Symmetry breaking and restoration in the Ginzburg-Landau model of nematic liquid crystals

Transcription

1 Symmetry breaking and restoration in the Ginzburg-Landau model of nematic liquid crystals Micha l Kowalczyk joint project with M. Clerc and P. Smyrnelis (LAFER/DIF and LSPS/CMM). March 1, 2018

2 Experimental set up a) x z y V0 b) c) 0.5mm Figure: Experiment, vortices, numerical simulations, vortex lattices, picture from Barboza et.al.

3 From 3d to 2d near the Freédericksz transition Free energy (Oseen-Frank) K1 F = 2 ( n)2 + K 2 ( ) 2+ K 3 ( ) 2+ ɛ a n ( n) n ( n) ( E n) 2, where n, n = 1 is the director, K j > 0 elastic constants, ɛ a < 0 dielectric constant. We assume E = E z ẑ + E r ˆr [V 0 + αi (r)] ẑ + zα d dω I (r)ˆr, electric field, d thickness of the sample. Freédericksz s transition threshold V FT = K 3 π 2 /ε a αi is modified by the laser beam. Gradient flow of F γ t n = δf δ n At the transition point n ẑ. + n( n δf δ n ).

4 Near the transition we assume that where n = n x ( r, πz/d, t)) n y ( r, πz/d, t)) 1 (n 2 x + n 2 y ), n x ( r, ζ, t) = d 2 u 0 ( r, t) cos(πζ) + d 4 u 1 ( r, t)ϑ(πζ) +..., n y ( r, ζ, t) = d 2 v 0 ( r, t) cos(πζ) + d 4 v 1 ( r, t)ϑ(πζ) +..., n z ( r, ζ, t) = 1 (nx 2 + ny 2 ), and ϑ(πζ) is a function to be determined. Solvability condition for ϑ gives an equation for w 0 = u 0 + iv 0 (earlier version of this calculation by Frisch).

5 The amplitude aquation and the energy After suitable scaling the amplitude w 0 of the deviation of the director n from the ẑ direction satisfies where t A = 4 z z A + 4δ 2 z zā + 1 ɛ 2 (µ A 2) A + α ɛ f, (1) z = 1 2 ( x i y ), z = 1 2 ( x + i y ), 4 z z = µ(x) = e x 2 χ, f (x) = e iθ x e x 2 and δ = (K 1 K 2 )/(K 1 + K 2 ) [ 1, 1] (elastic anisotropy). From now on we will consider the stationary problem, δ = 0 and ɛ > 0 a small parameter and we denote by u the order parameter resulting from the amplitude equation.

6 The energy We check that u is a critical point of the energy ɛ E(u) = 2 u 2 1 2ɛ µ(x) u ɛ u 4 αre (f (x)ū) R 2 Above calculation leading from the Ossen-Frank energy to E is quite flexible and allows for complex or real valued order parameter, respectively describing vortices, domain walls or kinks. These are well known examples of topological defects, often modeled by the Ginzburg-Landau energy ɛ G(u) = 2 u ɛ (1 u 2 ) 2 where u is a real valued (kinks, domain walls) or complex valued order parameter (vortices).

7 Examples of standard defects Consider the one dimensional version of the model with ɛ = u x (1 u2 ) 2. The Euler-Lagrange equation R u xx + u(1 u 2 ) = 0 has a solution connecting 1 to 1 H(x) = tanh ( x 2 ). By the Bogomolnyi trick it follows that H minimizes the energy among all connections of the two different stable states.

8 Consider G(u) = R N ɛ 2 u ɛ (1 u 2 ) 2 with u real valued. Take ɛ = 1. Domain wall in two dimensions is represented by u(x, y) = H(x), y R N 1. The zero level set of this function is the hyperplane x = 0. The energy of the domain wall is infinite, and it is known that there is no finite energy domain wall in R N, when N > 1. It is however a solution to the Euler-Lagrange equation. Planar domain walls correspond to local behavior of minimizers of G when ɛ 0 in R N, at least when 1 < N < 8. This even happens when mass constraint is added to the theory.

9 Figure: From left two right: standard kink, soliton as an example of non topological defect, two dimensional domain wall.

10 Minimizing G(u) = R 2 ɛ 2 u ɛ (1 u 2 ) 2 for u C leads to u 1. Standard vortex of degree n Z is given by a function of the form u(r, θ) = e inθ U(r) where U + 1 r U n2 r 2 U + U(1 U2 ) = 0, U(0) = 0, U( ) = 1 r (0, ), This is the Euler-Lagrange equation in the class of radial functions. The energy of a degree n vortex is infinite but locally vortex defects are described by this standard profiles suitably rescaled.

11 Figure: Standard vortex of degree 1 is given by η(x) = U(r)e iθ in polar coordinates. From left two right: the modulus u(x) = U(r), plot of the vector field u(x) = U(r)(cos θ, sin θ).

12 The Thomas-Fermi limit and relation with the theory of Bose-Einstein condesates We recall E(u) = R 2 ɛ 2 u 2 1 2ɛ µ(x) u ɛ u 4 αre (f (x)ū). The functional E shares some similarities with the standard Ginzburg-Landau functional (α = 0, µ const), the Allen-Cahn functional (u R) and the Gross-Pitaevski functional with harmonic trapping a(x) (Bose-Einstein condensates) 1 F (u) = R 2 2 u [ ( u 2 4ɛ 2 a(x) ) 2 ( a (x) ) ] 2 2 Ωx (iu, u), a(x) = Λ x 2.

13 When α = 0 then E(u) = R ɛ 2 u 2 1 2ɛ µ(x)u ɛ u 4 and when Ω = 0 then 1 F (u) = R 2 2 u [ ( u 2 4ɛ 2 a(x) ) 2 ( a (x) ) ] 2 2. The global minimizers should be respectively: u = µ + and u = a + (this is the Thomas-Fermi limit of Bose-Einstein condensate). The problem is that both of this functions are not smooth at their zero level sets. It is known that in this case the true minimizers will exhibit a boundary layer behavior whose local profile is governed by the special solution of the second Painlevé equation discovered by Hastings-McLeod 80 such that y xy y 3 = 0, in R, y(x) 0, x, y(x) x x.

14 This phenomenon is also known as the corner layer and it is present in the context of the Gross-Pitaevski energy for the Bose-Einstein condensates: Aftalion-Blanc 04, Karali-Sourdis 15, as well as in many other problems, see for example, Alikakos-Bates-Cahn-Fife-Tanoglu 06, Alikakos-Fife-Fusco-Sourdis 07, Karali-Sourdis 12, Lassoued-Mironescu 91. For the Gross-Pitaevski energy when Ω = O( ln ɛ ) is below a certain critical value Ω 1 global minimizers are vortex free (Ignat-Millot 06, Aftalion-Jerrard-Royo Letelier 11), while at some other critical values Ω 2 > Ω 1 global minimizers have at least one vortex (Ignat-Millot 06), which looks locally like the radially symmetric degree ±1 solution to the Ginzburg-Landau equation.

15 One dimensional problem We will consider the simplest version of the energy E(u) with u : R R, ɛ > 0, α > 0: ɛ E(u) = 2 u x 2 1 2ɛ µ(x)u ɛ u 4 αf (x)u. (2) R µ(x) = e x 2 χ f (x) = 1 2 µ (x) ξ ξ χ

16 Global minimizers varying α with ɛ small fixed, a summary a) b) u(x,t) u(x,t) 0.6 Standard kink 0.6 Corner layer x 0.0 x c) d) u(x,t) 0.6 Left shadow kink 0.1 Right shadow kink u(x,t) x 0.0 x Giant kink μ(x) Figure: Four types of minimizers of the energy E. In panel a), b) and c) the thin oval line represents the curve ± max{ µ, 0}.

17 We will discuss in details various types of minimizers of the energy E as ε 0 and α [0, ). The Euler-Lagrange equation of E is ɛ 2 u xx + µ(x)u u 3 + ɛαf (x) = 0, x R. (3) Since we are interested in finite energy solutions of (3) it is natural to expect that, as ɛ 0 their behavior is governed by the roots of the cubic equation µ(x)y y 3 + ɛαf (x) = 0. (4) This equation has three or one real root depending on x.

18 y y r ε,+ ξ r ε,0 ξ x ξ ξ x r ε,- Figure: The roots of equation (4). Left: with α > 0; Right: with α = 0. Energy minimizer should connect the stable root branches r ɛ,± and we see that this may happen only in the vicinity of the points ±ξ (roots of µ(x) = 0) or of the origin (root on 1 2 µ (x) = f (x) = 0).

19 General properties of the global minimizer (M. Clerc, J. Dávila, M. Kowalczyk, P. Smyrnelis, E. Vidal-Henríquez, Calc. Var. PDEs, 2017.) Theorem (i) When α = 0, the global minimizer v of E has constant sign or is identically zero, is even and is unique up to changing v by v. (ii) Assuming that α > 0, the global minimizer v of E has at most one zero, denoted by x. Furthermore, v(x) > 0, x > x, and v(x) < 0, x < x. (iii) If α is bounded and ɛ α then x ξ + O( ɛ/α), where ξ is the positive zero of µ. (iv) When α > 2 then the global minimizer is the standard kink and when 0 < ɛ α < 2 the global minimizer is the shadow kink.

20 Local profile of the standard kink The standard kink (Fig. 5a) is an odd function and connects r ɛ,+ and r ɛ, at x = 0. To understand the local profile of this solution we assume that when x 0: In stretched variable s = x ɛ u(x) = v ( x ɛ ). µ(ɛs) = µ(0) + O(ɛ), f (ɛs) = ɛsf (0) + O(ɛ 2 ). Then, ignoring O(ɛ) terms, we see that v satisfies which implies v ss + µ(0)v v 3 = 0, v(s) = ( s µ(0) µ(0) tanh ). 2

21 Figure: From left two right: simulation of the standard kink (large α), detail at the origin, detail near the point µ(x) = 0.

22 The shadow kink As soon α > 0 the linear term in the energy forces a minimizer to change sign exactly once and if α is not too large the shadow kink appears near the points x = ±ξ, which are the roots of µ(x) = 0 (Fig. 5c). We write for x ξ The equation for v reads u(x) = ɛ 1 3 v ( x ξ v ss + µ 1 sv v 3 + αf (ξ) = 0, s (, ), (5) where µ 1 = µ (ξ) < 0. The asymptotic behavior of the shadow kink v at ± should be v(s) ± µ 1 s, s αf (ξ) v(s) µ 1 s, s. (6) ɛ 2 3 ).

23 Figure: From left two right: simulation of the shadow kink (small α), blown-up image of the shadow kink, solution near the right point where µ(x) = 0 (no transition).

24 The second Painlevé equation After scaling variables the limiting profile of the shadow kink is given by y xy 2y 3 = a, x R, a = αf (ξ) 2µ1 < 0, known as the second Painlevé equation. Painlevé transcedents are (singular in general) solutions of certain second order ODEs in the complex plane were discovered by Picard, Painlevé, Fuchs and Gambier around They have in some sense universal character (Deift) and play important role in the theory of integrable systems (Ablowitz-Segur conjecture) and in the theory of random matrices (Claeys, Kuijlaars, and Vanlessen).

25 We are interested in real and entire (non singular) solutions of y xy 2y 3 = a, x R and it turns out that: when a = 0 there is only one such solution known as the Hastings-McLeod solution (1980) { Ai(x), x y(x) x /2, x when 1 2 < a < 0 there is a unique, entire sign changing solution and when a 0 there is a unique, entire, positive solution (Claeys, Kuijlaars, and Vanlessen 06; Troy 17) y(x) { a x, x ± x /2, x

26 Symmetry breaking: standard and shadow vortices One dimensional model shows symmetry breaking between the standard vortex (odd) and the shadow vortex as the forcing, measured by α, increases. Now we consider the problem of minimizing ɛ E(u) = 2 u 2 1 2ɛ µ(x) u ɛ u 4 αre (f (x)ū), where R 2 and u : R 2 C. µ(x) = e x 2 χ, f (x) = e iθ x e x 2 Because this functional is similar to the standard Ginzburg-Landau functional ɛ 2 u ɛ (1 u 2 ) 2 we expect to see localized vortex structures.

27 Numerical simulation of optical vortices R. Barboza, U. Bortolozzo, M. G. Clerc,1 J. D. Da vila, M. Kowalczyk, S. Residori, and E. Vidal-Henrı quez, Phys Rev E, a) b) 100 c) 0.9 A Y Im(A) Y X 50 Y c j h b l b b b i i f b X - i o f k m e 30 g Y a k -30 j m d sx q i h o l X d g ψ Re(A) -50 d) 90 A X e-4 90 Figure: Standard vortex, large α. a) b) Y o e g i p o b n 0.00 b b q j d m c b f b X e i Y g f c k -30 b h j o m r n l xd a o q t s p X max t: s: r: q: p: o: n: m: l: k: j: i: h: g: f: e: d: c: b: a: min Figure: Shadow vortex, small α. d) A X b o Re(A) b c g h x Y -100 Y i j k b c Im(A) max r: q: p: o: n: m: l: k: j: i: h: g: f: e: d: c: b: a: min c) ψ 60 X 105

28 General properties of the global minimizer M. Clerc, M. Kowalczyk, P. Smyrnelis, J. Nonlinear Science, 2018 Theorem Assume that 0 < α(ɛ) < +, and lim ɛ 0 ɛ 1 3γ 2 γ [0, 2/3). Let ρ be the root of µ(r) = 0. ln α = 0 for some (i) For ɛ 1, the global minimizer v ɛ,α has at least one zero x ɛ such that x ɛ ρ + o(ɛ γ ). (7) (ii) If α = o(ɛ ln ɛ ) then any limit point l R 2 of the set of zeros of the global minimizer satisfies l = ρ. (iii) If α ɛ ln ɛ 2 then l = 0.

29 (a) (b) (c) 100. x x x x x x μ v 0.9 v 0.6 μ 0.9 μ v r r 0. r Figure: (a) The standard vortex described in Theorem 2 (iii) (b) The shadow vortex near the boundary of the set µ > 0 and (c) The shadow vortex described in Theorem 2 (ii) in case α = o(ɛ ln ɛ ). The upper panel shows the global minimizer v = (v 1, v 2 ) as a vector field in R 2. In the lower panel a radial section of v taken at the angle θ indicated in the upper right corner and compared with the Thomas-Fermi limit µ +. Numerical simulations where performed after rescaling the original spacial variable x x/ɛ.

30 Symmetry breaking and restoration Theorem (i) When α = 0 the global minimizer can be written as v(x) = (v rad ( x ), 0) with v rad C (R) positive. It is unique up to change of v by gv with g SO(2). (ii) Given ɛ > 0, there exists A > 0 such that for every α > A, the global minimizer v ɛ,α is unique and radial i.e. v(x) = v rad ( x ) x x. (iii) There are non radial minimizers for small, positive α. Radially symmetric critical points of the energy functional exist for all α 0, but Theorem 2 shows that they are not in general minimizers. When α varies radial symmetry of the global minimizer is instantly broken at α = 0 and then restored for large values of α. There is a hysterisis phenomenon.

31 Standard vortex Let α ɛ ln ɛ 2. We recall the standard Ginzburg-Landau vortex of degree one η(x) = U(r)e iθ. If v ɛ,a ( x ɛ ) = 0 and x ɛ l then up to a subsequence lim v ɛ,a( x ɛ + ɛs) µ(l)(g η)( µ(l)s), ɛ 0 in C 2 loc (R2 ), for some g O(2). Simply speaking, locally near the defect after suitable scaling the global minimizer looks like the standard Ginzburg-Landau vortex of degree 1.

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33 Boundary behavior of the minimizers, the shadow vortex With µ(x) = µ rad ( x ) x x let ρ be the root of µ rad (r) = 0 and let µ 1 := µ rad (ρ) < 0. For every ξ = ρe iθ, we consider the local coordinates s = (s 1, s 2 ) in the basis (e iθ, ie iθ ), and the rescaled minimizers: ( w ɛ,α (s) = 2 1/2 ( µ 1 ɛ) 1/3 v ɛ,α ξ + ɛ 2/3 s ) ( µ 1 ) 1/3. As ɛ 0, the function w ɛ,α converges in C 2 loc (R2, R 2 ) up to subsequence, to a bounded in the half-planes [s 0, ) R solution of y(s) s 1 y(s) 2 y(s) 2 y(s) a = 0, s = (s 1, s 2 ) R 2, (8) α(ɛ)f (ξ) with a = lim ɛ 0 2µ1. The boundary layer behavior of the minimizers is determined by solutions of the generalized second Painlevé equation.

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35 Entire solutions of are also solutions of y xy 2y 3 a = 0, x R, y(s) s 1 y(s) 2 y(s) 2 y(s) a = 0, s = (s 1, s 2 ) R 2 but they do not describe the shadow vortex. It is a non trivial open problem to prove that the generalized Painlevé equation has an entire solution such that y(x) has exactly one zero in R 2. It is not even clear that there other than one dimensional solutions but next we will use the energy functional E to show that there are real valued solutions that depend truly on two variables.

36 Phase transitions and the Painlevé equation Now we consider the problem of minimizing ɛ E(u) = 2 u 2 1 2ɛ µ(x) u ɛ u 4 αf 1 (x)u, R 2 but assuming u : R 2 R and µ(x) = e x 2 χ, f 1 (x) = 1 2 x 1 µ(x). This functional resembles the Allen-Cahn functional ɛ 2 u 2 + W (u, x) where for each fixed x the function W (, x) is a double well, not necessarily balanced, potential e.g. W (u, x) = u 1 (s 2 1)(s g(x)) ds, 1 < g(x) < 1. It is known that in the Allen-Cahn theory transitions occur along the set of x where the wells of W (u, x) have equal depths (bistable region).

37 Transitions in the monostable region M. Clerc, M. Kowalczyk. P. Smyrnelis, in progress Theorem Let Z be the set of limit points of the set of zeros of v ɛ,α as ɛ 0. The following statements hold. (i) When α = 0 then Z =. (ii) There exists a constant α > 0 such that for all α (0, α ), we have up to change of (x 1, x 2 ) ( x 1, x 2 ): Z = {x 1 < 0, x = ρ} {x 1 = 0, x 2 ρ}, (iii) For all α > α we have Z = {x 1 = 0} and the global minimizer v ɛ,α is odd with respect to x 1.

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40 Conclusions The modified (by forcing) Ginzburg-Landau theory we proposed is a good model of optical vortices, predicting in particular the phenomenon of symmetry breaking as the forcing parameter (light intensity) increases. The shadow vortex, which is a new type of topological defect is well described by our theory. Its profile is given by special solutions of the (generalized) second Painlevé equation.