ 1 p κha ψ 2 κ ψ 2 H curla 1 2 ψ 2 ψ 2 4 ψ 2 Lemma 11. Let Ω be a bounded domain with Lipschitz boundary and let (ψ,a) be a (weak) solution to (1). Then curl (A F) = 0 on the unbounded component of R 2 \ Ω. Lemma 12. There exists a constant C 0 (depending only on Ω) such that if (ψ,a) is a (weak) solution of the Ginzburg-Landau equations (1), then A F 2 C 0 curla 1 R 2 dx, 2 Ω A F 2 W 1,2 (Ω) C 0 R 2 curla 1 2 dx.

70 Non-linear Agmon estimates Theorem 13 (Weak decay estimate). There exist C, C > 0, such that if (ψ,a) κ,h is a minimizer of E κ,h with κ(h κ) 1/2, then ψ 2 2 C { κ(h κ)dist(x, Ω) 1} ψ(x) 2 dx C κ(h κ).

71 Theorem 14 (Decay estimate on the boundary). For µ (Λ 1, Θ 0 ), define Σ := {s Σ µ 1 (α s ) µ}, and b := inf s Σ\Σ {µ 1(α s ) µ}. (in the case Σ = Σ, we set b := Θ 0 µ). There exist κ 0, C, C, M > 0, such that if (ψ,a) κ,h is a minimizer of E κ,h with H κ µ 1, κ κ 0, then ψ 2 2 C ψ(x) 2 dx C κ. 2 {κ dist(x,σ ) M}