Existence and Dynamics of Abrikosov Lattices I.M.Sigal based on the - PowerPoint PPT Presentation

Existence and Dynamics of Abrikosov Lattices I.M.Sigal based on the joint work with T. Tzaneteas Discussions with J¨ urg Fr¨ ohlich and Stephen Gustafson Western States Meeting, Caltech, February 2015

Ginzburg-Landau Equations Equilibrium states of superconductors (macroscopically) and of the U (1) Higgs model of particle physics are described by the Ginzburg-Landau equations: − ∆ A Ψ = κ 2 (1 − | Ψ | 2 )Ψ curl 2 A = Im(¯ Ψ ∇ A Ψ) where (Ψ , A ) : R d → C × R d , d = 2 , 3, ∇ A = ∇ − iA , ∆ A = ∇ 2 A , the covariant derivative and covariant Laplacian, respectively, and κ is the Ginzburg-Landau material constant.

Origin of Ginzburg-Landau Equations Superconductivity . Ψ : R d → C is called the order parameter ; | Ψ | 2 gives the density of (Cooper pairs of) superconducting electrons. A : R d → R d is the magnetic potential. Im(¯ Ψ ∇ A Ψ) is the superconducting current. Particle physics . Ψ and A are the Higgs and U (1) gauge (electro-magnetic) fields, respectively. (Part of Weinberg - Salam model of electro-weak interactions/ a standard model.) Geometrically, A is a connection on the principal U (1)- bundle R 2 × U (1), and Ψ, a section of the associated bundle. Similar equations appear in superfluidity, Bose-Einstein condensation and fractional quantum Hall effect.

Type I and II Superconductors Two types of superconductors: √ κ < 1 / 2: Type I superconductors, exhibit first-order phase transitions from the non-superconducting state to the superconducting state (essentially, all pure metals); √ κ > 1 / 2: Type II superconductors, exhibit second-order phase transitions and the formation of vortex lattices (dirty metals and alloys). √ For κ = 1 / 2, Bogomolnyi has shown that the Ginzburg-Landau equations are equivalent to a pair of first-order equations. Using this Taubes described completely solutions of a given degree.

Abrikosov Vortex Lattice States A pair (Ψ , A ) for which all the physical characteristics J ( x ) := Im(¯ | Ψ | 2 , B ( x ) := curl A ( x ) , Ψ ∇ A Ψ) are doubly periodic with respect to a lattice λ is called the Abrikosov (vortex) lattice state . � Quantization of magnetic flux: ω curl A = 2 π deg(Ψ) ∈ 2 π Z , where ω be an elementary cell of the lattice λ .

Existence of Abrikosov Lattices Let H c 1 and H c 2 = κ 2 be the 1st and 2nd critical magnetic fields and let ω be an elementary cell of the lattice λ . Theorem (Existence for high magnetic fields) � ≪ 1 and ( ∗ ) , ∃ an Abrikosov � � 2 π | ω | − κ 2 � For for every λ satisfying lattice sol., with this λ and 1 ω curl A = 2 π � b := | ω | (magnetic flux quantization). | ω | Theorem (Energy for high magnetic fields) √ If κ > 1 / 2 (Type II superconductors), then the minimum of the average energy per cell is achieved for the hexagonal lattice. Theorem (Existence for low magnetic fields) For every λ , with | ω | sufficiently large, ∃ an Abrikosov lattice 1 ω curl A = 2 π � solution, with this λ and b := | ω | . | ω |

References - Aver. magn. field ≈ H c 2 = κ 2 . Existence for ( ∗ ) b < κ 2 and κ > 1 2 : Odeh, Barany - √ Golubitsky - Tursky, Dutour, Tzaneteas - IMS Tzaneteas - IMS: Existence for ( ∗ ) b < κ 2 and κ > κ c ( λ ) or b > κ 2 and κ < κ c ( λ ), where κ c ( λ ) is a new threshold in κ : � � � 1 1 1 κ c ( λ ) := 1 − ( < 2 ). √ 2 β ( λ ) Energy minim. by triangular lattices: Dutour, Tzaneteas - IMS, using results of Aftalion - Blanc - Nier, Nonnenmacher - Voros. - Aver. magn. field ≈ H c 1 ( | ω | → ∞ ). Existence: Aydi - Sandier and others ( κ → ∞ ) and Tzaneteas - IMS (all κ ’s).

Time-Dependent Eqns. Superconductivity In the leading approximation the evolution of a superconductor is described by the gradient-flow-type equations γ ( ∂ t + i Φ)Ψ = ∆ A Ψ + κ 2 (1 − | Ψ | 2 )Ψ σ ( ∂ t A − ∇ Φ) = − curl 2 A + Im (¯ Ψ ∇ A Ψ) , Re γ ≥ 0, the time-dependent Ginzburg-Landau equations or the Gorkov-Eliashberg-Schmidt equations . (Earlier versions: Bardeen and Stephen and Anderson, Luttinger and Werthamer.) The last equation comes from two Maxwell equations, with − ∂ t E neglected, (Amp` ere’s and Faraday’s laws) and the relations J = J s + J n , where J s = Im(Ψ ∇ A Ψ), and J n = σ E .

Time-Dependent Eqns. U (1) Higgs Model The time-dependent U (1) Higgs model is described by U (1) − Higgs (or Maxwell-Higgs) equations (Φ = 0) ( ∂ t + i Φ) 2 Ψ = ∆ A Ψ + κ 2 (1 − | Ψ | 2 )Ψ ∂ t ( ∂ t A + ∇ Φ) = − curl 2 A + Im(¯ Ψ ∇ A Ψ) , coupled (covariant) wave equations describing the U (1)-gauge Higgs model of elementary particle physics. In what follows we use the temporal gauge Φ = 0.

Stability of Abrikosov Lattices Let (Ψ λ , A λ ) = Abrikosov lattice solution specified by a lattice λ and E Q (Ψ , A ) = Ginzburg-Landau energy functional . Finite-energy perturbations: perturbations satisfying, for some λ , � � lim E Q (Ψ , A ) − E Q (Ψ λ , A λ ) < ∞ . (1) Q → R 2 Theorem (Tzaneteas - IMS) 1 ω curl A = 2 π � Let b := | ω | ≈ H c 2 (high magnetic fields) and | ω | the i. c. satisfy (1) and (Ψ 0 ( − x ) , − A 0 ( − x )) = (Ψ 0 ( x ) , A 0 ( x )) . There is γ ( λ ) s.t. the Abrikosov vortex lattice solutions are 1 (i) asymptotically stable if κ > 2 and γ ( λ ) > 0 ; √ 1 1 (ii) unstable if either κ < 2 or κ > 2 and γ ( λ ) < 0 . √ √

Gamma Function Let λ = r ( Z + τ Z ), r > 0, τ ∈ C , Im τ > 0, and γ ( τ ) = γ ( λ ). Then the function γ ( τ ) is invariant under modular group SL (2 , Z ) e strip, Π + / SL (2 , Z ), and therefore can be reduced to the Poincar´ STRUCTURE OF THE GROUND STATE OF THE ELECTROWEAK. . . 45 3843 TABLE II. Sample of values for u (At/'(/eB, r) for lattices of par allelograms. rrrr~rrf 3$ 1! Wr~'i'rl l vq =0 ~rrrrrrrr JN/', t/eB 0. 25 0. 5 ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr 0. 5 1. 000 26 1. 1. ~rrrrrrrr 000 22 000 18 ~rrrrrrrr ~rrrrr rrr ~rrrrrrrr 1. 1. 1. 002 02 ~rrrrrrrr 002 26 002 20 ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr 1. 008 01 1. 1. 007 69 007 93 1. 003 30 1. 002 92 1. 002 34 ~rrrrrrra ~rrrrrrra 1. 024 28 1. 023 70 1. 021 96 ~rrrrrrra ~rrrrrrrr 1. 073 64 1. 073 02 1. 071 17 ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr 1. 5 1. 012 19 1. 010 89 1. 008 94 ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr ~rrrrrrrr 1. 072 60 1. 071 05 1. 066 42 ~rrrrrrrr grrrrrrrr ~rrr 1. 189 62 1. 188 27 1. 184 23 1. 026 59 1. 02401 1. 020 15 1. 130 93 1. 128 43 1. 120 89 1. 305 52 1. 303 64 1. 297 94 1. 061 25 1. 056 28 1. 048 90 1. 229 95 1. 226 21 1. 214 89 -1 -05 05 0 1 1. 47005 1. 467 61 1. 460 24 FIG. 1. The hatched area is the semifundamental domain of 1. 180 34 1. 171 95 1. 159 60 to the set of parameters of inequivalent ~ space corresponding 1. 424 80 1.419 71 1. 404 24 Symmetries: γ ( − ¯ τ ) = γ ( τ ) and γ (1 − ¯ lattice solutions. The solid squares indicate τ ) = γ ( τ ) points sampled in 1. 732 61 1. 729 61 1. 720 53 Table II. ⇒ critical points at τ = e i π/ 2 and τ = e i π/ 3 f d'p'Ip'I& (Atlp'I } f (p}l'IIv (p'+p}l' d'pllv cell (84} V(At } Work in progress: Estimating γ ( τ ) and checking the critical points. f I IV+ (p) I'd'p So far we have γ ( e i π/ 3 ) > 0 (numerics). only of the geometrical w. In this lim- function parameter with the parameter p„ in the Ginzburg- it, v coincides The integrals were carried out analytically [15]. Using eBA =2m, one ob. - of type-II Eqs. (47) and (48) and the condition Landau theory superconductivity near the point H, z [7]. The values of u in Table II for transition tains M=~ u =1. 18034 for and IvI=1, the and square v =1. 15960 for the (85) hexagonal lattice, agree with the values of p„calculated by Abrikosov [5] for the square „JK'/eB+(2~/~1 }1k~ Il— and and by Kleiner, Roth, and Autler [16] for the hexagonal V(At, r) =- d lattice. 1 dAt „, This treatment of the electroweak transition is phase of the Ginzburg-Landau mathematically a generalization by the Z theory in that the quartic interaction mediated X exp — . (8— and 4 fields is nonlocal. Ikr l 6) The propagator, in coordinate I space, for the respective interactions is given by the func- tion (I/2~)ICO(Mlp — p'I), where the inverse of the mass The sum converges very rapidly. M ( =Mz, M~ } determines parameter the range of the in- We have scanned this function for ~ in the numerically teraction. region 2). It has the following for all positive properties Our analysis supports the conclusion that the hexago- values of At: (i) For fixed nal lattice with k = 1 has the lowest average energy densi- I~l it decreases monotonically with rz, (ii} for fixed ~z it increases monotonically with ty. This result is valid for more general interac- quartic IvI =1, r„=0. 5 that corresponds The point to the tions of the form of this f d'p' f d'p hexagonal lattice gives the minimum function. I ~l = 1, rz =0, corresponding I Iv (p'}I'&( Ip' — pl ) I Iv (p) I' The point to the square lat- (87) tice, is a saddle point. where %'(p)= f o g(At)KO(Atp}dAt, with g(At) ~0 for The function V(At, v ) behaves as all A, . )v (At /YeB, r) (2eB/At ACKNOWLEDGMENTS v is a slowly varying function of the arguments where and v (0, ~} =— 1. of of A sample numerical values We would like to thank Feza Gursey, Hubert Saleur, v(Atl+eB, r) is given in Table II. The limit At~ao for useful dis- Subir Sachdev, and Charles Sommerfield corresponds to a local interaction and v becomes a I IVI cussions.