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Vortex dynamics in d-wave superconductors
Physica D 149 (2001) 293–305
Vortex dynamics in d-wave superconductors Tai-Chia Lin∗ Department of Mathematics, National Chung-Cheng University, Minghsiung, Chiayi 621, Taiwan Received 9 November 1999; received in revised form 2 October 2000; accepted 27 October 2000 Communicated by C.K.R.T. Jones
Abstract We study the dynamics of vortices in the time-dependent Ginzburg–Landau equation for d-wave superconductors. Under suitable assumptions on the coefficients of the time-dependent Ginzburg–Landau equation and some assumptions on the vortex structures, the asymptotic motion equations of vortices are derived as a system of ordinary differential equations. Furthermore, the asymptotic motion equations of vortices predict the attracting of two degree-one vortices in d-wave superconductors which is observed by Du [SIAM J. Appl. Math. 59 (1999) 1225] by numerical simulation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Vortex dynamics; d-Wave Superconductors; Ginzburg–Landau equation
1. Introduction In this paper, we study the dynamics of vortices in the time-dependent Ginzburg–Landau equation for d-wave superconductors as follows: ut = 1u +
1 (1 − |u|2 )u − η (䊐2 u + γ 12 u) for (x, y) ∈ R2 , t > 0, 2
(1.1)
couples with the initial condition u|t=0 = u0 (x, y)
for (x, y) ∈ R2 ,
(1.2)
where is a positive small parameter, η , γ are nonnegative constants depending on , and u = u(x, y, t) is a complex-valued d-wave order parameter. Hereafter, 䊐 = ∂x2 − ∂y2 and 1 = ∂x2 + ∂y2 . Eq. (1.1) comes from ut = −grad F (u), where F is the simplified Ginzburg–Landau free energy for d-wave superconductors given by Z 1 1 1 |∇u|2 + 2 (1 − |u|2 )2 + η (|䊐u|2 + γ |1u|2 ) dx dy, (1.3) F (u) = 2 2 2 4 R for u = u(x, y) is a complex-valued d-wave order parameter. Hereafter, the simplified Ginzburg–Landau free energy means that we neglect the magnetic field in the Ginzburg–Landau free energy. From [4,5], we learned that ∗
Fax: +886-5-2720497. E-mail address: [email protected] (T.-C. Lin). 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 2 0 0 - 1
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it is reasonable to ignore the magnetic field in strongly type II superconductors when the applied magnetic field is close to the lower critical field Hcl and the current temperature is close to the critical temperature Tc . The simplified Ginzburg–Landau free energy (1.3) is a model for high-temperature superconductors. The hightemperature superconductors were found in some copper-oxide materials (cf. [14]). To distinguish the low- and high-temperature superconductors, the s-wave and d-wave order parameters were introduced (cf. [15,20]). Ren et al. (cf. [23,24,29]) and Soininen et al. (cf. [2,25]) derived and investigated the Ginzburg–Landau free energy with an s-wave and a d-wave order parameter. As the current temperature is close to the critical temperature Tc , Franz et al. [8] observed that in a predominantly d-wave superconductor, the s-wave order parameter is generically very small. In the absence of magnetic field, Affleck et al. [1] obtained the leading order in (1 − T /Tc ) as Ψs = ξ(∂x2 − ∂y2 )Ψd ,
(1.4)
where Ψs = Ψs (x, y) and Ψd = Ψd (x, y) are an s-wave and a d-wave order parameter, respectively. Here ξ is a parameter satisfying that ξ → 0 as T → Tc . Du [6] derived (1.4) by the formal asymptotic analysis. In fact, Affleck et al. [1] deduced the Ginzburg–Landau free energy for d-wave superconductors which can be simplified as Z 1 1 1 ˜ (1.5) |∇u|2 + 2 (1 − |u|2 )2 + η |䊐u|2 dx dy, F (u) = 2 2 2 4 R where is a positive small parameter, η a constant depending on , and u = u(x, y) is a complex-valued d-wave order parameter. Rosenstein and co-workers [4,5] investigated the Euler–Lagrange equation of (1.5) and obtained the vortex structure of d-wave superconductors. To stabilize the Ginzburg–Landau free energy (1.5), Park and Huse [22] and Wilde et al. [28] introduced other Ginzburg–Landau free energies for d-wave superconductors which can be simplified as (1.3) (cf. [16]). In [16], we restricted the integral of (1.3) on a bounded smooth domain Ω. Then the free energy (1.3) became Z 1 1 1 |∇u|2 + 2 (1 − |u|2 )2 + η (|䊐u|2 + γ |1u|2 ) dx dy. (1.6) G (u) = 2 4 Ω2 We found the minimizer of the free energy (1.6) over Hg1 (Ω) ≡ {v ∈ H 1 (Ω; C) : v = g on ∂Ω}, where g : ∂Ω → S 1 is a smooth function with degree N ∈ N. Assume that 0 < η = O( 2 ) and 0 < γ = O(1) as → 0+. Then we showed that the minimizer of the free energy (1.6) has N degree-one vortices in Ω. Furthermore, we obtained a single vortex structure having fourfold symmetry in the inner vortex core by solving the Euler–Lagrange equation of (1.6). The single vortex structure may coincide with results in [4,5,26]. The vortex structures are set in motion by a variety of forces acting on them, including the electromagnetic Lorentz force, the hydrodynamic Magnus force and the viscous drag force. The motion of vortices causes the conversion of electromagnetic energy to kinetic energy and therefore generates resistance. Hence there is theoretical and practical interest in studying the dynamics of vortices. The main result of this paper is to derive the asymptotic motion equations of vortices in d-wave superconductors. As → 0+, under a suitable time scaling and some assumptions on the vortex structures (see hypotheses (1), (2) and Assumption 1 in Section 2), the leading order terms of the motion equations of vortices in d-wave superconductors are q˙j = ∇qj W (q1 , q2 , . . . , qN ),
j = 1, . . . , N,
(1.7)
provided that η and γ satisfy −2 η → ∞,
0 < γ = O(1)
as → 0 + .
(1.8)
Note that (1.8) includes the case that γ is arbitrarily small as goes to 0. Such a case implies that (1.3) is a small perturbation of (1.5). Moreover, (1.8) includes the case that η = a , 0 < a < 2 which tends to 0 as goes to 0.
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The time scale of (1.7) may depend on the coefficients γ , η and the radius R of the inner vortex core. Under suitable choices of γ , η and R , the time scale may be of order O((log(1/))−1 ). Here N ∈ N is the number of vortices, qj = (qjx , qjy )’s are centers of vortices, and X (1.9) W (q1 , q2 , . . . , qN ) = − ni nj log|qi − qj |, i6=j
where nj ∈ {±1} is the degree of the j th vortex. Hereafter, we assume that |qj − qk | > 2r0 for j 6= k, where r0 is the radius of each vortex core and is independent of . Such an assumption is essential for the derivation of (1.7) and it excludes the collision of vortices. From (1.7), it is remarkable that q1 and q2 attract each other if N = 2, n1 = n2 = 1. Hence two degree-one vortices may attract each other. Such a new phenomenon was observed in [6] by numerical simulation. Note that the attracting of two degree-one vortices in d-wave superconductors is opposite to the repelling of two degree-one vortices in conventional s-wave superconductors. From the Ginzburg–Landau theory of conventional superconductivity (cf. [9]), the simplified Ginzburg–Landau free energy for s-wave superconductors was given by Z 1 1 (1.10) |∇Ψ |2 + 2 (1 − |Ψ |2 )2 dx dy, E (Ψ ) = 4 R2 2 where is a positive small parameter and Ψ = Ψ (x, y) is a complex-valued s-wave order parameter. As is small, the Ginzburg–Landau theory predicted the existence of vortex state. To investigate the dynamics of vortices in s-wave superconductors, many people considered the equation Ψt = −grad E (Ψ ), i.e. a nonlinear heat equation given by 1 (1 − |Ψ |2 )Ψ for (x, y) ∈ R2 , t > 0, (1.11) 2 where Ψ = Ψ (x, y, t) is a complex-valued s-wave order parameter. By the standard asymptotic analysis, Neu [21] derived the dynamics of vortices from (1.11). Moreover, E [7] improved Neu’s results and obtained a simpler dynamic law of vortices. As → 0+ and under a suitable time scaling of order O((log(1/))−1 ), the leading order terms of the motion equations of vortices are Ψt = 1Ψ +
q˙j = −∇qj W (q1 , q2 , . . . , qN ),
j = 1, . . . , N,
(1.12)
where N ∈ N is the number of vortices and qj = (qjx , qjy )’s are centers of vortices. For the proof of (1.12), please refer to [17]. Note that (1.12) the asymptotic motion equations of vortices in s-wave superconductors are exactly up to a minus sign with (1.7) the asymptotic motion equations of vortices in d-wave superconductors. In the rest of this paper, we will prove (1.7) the asymptotic motion equations of vortices in d-wave superconductors. Firstly, we prove the asymptotic motion equations of a single vortex under some assumptions on the vortex structure. The main tool of our proof is the spectrum of the linearized operator about the symmetric vortex solution of the steady-state equation of (1.11). In Section 2, we will introduce the idea and the proof of (1.7). In Section 3, we will complete the proof of the asymptotic motion equations of a single vortex.
2. Preliminaries For the dynamics of a single vortex, we assume that the solution u of (1.1) has only one vortex center at q(t), where q(t) = (qx (t), qy (t)) is smooth in t, q(0) = (0, 0), and Br0 (q(t)) is the vortex core which moves along with the vortex trajectory (x, y) = q(t). Here Br0 (q(t)) is a disk on R2 with radius r0 and center at q(t), where r0 is a positive constant independent of . Moreover, there are two essential hypotheses given by
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1. On the outer vortex core OT , the amplitude of the solution u of (1.1) has almost perfect circular symmetry. 2. When a single vortex begins to move at the time t = 0, the vortex structure on the outer vortex core OT does not change much at the time t = 0, where OT (t) ≡ {(x, y) : |(x, y) − q(t)| ∈ [R , r0 )} and R is a positive constant satisfying R → ∞,
R → 0
as → 0 + .
(2.1)
The hypothesis (1) is a generic property of d-wave superconductors (cf. [5,8,26]). From the standard asymptotic analysis (cf. [7]), we learned that each of the vortices has a core of size , and they are O(1) distance apart. Furthermore, in the outer region outside the vortex cores with size , the amplitude of the solution u has almost perfect circular symmetry. Please note that the outer vortex core OT may be a portion of the outer region outside the vortex core of size . Hence the hypothesis (1) is consistent with the assumptions in [7]. In fact, the solution u of (1.1) may loose the full rotational symmetry and have fourfold symmetry on the inner vortex core {(x, y) : |(x, y) − q(t)| < R }. Thus the vortex structure on the inner vortex core may become quite complicated. However, the hypothesis (1) gives us a simpler vortex structure on the outer vortex core so we may use it to derive the vortex dynamics. The hypothesis (2) is to preserve the vortex structure on the outer vortex core when the vortex moves. We will use Assumption 1 in the last part of this section to fulfill the hypotheses (1) and (2). Now we focus on the outer vortex core OT and consider the following system of equations: ut = 1u + (1/ 2 )(1 − |u|2 )u − η (䊐2 u + γ 12 u) for (x, y) ∈ OT (t), t > 0, u|t=0 = u0 (x, y)
for (x, y) ∈ OT (0).
(2.2)
We introduce the stretched variables X=
x − qx (t) ,
Y=
y − qy (t) ,
(2.3)
and we set Ψ (X, Y, t, ) = u(x, y, t, ) for (x, y) ∈ OT , i.e. (X, Y) ∈ Ω , t ≥ 0, where Ω ≡ {(X, Y) : |(X, Y)| ∈ [R , (r0 /))}. Then we have ˜ + 1Ψ ˜ + (1 − |Ψ |2 )Ψ − −2 η P (Ψ ) for (X, Y) ∈ Ω , t > 0, 2 Ψt = q˙ · ∇Ψ Ψ |t=0 = u0
for (X, Y) ∈ Ω ,
(2.4)
where ˜ 2 ), P = (䊐2∗ + γ 1
∇˜ = (∂X , ∂Y ),
˜ = ∂2 + ∂2 , 1 X Y
2 䊐∗ = ∂X − ∂Y2 ,
(2.5)
and Ψt = (∂/∂t)Ψ (·, ·, t, ). We take an expansion form of Ψ as follows: Ψ (X, Y, t, ) = Ψ0 (X, Y)eiH + Ψ1 (X, Y, t, )eiH ,
(2.6)
where Ψ0 = Ψ0 (X, Y) satisfies Ψ0 (X, Y) = f0 (R)eiθ ,
R = |(X, Y)|, θ = arg(X, Y),
(2.7)
and f0 (R) is the solution of −f 00 −
1 0 1 f + 2 f = (1 − f 2 )f R R
for R > 0, f (+∞) = 1, f (0) = 0, f ≥ 0.
(2.8)
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From [3,10,12], we learned that (2.8) has a unique solution and Ψ0 is called the symmetric vortex solution of the steady-state equation of (1.11) in R2 . In addition, we assume that H = H (x, y, t, ) is a smooth real-valued function and satisfies 1H = 0,
|∇ n H |, |Ht |, |∇Ht | ≤ K
for (x, y) ∈ Br0 (q(t)), t > 0, n = 1, . . . , 5,
(2.9)
where Ht = (∂/∂t)H and K is a positive constant independent of . By (2.6)–(2.9), (2.4) becomes ˜ 1 ) − iΨ0 Ht − i 2 Ψ1 Ht − 2 Ψ1,t ˜ 0 + ∇Ψ q˙ · (∇Ψ ˜ 0 · ∇H ) + L˜ (Ψ1 ) + Nˆ (Ψ1 ) + −3 η e−iH P (Ψ0 eiH ) + −2 η e−iH P (Ψ1 eiH ) = Ψ0 |∇H |2 − 2i(∇Ψ for (X, Y) ∈ Ω , t > 0,
(2.10)
where ˜ 1 + (1 − |Ψ0 |2 )Ψ1 − 2(Ψ0 · Ψ1 )Ψ0 , −L˜ (Ψ1 ) = 1Ψ ˜ 1 · ∇H ) + 2 |Ψ1 |2 Ψ1 + [2(Ψ0 · Ψ1 )Ψ1 + |Ψ1 |2 Ψ0 ], Nˆ (Ψ1 ) = 2 Ψ1 |∇H |2 − 2i(∇Ψ
(2.11)
and Ψ1,t = (∂/∂t)Ψ1 (·, t, ). Note that in the first term of (2.11), (Ψ0 · Ψ1 ) = 21 (Ψ¯ 0 Ψ1 + Ψ¯ 1 Ψ0 ) and (¯·) denotes the complex conjugate. The spectrum information of the linear operator L˜ is essential for our argument. From [18,19], we have estimates on the eigenvalues of L˜ as follows. Theorem 1. Let λ1 and λ2 be the first and the second eigenvalue of L˜ . Then 1. The eigenvalue λ1 has only two associated eigenfunctions e˜1 = a (R)+b (R) e2iθ and ˜2 = ia (R)−ib (R) e2iθ , where a and b are real-valued. 2. 0 < λ1 = O( 2 (log(r0 /))−1 ) as → 0+. 3. There exist 1 > 0 and c1 > 0 independent of such that λ2 ≥ 2 c1 for 0 < ≤ 1 . Let w˜ 1 =
∂X Ψ0 , k∂X Ψ0 kL2
w˜ 2 =
∂Y Ψ0 , k∂Y Ψ0 kL2
(2.12)
where k · kL2 is the L2 norm on the disk B(r0 /) (0). We observe that w˜ j ’s are from the translation invariance. In [19], we used Theorem 1 to derive a useful estimate of eigenfunctions e˜j ’s as follows. Proposition 1. Assume that hw˜ j , e˜j i > 0,
ke˜j kL2 = 1, j = 1, 2.
Then the eigenfunctions e˜j ’s satisfy e˜j = w˜ j + νj, ,
kνj, kL2 = O
1 log
−1/2 !
as → 0+, for j = 1, 2.
Hereafter, we use k · kL2 and h·, ·i to denote the L2 norm and the L2 inner product, respectively, on the disk Br0 / (0). Let V˜1 = {a(R) + b(R) e2iθ ∈ H01 (Br0 / (0); C)}. Then it is easy to check that V˜1 is invariant under L˜ −1 . Hence we
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set e˜j,k ∈ V˜1 ’s the unit L2 norm eigenfunctions of L˜ corresponding to the eigenvalue λ˜ k , respectively, for k ≥ 2. Then e˜j ’s and e˜j,k ’s are dense in V1 = {a(R) + b(R) e2iθ ∈ L2 (Br0 / (0); C)}. Since w˜ l ∈ V1 , l = 1, 2, then w˜ l ’s can be represented by e˜j ’s and e˜j,k ’s as follows: (k) 2 ∞ JX X X hw˜ l , e˜j ie˜j + hw˜ l , e˜j,k ie˜j,k , w˜ l = j =1
(2.13)
k=2 j =1
where J (k) is the multiplicity of the eigenvalue λ˜ k , k ≥ 2. Hereafter, we denote as h·, ·i∗ the L2 inner product on the outer core Ω . Using integration by parts, we have Z Z BD(Ψ1 , e˜j ) + λ1 hΨ1 , e˜j i∗ , hL˜ (Ψ1 ), e˜j,k i∗ = BD(Ψ1 , e˜j,k ) + λ˜ k hΨ1 , e˜j,k i∗ , hL˜ (Ψ1 ), e˜j i∗ = ∂Ω
∂Ω
(2.14) where BD(Ψ1 , ·) is defined by BD(Ψ1 , v) = (Ψ1 · ∂nˆ v) − (∂nˆ Ψ1 · v) for v is a smooth complex-valued function, ¯ for α, β ∈ C. Hence by (2.13) and (2.14) and and ∂nˆ is the normal derivative. Hereafter, (α · β) ≡ 21 (αβ ¯ + βα) Proposition 1, we obtain 2 X
hL˜ (Ψ1 ), w˜ l i∗ =
λ1 hΨ1 , e˜j i∗ +
j =1
+
Z ∂Ω
BD(Ψ1 , e˜j ) hw˜ l , e˜j i
(k) ∞ JX X
λ˜ k hΨ1 , e˜j,k i∗ +
k=2 j =1
= o
log
1
−1/2
! R−2 ,
Z ∂Ω
BD(Ψ1 , e˜j,k ) hw˜ l , e˜j,k i
l = 1, 2,
(2.15)
provided that Ψ1 satisfies |hΨ1 , e˜j i∗ | = o Z
−1
1 log
1/2
! R−2
,
j = 1, 2,
! −1/2 1 −2 BD(Ψ1 , e˜j ) = o R log , ∂Ω
(k) ∞ JX X λ˜ k |hΨ1 , e˜j,k i∗ | = o (R−2 ), k=2 j =1
j = 1, 2,
(k) Z ∞ JX X k=2 j =1
∂Ω
BD(Ψ1 , e˜j,k ) = o (R−2 ). (2.16)
Hereafter, we denote as o (1) a small quantity, independent of time t, and tending to 0 as → 0+. In Section 3, we will derive the asymptotic motion equations of the vortex trajectory (x, y) = q(t). Under a suitable time scaling, the leading order terms of the asymptotic motion equations are
1 q˙ = log
−1
(c3 + γ c4 )
where J =
0 −1
1 0
,
−2
R−2 η ∇ H˜ (q, t, ) +
1 2 ˜ r ∇ Ht (q, t, ) , 2 0
(2.17)
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c3 and c4 are positive constants independent of . In addition, H and H˜ are harmonic conjugates. The main idea of the proof of (2.17) is to make the L2 inner product with (2.10) and w˜ j , j = 1, 2. For details, we refer to Section 3. We require that Ψ1 satisfies the “small” perturbation condition (2.16) and ! ! 1 −1/2 −2 1 −1/2 −2 −1 −1 R R , kΨ1 kL2 (Ω ) = o , log log kΨ1,t kL2 (Ω ) = o ! ! −1/6 −1/4 1 1 R−2/3 , R−1 , kΨ1 kL4 (Ω ) = o log kΨ1 kL6 (Ω ) = o −1/3 log ! ! −1/2 −1/2 1 1 −2 n n−4 −2 ˜ 1 kL2 (Ω ) = o R R log , k∇˜ Ψ1 kL2 (Ω ) = o , n = 2, 3, 4. log k∇Ψ (2.18) Suppose that R satisfies R → ∞,
1 1/2 R2 log →0
as → 0 + .
(2.19)
Then the upper bound of the first term of (2.16), the first three terms of (2.18) and the last term of (2.18) as n = 2, 3 may tend to infinity as goes to 0. From [11], Eq. (1.1) is well posed. Then Ψ1 is smooth in both space and time variables. Hence (2.16) and (2.18) can be fulfilled at least in a short time when Ψ1 |t=0 satisfies. Assumption 1. Ψ1 = Ψ1 (X, Y, t, ) satisfies 1. kΨ1 (·, ·, 0, )kC 4 (Ω˜ ) is sufficiently small, where Ω˜ ≡ {(X, Y) : 21 R ≤ |(X, Y)| ≤ 2r0 /} at t = 0, 2. k∂t Ψ1 (·, ·, 0, )kL2 (Ω˜ ) = o ( −1 (log(1/))−1/2 R−2 ). By the suitable choice of initial data u0 in (1.2), Assumption 1(1) can be satisfied. Assumption 1(2) fulfills the requirement that the vortex structure, i.e. the solution u on the outer vortex core does not change much at t = 0 when the associated vortex begins to move at t = 0. Note that by (2.19), the upper bound of Assumption 1(2) may tend to infinity as goes to 0. Actually, Assumption 1 which may assure (2.16) and (2.18) are more generalized than the assumptions of the standard asymptotic analysis. To derive the dynamics of vortices, the standard asymptotic analysis is well-accepted (cf. [7,21,27]). E [7] and Neu [21] used a specific asymptotic expansion formula and some pointwise conditions for the solution on vortex cores to derive the dynamics of vortices. However, (2.16) and (2.18) are not pointwise. This is a kind of generalization for the assumptions of the standard asymptotic analysis. For the motion of N -vortices, we restrict (1.1) on the vortex cores Br0 (qj )’s and we consider the following system of equations: ut = 1u + (1/ 2 )u(1 − |u|2 ) − η (䊐2 u + γ 12 u) for (x, y) ∈ OTj (t), t > 0, u|t=0 = uj 0 (x)
for (x, y) ∈ OTj (0),
(2.20)
where (x, y) = qj (t) = (qjx (t), qjy (t)) is the j th vortex trajectory, qj the smooth in t, |qj − qk | > 2r0 for j 6= k, j and OT (t) = {(x, y) : |(x, y) − qj (t)| ∈ (R , r0 )} the j th outer vortex core which moves along with the j th vortex trajectory (x, y) = qj (t), j = 1, . . . , N. Now we assume that Xj =
x − qjx (t) ,
Yj =
y − qjy (t) ,
Ψ (Xj , Yj , t, ) = u(x, y, t, )
(2.21)
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T.-C. Lin / Physica D 149 (2001) 293–305 j
j
for (x, y) ∈ OT , i.e. (Xj , Yj ) ∈ Ω = {(Xj , Yj ) : |(Xj , Yj )| ∈ (R , r0 /)}, j = 1, . . . , N. As for (2.6), we take a similar expansion form of Ψ on each outer vortex core as follows: Ψ (Xj , Yj , t, ) = Ψ0 (Xj , Yj )eiHj + Ψ1 (Xj , Yj , t, ) eiHj
for (Xj , Yj ) ∈ Ωj ,
(2.22)
where Ψ0 (Xj , Yj ) = f0 (Rj )einj φj , Here we assume that X nk φk + H, 1H = 0 Hj =
nj ∈ {±1},
Rj = |(Xj , Yj )|,
φj = arg(Xj , Yj ).
for (x, y) ∈ Br0 (qj (t)), t > 0, j = 1, . . . , N,
k6=j
|∇ n H |, |Ht |, |∇Ht | ≤ K
for (x, y) ∈ Br0 (qj (t)), t > 0, n = 1, . . . , 5,
where nk ∈ {±1}. As for (2.17), we obtain the equations of qj given by 1 −1 1 2 −2 −2 ˜ ˜ nj (c3 + γ c4 ) R η ∇ Hj (qj , t, ) + r0 ∇∂t Hj (qj , t, ) , q˙j = log 2
(2.23)
(2.24)
where Hj and H˜ j ’s are harmonic conjugates. Here we require that Ψ1 satisfies the “small” perturbation conditions j (2.16) and (2.18) for (Xj , Yj ) ∈ Ω , j = 1, . . . , N. Of course, such a requirement can be fulfilled by Assumption j 1 on outer vortex cores Ω ’s. In particular, suppose H ≡ 0. Then (2.23) and (2.24) imply that 1 −1 (c3 + γ c4 ) −2 R−2 η nj ∇ H˜ j (qj , t), j = 1, . . . , N, (2.25) q˙j = log Under a suitable time scaling, the leading order terms of (2.25) becomes (1.7) and we complete the proof of (1.7). Remark 1. The time scale of (2.25) is of order ! 1 −1 −2 −2 (c3 + γ c4 ) R η , O log which depends on γ , η and R . In particular, we may choose suitable γ , η and R (e.g. γ = 1, η = 3/2 , R = −1/4 ) such that the time scale of (2.25) is of order O((log(1/))−1 ). Hence the time scale for the dynamics of d-wave Ginzburg–Landau vortices may be equal to the time scale for the dynamics of s-wave Ginzburg–Landau vortices.
3. Proof of the vortex dynamics in d -wave superconductors In this section, we prove (2.17) as follows. From (2.10), we have q˙x (∂X Ψ0 + ∂X Ψ1 ) + q˙y (∂Y Ψ0 + ∂Y Ψ1 ) − iΨ0 Ht − i 2 Ψ1 Ht − 2 Ψ1,t = Ψ0 |∇H |2 − 2i(∂X Ψ0 ∂x H + ∂Y Ψ0 ∂y H ) + −3 η e−iH P (Ψ0 eiH ) + −2 η e−iH P (Ψ1 eiH ) + L˜ Ψ1 + Nˆ (Ψ1 ) for (X, Y) ∈ B(r0 /) (0), t > 0.
(3.1)
Making the inner product with (3.1) and w˜ j , j = 1, 2, we have q˙x [h∂X Ψ0 , w˜ j i∗ + h∂X Ψ1 , w˜ j i∗ ] + q˙y [h∂Y Ψ0 , w˜ j i∗ + h∂Y Ψ1 , w˜ j i∗ ] = 2 (hΨ1,t , w˜ j i∗ + hiΨ1 Ht , w˜ j i∗ ) + hNˆ (Ψ1 ), w˜ j i∗ + Γj + hL˜ Ψ1 , w˜ j i∗
(3.2)
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301
for j = 1, 2, where Γj = −2hi∂X Ψ0 ∂x H, w˜ j i∗ − 2hi∂Y Ψ0 ∂y H, w˜ j i∗ + hΨ0 |∇H |2 , w˜ j i∗ + hiΨ0 Ht , w˜ j i∗ + −3 η [he−iH P (Ψ0 eiH ), w˜ j i∗ + he−iH P (Ψ1 eiH ), w˜ j i∗ ].
(3.3)
It is obvious that e−iH 䊐2∗ (Ψk eiH ) = 䊐2∗ Ψk + 4i(HX ∂X 䊐∗ Ψk − HY ∂Y 䊐∗ Ψk ) + S(Ψk ), ˜ 2 (Ψk eiH ) = 1 ˜ k + HY ∂Y 1Ψ ˜ k ) + T (Ψk ), k = 0, 1, ˜ 2 Ψk + 4i(HX ∂X 1Ψ (3.4) e−iH 1 Q α β j j where S(Ψk ) and T (Ψk ) are linear combinations of ∂Xl ∂Ym Ψk 4j =1 (∂X ∂Y H )γj ’s, for l, m ≥ 0, l + m ≤ 2, αj , βj , P γj ≥ 0, j = 1, . . . , 4 and l + m + 4j =1 (αj + βj )γj = 4, respectively. Hence (2.5) and (3.4) imply that e−iH P (eiH Ψk ) = 4i[(HX ∂X 䊐∗ Ψk − HY ∂Y 䊐∗ Ψk ) ˜ k + HY ∂Y 1Ψ ˜ k )] + P (Ψk ) + S(Ψk ) + γ T (Ψk ). +γ (HX ∂X 1Ψ From (2.7), it is easy to check that 1 1 1 1 f0 + f00 + f00 − f0 e2iθ , ∂X Ψ0 = 2 R 2 R
∂Y Ψ0 =
i 2
(3.5)
1 i 1 f0 + f00 + f0 − f00 e2iθ . (3.6) R 2 R
Moreover,
1 1 3 1 1 1 3 1 0 0 2iθ 0 −2iθ 0 + g + g1 + g + g3 e + g − g1 e g − g3 e4iθ , ∂ X 䊐∗ Ψ0 = 4 1 R 4 3 R 4 1 R 4 3 R i 1 3 i 1 i 3 i g10 + g1 + g30 + g3 e2iθ + g10 − g1 e−2iθ − g30 − g3 e4iθ , ∂ Y 䊐∗ Ψ0 = − 4 R 4 R 4 R 4 R 䊐2∗ Ψ0 = a1 (R)e−3iθ + a2 (R)eiθ + a3 (R)e5iθ ,
(3.7)
where aj , j = 1, 2, 3 are smooth real-valued functions and 0 0 1 1 2 1 0 0 0 f0 − f0 . g3 = g3 (R) = f0 − f0 − g1 = g1 (R) = f0 + f0 , R R R R Furthermore, ˜ 0= ∂X 1Ψ
1 1 1 1 g10 + g1 + g10 − g1 e2iθ , 2 R 2 R
˜ 0= ∂Y 1Ψ
(3.8)
i 1 1 i g10 + g1 − g10 − g1 e2iθ , 2 R 2 R
˜ 2 Ψ0 = b(R)eiθ , 1
(3.9)
where b is a smooth real-valued function. Thus by (2.1), (2.12), (3.6)–(3.9), we obtain that h∂X Ψ0 , w˜ 2 i∗ = h∂Y Ψ0 , w˜ 1 i∗ = 0,
hiHX ∂X Ψ0 , w˜ 1 i∗ = hiHY ∂Y Ψ0 , w˜ 2 i∗ = 0, 1 1/2 (1 + o (1)), log
h∂X Ψ0 , w˜ 1 i∗ = h∂Y Ψ0 , w˜ 2 i∗ = c2 ˜ 2 Ψ0 , w˜ j i∗ = 0, h䊐2∗ Ψ0 , w˜ j i∗ = h1
j = 1, 2,
(3.10)
where c2 is a positive constant independent of . Here we have used the fact that f0 (R) = 1 − 21 R −2 − 98 R −4 + · · ·
as R → +∞
(3.11)
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(cf. [3,12]). Note that by (3.8) and (3.11), we obtain that g1 (R) ∼ −R −2 ,
g3 (R) ∼ 3R −2
as R → +∞.
(3.12)
By (2.1), (3.6) and (3.11) and the mean-valued theorem of harmonic functions, we have hiHx ∂X Ψ0 , ∂Y Ψ0 i∗ = c0 R−2 (Hx (q, t, ) + o (1)),
hiHy ∂Y Ψ0 , ∂X Ψ0 i∗ = −c0 R−2 (Hy (q, t, ) + o (1)),
(3.13)
where c0 is a positive constant independent of . From (3.6) and (3.7), we obtain that Z r0 / Z 2π R[α1 (R) + α2 (R) cos(2θ ) + α3 (R) cos(4θ )]HX dθ dR, hiHX ∂X 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = R
Z
h−iHY ∂Y 䊐∗ Ψ0 , ∂X Ψ0 i∗ = Z hiHX ∂X 䊐∗ Ψ0 , ∂X Ψ0 i∗ =
R 0 r0 / Z 2π
R
Z
h−iHY ∂Y 䊐∗ Ψ0 , ∂Y Ψ0 i∗ =
0 r0 / Z 2π
R[β1 (R) sin(2θ ) − α3 (R) sin(4θ )]HX dθ dR,
0 r0 / Z 2π
R
0
R[−α1 (R) + α2 (R) cos(2θ ) − α3 (R) cos(4θ )]HY dθ dR,
R[−β1 (R) sin(2θ ) − α3 (R) sin(4θ )]HY dθ dR,
where
1 1 1 1 g10 + g1 f0 − f00 f00 + f0 + g30 + 8 R R R 1 α2 (R) = R −1 [(g1 + g3 )0 f0 + (3g3 − g1 )f00 ], 4 1 3 1 1 g30 − g3 f0 − f00 f00 + f0 + g10 − α3 (R) = 8 R R R 1 β1 (R) = [(g1 − g3 )0 f00 − R −2 (g1 + 3g3 )f0 ]. 4 Moreover, by (3.11), (3.12) and (3.15), we have α1 (R) =
α1 (R) ∼ 21 R −4 ,
α2 (R) ∼ −R −4 ,
α3 (R) ∼ − 23 R −4 ,
3 g3 R
1 g1 R
(3.14)
,
,
β1 (R) ∼ −2R −4
(3.15)
as R → +∞.
By (2.9) and the standard Poisson integral formulas for harmonic functions (cf. [13]), we may derive that Z 1 2π HX (R, θ, t, ) dθ, HX |X=Y=0 = 2π 0 Z 1 2π HY (R, θ, t, ) dθ, HY |X=Y=0 = 2π 0 Z 2π 2 HX (R, θ, t, ) cos(2θ ) dθ, ∂X3 H |X=Y=0 = R −2 π 0 Z 2π Z 2π 2 2 HY (R, θ, t, ) cos(2θ ) dθ = R −2 HX (R, θ, t, ) sin(2θ ) dθ, ∂X2 ∂Y H |X=Y=0 = R −2 π π 0 0 Z 24 −4 2π HX (R, θ, t, ) cos(4θ ) dθ, R ∂X5 H |X=Y=0 = π 0 Z Z 24 −4 2π 24 −4 2π R R ∂X4 ∂Y H |X=Y=0 = HY (R, θ, t, ) cos(4θ ) dθ = HX (R, θ, t, ) sin(4θ ) dθ π π 0 0
(3.16)
(3.17)
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for 0 < R < (r0 /). Hence (2.9), (3.14), (3.16) and (3.17) imply that hiHX ∂X 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = c3 R−2 (Hx (q, t, ) + o (1)),
h−iHY ∂Y 䊐∗ Ψ0 , ∂X Ψ0 i∗ = −c3 R−2 (Hy (q, t, ) + o (1)),
hiHX ∂X 䊐∗ Ψ0 , ∂X Ψ0 i∗ = R−2 o (1),
h−iHY ∂Y 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = R−2 o (1),
(3.18)
where c3 is a positive constant independent of . Here we have used the rescaling (2.3) on spatial variables such that j
j
∂X ∂Yk H = j +k ∂x ∂yk H
for j, k ≥ 0.
(3.19)
Similarly, ˜ 0 , ∂Y Ψ0 i∗ = c4 R−2 (Hx (q, t, ) + o (1)), hiHX ∂X 1Ψ ˜ 0 , ∂X Ψ0 i∗ = −c4 R−2 (Hy (q, t, ) + o (1)), hiHY ∂Y 1Ψ ˜ 0 , ∂X Ψ0 i∗ = R−2 o (1), hiHX ∂X 1Ψ ˜ 0 , ∂Y Ψ0 i∗ = R−2 o (1), hiHY ∂Y 1Ψ
(3.20)
where c4 is a positive constant independent of . From (2.1), (2.9), (3.5), (3.10), (3.18)–(3.20), we obtain that he−iH P (eiH Ψ0 ), ∂X Ψ0 i∗ = −4(c3 + γ c4 )R−2 (Hy (q, t, ) + o (1)),
he−iH P (eiH Ψ0 ), ∂Y Ψ0 i∗ = 4(c3 + γ c4 )R−2 (Hx (q, t, ) + o (1)).
(3.21)
By (2.15) and (3.2), we have q˙x (Γ + α1 ) + q˙y β1 = γ1 ,
q˙x β2 + q˙y (Γ + α2 ) = γ2 ,
(3.22)
where = k∂Y Ψ0 k2L2 (Ω ) k∂Y Ψ0 k−1 , Γ ≡ k∂X Ψ0 k2L2 (Ω ) k∂X Ψ0 k−1 L2 L2
α2 = h∂Y Ψ1 , w˜ 2 i∗ ,
β1 = h∂Y Ψ1 , w˜ 1 i∗ ,
α1 = h∂X Ψ1 , w˜ 1 i∗ ,
β2 = h∂X Ψ1 , w˜ 2 i∗ ,
ηj = hNˆ (Ψ1 ), w˜ j i∗ ,
and γj = Γj + (hΨ1,t , w˜ j i∗ + hiΨ1 Ht , w˜ j i∗ ) + ηj + o 2
1 log
−1/2
! R−2
,
for j = 1, 2. From (3.6) and (3.11), we obtain 1 1/2 (κ0 + o (1)) as → 0+, Γ = log where κ0 is a positive constant independent of . By (2.9) and (2.18) and the Hölder inequality, we have ! ! 1 −1/2 −2 1 −1/2 −2 R R |αj | = o log , |βj | = o log , ! ! 1 −1/2 −2 1 −1/2 −2 log , γj = Γj + o log , j = 1, 2, R R |ηj | = o
(3.23)
(3.24)
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and he−iH P (eiH Ψ1 ), ∂X Ψ0 i∗ = o (R−2 ),
he−iH P (eiH Ψ1 ), ∂Y Ψ0 i∗ = o (R−2 ).
(3.25)
Hence (3.22) implies that q˙x =
1 [(Γ + α2 )γ1 − β1 γ2 ], γ
q˙y =
1 [(Γ + α1 )γ2 − β2 γ1 ], γ
(3.26)
where γ = Γ2 + Γ (α1 + α2 ) + 2 (α1 α2 − β1 β2 ). Thus by (2.9), (3.6) and (3.11) and the mean-valued theorem of harmonic functions, we have Z r0 / C , C = 2πK 2 Rf0 f00 dR ∼ R−1 , |hΨ0 |∇H |2 , w˜ j i| ≤ Γ R Z r0 / π Rf 20 dR ∂y Ht (q, t, ), 2 hiΨ0 Ht , w˜ 1 i = − Γ R Z r0 / π 2 2 Rf 0 dR ∂x Ht (q, t, ). (3.27) hiΨ0 Ht , w˜ 2 i = Γ R By (1.8), (3.3), (3.13), (3.21), (3.25) and (3.27), we obtain Γ1 = −c5 Γ−1 [(c3 + γ c4 ) −2 R−2 η (∂y H (q, t, ) + o (1)) + c ∂y Ht (q, t, )],
(3.28) Γ2 = c5 Γ−1 [(c3 + γ c4 ) −2 R−2 η (∂x H (q, t, ) + o (1)) + c ∂x Ht (q, t, )], R r / where c5 is a positive constant independent of and c = 2 R0 Rf 20 dR > 0. Note that by (2.1) and (3.11), c ∼ 21 r02
as → 0 + .
(3.29)
Therefore by (3.23), (3.24), (3.26), (3.28) and (3.29), we complete the proof of (2.17). References [1] I. Affleck, M. Franz, M.H.S. Amin, Generalized London free energy for high-Tc vortex lattices, Phys. Rev. B 55 (1997) R704. [2] A.J. Berlinsky, A.L. Fetter, M. Franz, C. Kallin, P.I. Soininen, Ginzburg–Landau theory of vortices in d-wave superconductors, Phys. Rev. Lett. 75 (1995) 2200. [3] X. Chen, C.M. Elliott, T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg–Landau equation, Proc. Roy. Soc. Edinburgh 124A (1994) 1075–1088. [4] D. Chang, C.Y. Mu, B. Rosenstein, C.L. Wu, The effect of anisotropy on vortex lattice structure and flux flow in d-wave superconductors, Chinese J. Phys. 36 (2) (1998) 238–244. [5] D. Chang, C.Y. Mu, B. Rosenstein, C.L. Wu, Static and dynamical anisotropy effects in mixed state of d-wave superconductors, Phys. Rev. B 57 (13) (1998) 7955–7969. [6] Q. Du, Studies of a Ginzburg–Landau model for d-wave superconductors, SIAM J. Appl. Math. 59 (4) (1999) 1225–1250. [7] W. E, Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity, Physica D 77 (1994) 383–404. [8] M. Franz, C. Kallin, P.I. Soininen, A.J. Berlinsky, A.L. Fetter, Vortex state in a d-wave superconductor, Phys. Rev. B 53 (9) (1996) 5795. [9] P.G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, Reading, MA, 1989. [10] P. Hagan, Spiral waves in reaction diffusion equations, SIAM J. Appl. Math. 42 (4) (1982) 762–786. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981. [12] R.M. Hervé, M. Hervé, Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg–Landau, Ann. Inst. Henri Poincaré 11 (1994) 427–440. [13] F. John, Partial Differential Equations, 4th Edition, Springer, New York, 1982. [14] J.D. Jorgensen, et al., Structural features that optimize high temperature superconductivity, in: B.W. Veal, B.M. Dabrowski, P.W. Klamut (Eds.), Proceedings of the First Polish US Conference on Recent Developments in High Temperature Superconductivity, Springer, Berlin, 1996, pp. 1–15.
T.-C. Lin / Physica D 149 (2001) 293–305 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
305
G. Kotliar, Resonating valence bonds and d-wave superconductivity, Phys. Rev. B 37 (1988) 3664. F.H. Lin, T.C. Lin, Vortex state of d-wave superconductors in the Ginzburg–Landau energy, SIAM J. Math. Anal. 32 (3) (2000) 493–503. F.H. Lin, J.X. Xin, On the dynamical law of the Ginzburg–Landau vortices on the plane, Comm. Pure Appl. Math. 52 (10) (1999) 1189–1212. T.C. Lin, The stability of the radial solution to the Ginzburg–Landau equation, Comm. PDE 22 (1997) 619–632. T.C. Lin, Spectrum of the linearized operator for the Ginzburg–Landau equation, Electron. J. Diff. Eqns. 2000 (42) (2000) 1–25. K.A. Müller, Possible coexistence of s- and d-wave condensates in copper oxide superconductors, Nature 377 (1995) 133–135. J.C. Neu, Vortices in complex scalar fields, Physica D 43 (1990) 385–406. K. Park, D.A. Huse, The phase transition to a square vortex lattice in type-II superconductors with fourfold anisotropy, Phys. Rev. B 58 (14) (1998) 9427–9432. Y. Ren, J.H. Xu, C.S. Ting, Ginzburg–Landau equations and vortex structure of a dx 2 −y 2 superconductor, Phys. Rev. Lett. 74 (1995) 3680. Y. Ren, J.H. Xu, C.S. Ting, Ginzburg–Landau equations for mixed s + d symmetry superconductors, Phys. Rev. B 53 (5) (1996) 2249. P.I. Soininen, C. Kallin, A.J. Berlinsky, Structure of a vortex line in a dx 2 −y 2 superconductor, Phys. Rev. B 50 (1994) 13883. J. Shiraishi, M. Kohmoto, K. Maki, Vortex lattice transition in d-wave superconductors, Phys. Rev. B 59 (6) (1999) 4497–4503. L. Ting, R. Klein, Viscous Vortical Flows, Lecture Notes in Physics, Vol. 374, Springer, Berlin, 1991. Y. De Wilde, M. Iavarone, U. Welp, V. Metlushko, A.E. Koshelev, I. Aranson, G.W. Crabtree, P.C. Canfield, Scanning tunneling microscopy observation of a square Abrikosov lattice in LuNi2 B2 C, Phys. Rev. Lett. 78 (1997) 4273. J.H. Xu, Y. Ren, C.S. Ting, Structures of single vortex and vortex lattice in a d-wave superconductor, Phys. Rev. B 53 (6) (1996) 2991.
Vortex dynamics in d-wave superconductors Tai-Chia Lin∗ Department of Mathematics, National Chung-Cheng University, Minghsiung, Chiayi 621, Taiwan Received 9 November 1999; received in revised form 2 October 2000; accepted 27 October 2000 Communicated by C.K.R.T. Jones
Abstract We study the dynamics of vortices in the time-dependent Ginzburg–Landau equation for d-wave superconductors. Under suitable assumptions on the coefficients of the time-dependent Ginzburg–Landau equation and some assumptions on the vortex structures, the asymptotic motion equations of vortices are derived as a system of ordinary differential equations. Furthermore, the asymptotic motion equations of vortices predict the attracting of two degree-one vortices in d-wave superconductors which is observed by Du [SIAM J. Appl. Math. 59 (1999) 1225] by numerical simulation. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Vortex dynamics; d-Wave Superconductors; Ginzburg–Landau equation
1. Introduction In this paper, we study the dynamics of vortices in the time-dependent Ginzburg–Landau equation for d-wave superconductors as follows: ut = 1u +
1 (1 − |u|2 )u − η (䊐2 u + γ 12 u) for (x, y) ∈ R2 , t > 0, 2
(1.1)
couples with the initial condition u|t=0 = u0 (x, y)
for (x, y) ∈ R2 ,
(1.2)
where is a positive small parameter, η , γ are nonnegative constants depending on , and u = u(x, y, t) is a complex-valued d-wave order parameter. Hereafter, 䊐 = ∂x2 − ∂y2 and 1 = ∂x2 + ∂y2 . Eq. (1.1) comes from ut = −grad F (u), where F is the simplified Ginzburg–Landau free energy for d-wave superconductors given by Z 1 1 1 |∇u|2 + 2 (1 − |u|2 )2 + η (|䊐u|2 + γ |1u|2 ) dx dy, (1.3) F (u) = 2 2 2 4 R for u = u(x, y) is a complex-valued d-wave order parameter. Hereafter, the simplified Ginzburg–Landau free energy means that we neglect the magnetic field in the Ginzburg–Landau free energy. From [4,5], we learned that ∗
Fax: +886-5-2720497. E-mail address: [email protected] (T.-C. Lin). 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 2 0 0 - 1
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it is reasonable to ignore the magnetic field in strongly type II superconductors when the applied magnetic field is close to the lower critical field Hcl and the current temperature is close to the critical temperature Tc . The simplified Ginzburg–Landau free energy (1.3) is a model for high-temperature superconductors. The hightemperature superconductors were found in some copper-oxide materials (cf. [14]). To distinguish the low- and high-temperature superconductors, the s-wave and d-wave order parameters were introduced (cf. [15,20]). Ren et al. (cf. [23,24,29]) and Soininen et al. (cf. [2,25]) derived and investigated the Ginzburg–Landau free energy with an s-wave and a d-wave order parameter. As the current temperature is close to the critical temperature Tc , Franz et al. [8] observed that in a predominantly d-wave superconductor, the s-wave order parameter is generically very small. In the absence of magnetic field, Affleck et al. [1] obtained the leading order in (1 − T /Tc ) as Ψs = ξ(∂x2 − ∂y2 )Ψd ,
(1.4)
where Ψs = Ψs (x, y) and Ψd = Ψd (x, y) are an s-wave and a d-wave order parameter, respectively. Here ξ is a parameter satisfying that ξ → 0 as T → Tc . Du [6] derived (1.4) by the formal asymptotic analysis. In fact, Affleck et al. [1] deduced the Ginzburg–Landau free energy for d-wave superconductors which can be simplified as Z 1 1 1 ˜ (1.5) |∇u|2 + 2 (1 − |u|2 )2 + η |䊐u|2 dx dy, F (u) = 2 2 2 4 R where is a positive small parameter, η a constant depending on , and u = u(x, y) is a complex-valued d-wave order parameter. Rosenstein and co-workers [4,5] investigated the Euler–Lagrange equation of (1.5) and obtained the vortex structure of d-wave superconductors. To stabilize the Ginzburg–Landau free energy (1.5), Park and Huse [22] and Wilde et al. [28] introduced other Ginzburg–Landau free energies for d-wave superconductors which can be simplified as (1.3) (cf. [16]). In [16], we restricted the integral of (1.3) on a bounded smooth domain Ω. Then the free energy (1.3) became Z 1 1 1 |∇u|2 + 2 (1 − |u|2 )2 + η (|䊐u|2 + γ |1u|2 ) dx dy. (1.6) G (u) = 2 4 Ω2 We found the minimizer of the free energy (1.6) over Hg1 (Ω) ≡ {v ∈ H 1 (Ω; C) : v = g on ∂Ω}, where g : ∂Ω → S 1 is a smooth function with degree N ∈ N. Assume that 0 < η = O( 2 ) and 0 < γ = O(1) as → 0+. Then we showed that the minimizer of the free energy (1.6) has N degree-one vortices in Ω. Furthermore, we obtained a single vortex structure having fourfold symmetry in the inner vortex core by solving the Euler–Lagrange equation of (1.6). The single vortex structure may coincide with results in [4,5,26]. The vortex structures are set in motion by a variety of forces acting on them, including the electromagnetic Lorentz force, the hydrodynamic Magnus force and the viscous drag force. The motion of vortices causes the conversion of electromagnetic energy to kinetic energy and therefore generates resistance. Hence there is theoretical and practical interest in studying the dynamics of vortices. The main result of this paper is to derive the asymptotic motion equations of vortices in d-wave superconductors. As → 0+, under a suitable time scaling and some assumptions on the vortex structures (see hypotheses (1), (2) and Assumption 1 in Section 2), the leading order terms of the motion equations of vortices in d-wave superconductors are q˙j = ∇qj W (q1 , q2 , . . . , qN ),
j = 1, . . . , N,
(1.7)
provided that η and γ satisfy −2 η → ∞,
0 < γ = O(1)
as → 0 + .
(1.8)
Note that (1.8) includes the case that γ is arbitrarily small as goes to 0. Such a case implies that (1.3) is a small perturbation of (1.5). Moreover, (1.8) includes the case that η = a , 0 < a < 2 which tends to 0 as goes to 0.
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The time scale of (1.7) may depend on the coefficients γ , η and the radius R of the inner vortex core. Under suitable choices of γ , η and R , the time scale may be of order O((log(1/))−1 ). Here N ∈ N is the number of vortices, qj = (qjx , qjy )’s are centers of vortices, and X (1.9) W (q1 , q2 , . . . , qN ) = − ni nj log|qi − qj |, i6=j
where nj ∈ {±1} is the degree of the j th vortex. Hereafter, we assume that |qj − qk | > 2r0 for j 6= k, where r0 is the radius of each vortex core and is independent of . Such an assumption is essential for the derivation of (1.7) and it excludes the collision of vortices. From (1.7), it is remarkable that q1 and q2 attract each other if N = 2, n1 = n2 = 1. Hence two degree-one vortices may attract each other. Such a new phenomenon was observed in [6] by numerical simulation. Note that the attracting of two degree-one vortices in d-wave superconductors is opposite to the repelling of two degree-one vortices in conventional s-wave superconductors. From the Ginzburg–Landau theory of conventional superconductivity (cf. [9]), the simplified Ginzburg–Landau free energy for s-wave superconductors was given by Z 1 1 (1.10) |∇Ψ |2 + 2 (1 − |Ψ |2 )2 dx dy, E (Ψ ) = 4 R2 2 where is a positive small parameter and Ψ = Ψ (x, y) is a complex-valued s-wave order parameter. As is small, the Ginzburg–Landau theory predicted the existence of vortex state. To investigate the dynamics of vortices in s-wave superconductors, many people considered the equation Ψt = −grad E (Ψ ), i.e. a nonlinear heat equation given by 1 (1 − |Ψ |2 )Ψ for (x, y) ∈ R2 , t > 0, (1.11) 2 where Ψ = Ψ (x, y, t) is a complex-valued s-wave order parameter. By the standard asymptotic analysis, Neu [21] derived the dynamics of vortices from (1.11). Moreover, E [7] improved Neu’s results and obtained a simpler dynamic law of vortices. As → 0+ and under a suitable time scaling of order O((log(1/))−1 ), the leading order terms of the motion equations of vortices are Ψt = 1Ψ +
q˙j = −∇qj W (q1 , q2 , . . . , qN ),
j = 1, . . . , N,
(1.12)
where N ∈ N is the number of vortices and qj = (qjx , qjy )’s are centers of vortices. For the proof of (1.12), please refer to [17]. Note that (1.12) the asymptotic motion equations of vortices in s-wave superconductors are exactly up to a minus sign with (1.7) the asymptotic motion equations of vortices in d-wave superconductors. In the rest of this paper, we will prove (1.7) the asymptotic motion equations of vortices in d-wave superconductors. Firstly, we prove the asymptotic motion equations of a single vortex under some assumptions on the vortex structure. The main tool of our proof is the spectrum of the linearized operator about the symmetric vortex solution of the steady-state equation of (1.11). In Section 2, we will introduce the idea and the proof of (1.7). In Section 3, we will complete the proof of the asymptotic motion equations of a single vortex.
2. Preliminaries For the dynamics of a single vortex, we assume that the solution u of (1.1) has only one vortex center at q(t), where q(t) = (qx (t), qy (t)) is smooth in t, q(0) = (0, 0), and Br0 (q(t)) is the vortex core which moves along with the vortex trajectory (x, y) = q(t). Here Br0 (q(t)) is a disk on R2 with radius r0 and center at q(t), where r0 is a positive constant independent of . Moreover, there are two essential hypotheses given by
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1. On the outer vortex core OT , the amplitude of the solution u of (1.1) has almost perfect circular symmetry. 2. When a single vortex begins to move at the time t = 0, the vortex structure on the outer vortex core OT does not change much at the time t = 0, where OT (t) ≡ {(x, y) : |(x, y) − q(t)| ∈ [R , r0 )} and R is a positive constant satisfying R → ∞,
R → 0
as → 0 + .
(2.1)
The hypothesis (1) is a generic property of d-wave superconductors (cf. [5,8,26]). From the standard asymptotic analysis (cf. [7]), we learned that each of the vortices has a core of size , and they are O(1) distance apart. Furthermore, in the outer region outside the vortex cores with size , the amplitude of the solution u has almost perfect circular symmetry. Please note that the outer vortex core OT may be a portion of the outer region outside the vortex core of size . Hence the hypothesis (1) is consistent with the assumptions in [7]. In fact, the solution u of (1.1) may loose the full rotational symmetry and have fourfold symmetry on the inner vortex core {(x, y) : |(x, y) − q(t)| < R }. Thus the vortex structure on the inner vortex core may become quite complicated. However, the hypothesis (1) gives us a simpler vortex structure on the outer vortex core so we may use it to derive the vortex dynamics. The hypothesis (2) is to preserve the vortex structure on the outer vortex core when the vortex moves. We will use Assumption 1 in the last part of this section to fulfill the hypotheses (1) and (2). Now we focus on the outer vortex core OT and consider the following system of equations: ut = 1u + (1/ 2 )(1 − |u|2 )u − η (䊐2 u + γ 12 u) for (x, y) ∈ OT (t), t > 0, u|t=0 = u0 (x, y)
for (x, y) ∈ OT (0).
(2.2)
We introduce the stretched variables X=
x − qx (t) ,
Y=
y − qy (t) ,
(2.3)
and we set Ψ (X, Y, t, ) = u(x, y, t, ) for (x, y) ∈ OT , i.e. (X, Y) ∈ Ω , t ≥ 0, where Ω ≡ {(X, Y) : |(X, Y)| ∈ [R , (r0 /))}. Then we have ˜ + 1Ψ ˜ + (1 − |Ψ |2 )Ψ − −2 η P (Ψ ) for (X, Y) ∈ Ω , t > 0, 2 Ψt = q˙ · ∇Ψ Ψ |t=0 = u0
for (X, Y) ∈ Ω ,
(2.4)
where ˜ 2 ), P = (䊐2∗ + γ 1
∇˜ = (∂X , ∂Y ),
˜ = ∂2 + ∂2 , 1 X Y
2 䊐∗ = ∂X − ∂Y2 ,
(2.5)
and Ψt = (∂/∂t)Ψ (·, ·, t, ). We take an expansion form of Ψ as follows: Ψ (X, Y, t, ) = Ψ0 (X, Y)eiH + Ψ1 (X, Y, t, )eiH ,
(2.6)
where Ψ0 = Ψ0 (X, Y) satisfies Ψ0 (X, Y) = f0 (R)eiθ ,
R = |(X, Y)|, θ = arg(X, Y),
(2.7)
and f0 (R) is the solution of −f 00 −
1 0 1 f + 2 f = (1 − f 2 )f R R
for R > 0, f (+∞) = 1, f (0) = 0, f ≥ 0.
(2.8)
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From [3,10,12], we learned that (2.8) has a unique solution and Ψ0 is called the symmetric vortex solution of the steady-state equation of (1.11) in R2 . In addition, we assume that H = H (x, y, t, ) is a smooth real-valued function and satisfies 1H = 0,
|∇ n H |, |Ht |, |∇Ht | ≤ K
for (x, y) ∈ Br0 (q(t)), t > 0, n = 1, . . . , 5,
(2.9)
where Ht = (∂/∂t)H and K is a positive constant independent of . By (2.6)–(2.9), (2.4) becomes ˜ 1 ) − iΨ0 Ht − i 2 Ψ1 Ht − 2 Ψ1,t ˜ 0 + ∇Ψ q˙ · (∇Ψ ˜ 0 · ∇H ) + L˜ (Ψ1 ) + Nˆ (Ψ1 ) + −3 η e−iH P (Ψ0 eiH ) + −2 η e−iH P (Ψ1 eiH ) = Ψ0 |∇H |2 − 2i(∇Ψ for (X, Y) ∈ Ω , t > 0,
(2.10)
where ˜ 1 + (1 − |Ψ0 |2 )Ψ1 − 2(Ψ0 · Ψ1 )Ψ0 , −L˜ (Ψ1 ) = 1Ψ ˜ 1 · ∇H ) + 2 |Ψ1 |2 Ψ1 + [2(Ψ0 · Ψ1 )Ψ1 + |Ψ1 |2 Ψ0 ], Nˆ (Ψ1 ) = 2 Ψ1 |∇H |2 − 2i(∇Ψ
(2.11)
and Ψ1,t = (∂/∂t)Ψ1 (·, t, ). Note that in the first term of (2.11), (Ψ0 · Ψ1 ) = 21 (Ψ¯ 0 Ψ1 + Ψ¯ 1 Ψ0 ) and (¯·) denotes the complex conjugate. The spectrum information of the linear operator L˜ is essential for our argument. From [18,19], we have estimates on the eigenvalues of L˜ as follows. Theorem 1. Let λ1 and λ2 be the first and the second eigenvalue of L˜ . Then 1. The eigenvalue λ1 has only two associated eigenfunctions e˜1 = a (R)+b (R) e2iθ and ˜2 = ia (R)−ib (R) e2iθ , where a and b are real-valued. 2. 0 < λ1 = O( 2 (log(r0 /))−1 ) as → 0+. 3. There exist 1 > 0 and c1 > 0 independent of such that λ2 ≥ 2 c1 for 0 < ≤ 1 . Let w˜ 1 =
∂X Ψ0 , k∂X Ψ0 kL2
w˜ 2 =
∂Y Ψ0 , k∂Y Ψ0 kL2
(2.12)
where k · kL2 is the L2 norm on the disk B(r0 /) (0). We observe that w˜ j ’s are from the translation invariance. In [19], we used Theorem 1 to derive a useful estimate of eigenfunctions e˜j ’s as follows. Proposition 1. Assume that hw˜ j , e˜j i > 0,
ke˜j kL2 = 1, j = 1, 2.
Then the eigenfunctions e˜j ’s satisfy e˜j = w˜ j + νj, ,
kνj, kL2 = O
1 log
−1/2 !
as → 0+, for j = 1, 2.
Hereafter, we use k · kL2 and h·, ·i to denote the L2 norm and the L2 inner product, respectively, on the disk Br0 / (0). Let V˜1 = {a(R) + b(R) e2iθ ∈ H01 (Br0 / (0); C)}. Then it is easy to check that V˜1 is invariant under L˜ −1 . Hence we
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set e˜j,k ∈ V˜1 ’s the unit L2 norm eigenfunctions of L˜ corresponding to the eigenvalue λ˜ k , respectively, for k ≥ 2. Then e˜j ’s and e˜j,k ’s are dense in V1 = {a(R) + b(R) e2iθ ∈ L2 (Br0 / (0); C)}. Since w˜ l ∈ V1 , l = 1, 2, then w˜ l ’s can be represented by e˜j ’s and e˜j,k ’s as follows: (k) 2 ∞ JX X X hw˜ l , e˜j ie˜j + hw˜ l , e˜j,k ie˜j,k , w˜ l = j =1
(2.13)
k=2 j =1
where J (k) is the multiplicity of the eigenvalue λ˜ k , k ≥ 2. Hereafter, we denote as h·, ·i∗ the L2 inner product on the outer core Ω . Using integration by parts, we have Z Z BD(Ψ1 , e˜j ) + λ1 hΨ1 , e˜j i∗ , hL˜ (Ψ1 ), e˜j,k i∗ = BD(Ψ1 , e˜j,k ) + λ˜ k hΨ1 , e˜j,k i∗ , hL˜ (Ψ1 ), e˜j i∗ = ∂Ω
∂Ω
(2.14) where BD(Ψ1 , ·) is defined by BD(Ψ1 , v) = (Ψ1 · ∂nˆ v) − (∂nˆ Ψ1 · v) for v is a smooth complex-valued function, ¯ for α, β ∈ C. Hence by (2.13) and (2.14) and and ∂nˆ is the normal derivative. Hereafter, (α · β) ≡ 21 (αβ ¯ + βα) Proposition 1, we obtain 2 X
hL˜ (Ψ1 ), w˜ l i∗ =
λ1 hΨ1 , e˜j i∗ +
j =1
+
Z ∂Ω
BD(Ψ1 , e˜j ) hw˜ l , e˜j i
(k) ∞ JX X
λ˜ k hΨ1 , e˜j,k i∗ +
k=2 j =1
= o
log
1
−1/2
! R−2 ,
Z ∂Ω
BD(Ψ1 , e˜j,k ) hw˜ l , e˜j,k i
l = 1, 2,
(2.15)
provided that Ψ1 satisfies |hΨ1 , e˜j i∗ | = o Z
−1
1 log
1/2
! R−2
,
j = 1, 2,
! −1/2 1 −2 BD(Ψ1 , e˜j ) = o R log , ∂Ω
(k) ∞ JX X λ˜ k |hΨ1 , e˜j,k i∗ | = o (R−2 ), k=2 j =1
j = 1, 2,
(k) Z ∞ JX X k=2 j =1
∂Ω
BD(Ψ1 , e˜j,k ) = o (R−2 ). (2.16)
Hereafter, we denote as o (1) a small quantity, independent of time t, and tending to 0 as → 0+. In Section 3, we will derive the asymptotic motion equations of the vortex trajectory (x, y) = q(t). Under a suitable time scaling, the leading order terms of the asymptotic motion equations are
1 q˙ = log
−1
(c3 + γ c4 )
where J =
0 −1
1 0
,
−2
R−2 η ∇ H˜ (q, t, ) +
1 2 ˜ r ∇ Ht (q, t, ) , 2 0
(2.17)
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299
c3 and c4 are positive constants independent of . In addition, H and H˜ are harmonic conjugates. The main idea of the proof of (2.17) is to make the L2 inner product with (2.10) and w˜ j , j = 1, 2. For details, we refer to Section 3. We require that Ψ1 satisfies the “small” perturbation condition (2.16) and ! ! 1 −1/2 −2 1 −1/2 −2 −1 −1 R R , kΨ1 kL2 (Ω ) = o , log log kΨ1,t kL2 (Ω ) = o ! ! −1/6 −1/4 1 1 R−2/3 , R−1 , kΨ1 kL4 (Ω ) = o log kΨ1 kL6 (Ω ) = o −1/3 log ! ! −1/2 −1/2 1 1 −2 n n−4 −2 ˜ 1 kL2 (Ω ) = o R R log , k∇˜ Ψ1 kL2 (Ω ) = o , n = 2, 3, 4. log k∇Ψ (2.18) Suppose that R satisfies R → ∞,
1 1/2 R2 log →0
as → 0 + .
(2.19)
Then the upper bound of the first term of (2.16), the first three terms of (2.18) and the last term of (2.18) as n = 2, 3 may tend to infinity as goes to 0. From [11], Eq. (1.1) is well posed. Then Ψ1 is smooth in both space and time variables. Hence (2.16) and (2.18) can be fulfilled at least in a short time when Ψ1 |t=0 satisfies. Assumption 1. Ψ1 = Ψ1 (X, Y, t, ) satisfies 1. kΨ1 (·, ·, 0, )kC 4 (Ω˜ ) is sufficiently small, where Ω˜ ≡ {(X, Y) : 21 R ≤ |(X, Y)| ≤ 2r0 /} at t = 0, 2. k∂t Ψ1 (·, ·, 0, )kL2 (Ω˜ ) = o ( −1 (log(1/))−1/2 R−2 ). By the suitable choice of initial data u0 in (1.2), Assumption 1(1) can be satisfied. Assumption 1(2) fulfills the requirement that the vortex structure, i.e. the solution u on the outer vortex core does not change much at t = 0 when the associated vortex begins to move at t = 0. Note that by (2.19), the upper bound of Assumption 1(2) may tend to infinity as goes to 0. Actually, Assumption 1 which may assure (2.16) and (2.18) are more generalized than the assumptions of the standard asymptotic analysis. To derive the dynamics of vortices, the standard asymptotic analysis is well-accepted (cf. [7,21,27]). E [7] and Neu [21] used a specific asymptotic expansion formula and some pointwise conditions for the solution on vortex cores to derive the dynamics of vortices. However, (2.16) and (2.18) are not pointwise. This is a kind of generalization for the assumptions of the standard asymptotic analysis. For the motion of N -vortices, we restrict (1.1) on the vortex cores Br0 (qj )’s and we consider the following system of equations: ut = 1u + (1/ 2 )u(1 − |u|2 ) − η (䊐2 u + γ 12 u) for (x, y) ∈ OTj (t), t > 0, u|t=0 = uj 0 (x)
for (x, y) ∈ OTj (0),
(2.20)
where (x, y) = qj (t) = (qjx (t), qjy (t)) is the j th vortex trajectory, qj the smooth in t, |qj − qk | > 2r0 for j 6= k, j and OT (t) = {(x, y) : |(x, y) − qj (t)| ∈ (R , r0 )} the j th outer vortex core which moves along with the j th vortex trajectory (x, y) = qj (t), j = 1, . . . , N. Now we assume that Xj =
x − qjx (t) ,
Yj =
y − qjy (t) ,
Ψ (Xj , Yj , t, ) = u(x, y, t, )
(2.21)
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j
for (x, y) ∈ OT , i.e. (Xj , Yj ) ∈ Ω = {(Xj , Yj ) : |(Xj , Yj )| ∈ (R , r0 /)}, j = 1, . . . , N. As for (2.6), we take a similar expansion form of Ψ on each outer vortex core as follows: Ψ (Xj , Yj , t, ) = Ψ0 (Xj , Yj )eiHj + Ψ1 (Xj , Yj , t, ) eiHj
for (Xj , Yj ) ∈ Ωj ,
(2.22)
where Ψ0 (Xj , Yj ) = f0 (Rj )einj φj , Here we assume that X nk φk + H, 1H = 0 Hj =
nj ∈ {±1},
Rj = |(Xj , Yj )|,
φj = arg(Xj , Yj ).
for (x, y) ∈ Br0 (qj (t)), t > 0, j = 1, . . . , N,
k6=j
|∇ n H |, |Ht |, |∇Ht | ≤ K
for (x, y) ∈ Br0 (qj (t)), t > 0, n = 1, . . . , 5,
where nk ∈ {±1}. As for (2.17), we obtain the equations of qj given by 1 −1 1 2 −2 −2 ˜ ˜ nj (c3 + γ c4 ) R η ∇ Hj (qj , t, ) + r0 ∇∂t Hj (qj , t, ) , q˙j = log 2
(2.23)
(2.24)
where Hj and H˜ j ’s are harmonic conjugates. Here we require that Ψ1 satisfies the “small” perturbation conditions j (2.16) and (2.18) for (Xj , Yj ) ∈ Ω , j = 1, . . . , N. Of course, such a requirement can be fulfilled by Assumption j 1 on outer vortex cores Ω ’s. In particular, suppose H ≡ 0. Then (2.23) and (2.24) imply that 1 −1 (c3 + γ c4 ) −2 R−2 η nj ∇ H˜ j (qj , t), j = 1, . . . , N, (2.25) q˙j = log Under a suitable time scaling, the leading order terms of (2.25) becomes (1.7) and we complete the proof of (1.7). Remark 1. The time scale of (2.25) is of order ! 1 −1 −2 −2 (c3 + γ c4 ) R η , O log which depends on γ , η and R . In particular, we may choose suitable γ , η and R (e.g. γ = 1, η = 3/2 , R = −1/4 ) such that the time scale of (2.25) is of order O((log(1/))−1 ). Hence the time scale for the dynamics of d-wave Ginzburg–Landau vortices may be equal to the time scale for the dynamics of s-wave Ginzburg–Landau vortices.
3. Proof of the vortex dynamics in d -wave superconductors In this section, we prove (2.17) as follows. From (2.10), we have q˙x (∂X Ψ0 + ∂X Ψ1 ) + q˙y (∂Y Ψ0 + ∂Y Ψ1 ) − iΨ0 Ht − i 2 Ψ1 Ht − 2 Ψ1,t = Ψ0 |∇H |2 − 2i(∂X Ψ0 ∂x H + ∂Y Ψ0 ∂y H ) + −3 η e−iH P (Ψ0 eiH ) + −2 η e−iH P (Ψ1 eiH ) + L˜ Ψ1 + Nˆ (Ψ1 ) for (X, Y) ∈ B(r0 /) (0), t > 0.
(3.1)
Making the inner product with (3.1) and w˜ j , j = 1, 2, we have q˙x [h∂X Ψ0 , w˜ j i∗ + h∂X Ψ1 , w˜ j i∗ ] + q˙y [h∂Y Ψ0 , w˜ j i∗ + h∂Y Ψ1 , w˜ j i∗ ] = 2 (hΨ1,t , w˜ j i∗ + hiΨ1 Ht , w˜ j i∗ ) + hNˆ (Ψ1 ), w˜ j i∗ + Γj + hL˜ Ψ1 , w˜ j i∗
(3.2)
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301
for j = 1, 2, where Γj = −2hi∂X Ψ0 ∂x H, w˜ j i∗ − 2hi∂Y Ψ0 ∂y H, w˜ j i∗ + hΨ0 |∇H |2 , w˜ j i∗ + hiΨ0 Ht , w˜ j i∗ + −3 η [he−iH P (Ψ0 eiH ), w˜ j i∗ + he−iH P (Ψ1 eiH ), w˜ j i∗ ].
(3.3)
It is obvious that e−iH 䊐2∗ (Ψk eiH ) = 䊐2∗ Ψk + 4i(HX ∂X 䊐∗ Ψk − HY ∂Y 䊐∗ Ψk ) + S(Ψk ), ˜ 2 (Ψk eiH ) = 1 ˜ k + HY ∂Y 1Ψ ˜ k ) + T (Ψk ), k = 0, 1, ˜ 2 Ψk + 4i(HX ∂X 1Ψ (3.4) e−iH 1 Q α β j j where S(Ψk ) and T (Ψk ) are linear combinations of ∂Xl ∂Ym Ψk 4j =1 (∂X ∂Y H )γj ’s, for l, m ≥ 0, l + m ≤ 2, αj , βj , P γj ≥ 0, j = 1, . . . , 4 and l + m + 4j =1 (αj + βj )γj = 4, respectively. Hence (2.5) and (3.4) imply that e−iH P (eiH Ψk ) = 4i[(HX ∂X 䊐∗ Ψk − HY ∂Y 䊐∗ Ψk ) ˜ k + HY ∂Y 1Ψ ˜ k )] + P (Ψk ) + S(Ψk ) + γ T (Ψk ). +γ (HX ∂X 1Ψ From (2.7), it is easy to check that 1 1 1 1 f0 + f00 + f00 − f0 e2iθ , ∂X Ψ0 = 2 R 2 R
∂Y Ψ0 =
i 2
(3.5)
1 i 1 f0 + f00 + f0 − f00 e2iθ . (3.6) R 2 R
Moreover,
1 1 3 1 1 1 3 1 0 0 2iθ 0 −2iθ 0 + g + g1 + g + g3 e + g − g1 e g − g3 e4iθ , ∂ X 䊐∗ Ψ0 = 4 1 R 4 3 R 4 1 R 4 3 R i 1 3 i 1 i 3 i g10 + g1 + g30 + g3 e2iθ + g10 − g1 e−2iθ − g30 − g3 e4iθ , ∂ Y 䊐∗ Ψ0 = − 4 R 4 R 4 R 4 R 䊐2∗ Ψ0 = a1 (R)e−3iθ + a2 (R)eiθ + a3 (R)e5iθ ,
(3.7)
where aj , j = 1, 2, 3 are smooth real-valued functions and 0 0 1 1 2 1 0 0 0 f0 − f0 . g3 = g3 (R) = f0 − f0 − g1 = g1 (R) = f0 + f0 , R R R R Furthermore, ˜ 0= ∂X 1Ψ
1 1 1 1 g10 + g1 + g10 − g1 e2iθ , 2 R 2 R
˜ 0= ∂Y 1Ψ
(3.8)
i 1 1 i g10 + g1 − g10 − g1 e2iθ , 2 R 2 R
˜ 2 Ψ0 = b(R)eiθ , 1
(3.9)
where b is a smooth real-valued function. Thus by (2.1), (2.12), (3.6)–(3.9), we obtain that h∂X Ψ0 , w˜ 2 i∗ = h∂Y Ψ0 , w˜ 1 i∗ = 0,
hiHX ∂X Ψ0 , w˜ 1 i∗ = hiHY ∂Y Ψ0 , w˜ 2 i∗ = 0, 1 1/2 (1 + o (1)), log
h∂X Ψ0 , w˜ 1 i∗ = h∂Y Ψ0 , w˜ 2 i∗ = c2 ˜ 2 Ψ0 , w˜ j i∗ = 0, h䊐2∗ Ψ0 , w˜ j i∗ = h1
j = 1, 2,
(3.10)
where c2 is a positive constant independent of . Here we have used the fact that f0 (R) = 1 − 21 R −2 − 98 R −4 + · · ·
as R → +∞
(3.11)
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(cf. [3,12]). Note that by (3.8) and (3.11), we obtain that g1 (R) ∼ −R −2 ,
g3 (R) ∼ 3R −2
as R → +∞.
(3.12)
By (2.1), (3.6) and (3.11) and the mean-valued theorem of harmonic functions, we have hiHx ∂X Ψ0 , ∂Y Ψ0 i∗ = c0 R−2 (Hx (q, t, ) + o (1)),
hiHy ∂Y Ψ0 , ∂X Ψ0 i∗ = −c0 R−2 (Hy (q, t, ) + o (1)),
(3.13)
where c0 is a positive constant independent of . From (3.6) and (3.7), we obtain that Z r0 / Z 2π R[α1 (R) + α2 (R) cos(2θ ) + α3 (R) cos(4θ )]HX dθ dR, hiHX ∂X 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = R
Z
h−iHY ∂Y 䊐∗ Ψ0 , ∂X Ψ0 i∗ = Z hiHX ∂X 䊐∗ Ψ0 , ∂X Ψ0 i∗ =
R 0 r0 / Z 2π
R
Z
h−iHY ∂Y 䊐∗ Ψ0 , ∂Y Ψ0 i∗ =
0 r0 / Z 2π
R[β1 (R) sin(2θ ) − α3 (R) sin(4θ )]HX dθ dR,
0 r0 / Z 2π
R
0
R[−α1 (R) + α2 (R) cos(2θ ) − α3 (R) cos(4θ )]HY dθ dR,
R[−β1 (R) sin(2θ ) − α3 (R) sin(4θ )]HY dθ dR,
where
1 1 1 1 g10 + g1 f0 − f00 f00 + f0 + g30 + 8 R R R 1 α2 (R) = R −1 [(g1 + g3 )0 f0 + (3g3 − g1 )f00 ], 4 1 3 1 1 g30 − g3 f0 − f00 f00 + f0 + g10 − α3 (R) = 8 R R R 1 β1 (R) = [(g1 − g3 )0 f00 − R −2 (g1 + 3g3 )f0 ]. 4 Moreover, by (3.11), (3.12) and (3.15), we have α1 (R) =
α1 (R) ∼ 21 R −4 ,
α2 (R) ∼ −R −4 ,
α3 (R) ∼ − 23 R −4 ,
3 g3 R
1 g1 R
(3.14)
,
,
β1 (R) ∼ −2R −4
(3.15)
as R → +∞.
By (2.9) and the standard Poisson integral formulas for harmonic functions (cf. [13]), we may derive that Z 1 2π HX (R, θ, t, ) dθ, HX |X=Y=0 = 2π 0 Z 1 2π HY (R, θ, t, ) dθ, HY |X=Y=0 = 2π 0 Z 2π 2 HX (R, θ, t, ) cos(2θ ) dθ, ∂X3 H |X=Y=0 = R −2 π 0 Z 2π Z 2π 2 2 HY (R, θ, t, ) cos(2θ ) dθ = R −2 HX (R, θ, t, ) sin(2θ ) dθ, ∂X2 ∂Y H |X=Y=0 = R −2 π π 0 0 Z 24 −4 2π HX (R, θ, t, ) cos(4θ ) dθ, R ∂X5 H |X=Y=0 = π 0 Z Z 24 −4 2π 24 −4 2π R R ∂X4 ∂Y H |X=Y=0 = HY (R, θ, t, ) cos(4θ ) dθ = HX (R, θ, t, ) sin(4θ ) dθ π π 0 0
(3.16)
(3.17)
T.-C. Lin / Physica D 149 (2001) 293–305
303
for 0 < R < (r0 /). Hence (2.9), (3.14), (3.16) and (3.17) imply that hiHX ∂X 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = c3 R−2 (Hx (q, t, ) + o (1)),
h−iHY ∂Y 䊐∗ Ψ0 , ∂X Ψ0 i∗ = −c3 R−2 (Hy (q, t, ) + o (1)),
hiHX ∂X 䊐∗ Ψ0 , ∂X Ψ0 i∗ = R−2 o (1),
h−iHY ∂Y 䊐∗ Ψ0 , ∂Y Ψ0 i∗ = R−2 o (1),
(3.18)
where c3 is a positive constant independent of . Here we have used the rescaling (2.3) on spatial variables such that j
j
∂X ∂Yk H = j +k ∂x ∂yk H
for j, k ≥ 0.
(3.19)
Similarly, ˜ 0 , ∂Y Ψ0 i∗ = c4 R−2 (Hx (q, t, ) + o (1)), hiHX ∂X 1Ψ ˜ 0 , ∂X Ψ0 i∗ = −c4 R−2 (Hy (q, t, ) + o (1)), hiHY ∂Y 1Ψ ˜ 0 , ∂X Ψ0 i∗ = R−2 o (1), hiHX ∂X 1Ψ ˜ 0 , ∂Y Ψ0 i∗ = R−2 o (1), hiHY ∂Y 1Ψ
(3.20)
where c4 is a positive constant independent of . From (2.1), (2.9), (3.5), (3.10), (3.18)–(3.20), we obtain that he−iH P (eiH Ψ0 ), ∂X Ψ0 i∗ = −4(c3 + γ c4 )R−2 (Hy (q, t, ) + o (1)),
he−iH P (eiH Ψ0 ), ∂Y Ψ0 i∗ = 4(c3 + γ c4 )R−2 (Hx (q, t, ) + o (1)).
(3.21)
By (2.15) and (3.2), we have q˙x (Γ + α1 ) + q˙y β1 = γ1 ,
q˙x β2 + q˙y (Γ + α2 ) = γ2 ,
(3.22)
where = k∂Y Ψ0 k2L2 (Ω ) k∂Y Ψ0 k−1 , Γ ≡ k∂X Ψ0 k2L2 (Ω ) k∂X Ψ0 k−1 L2 L2
α2 = h∂Y Ψ1 , w˜ 2 i∗ ,
β1 = h∂Y Ψ1 , w˜ 1 i∗ ,
α1 = h∂X Ψ1 , w˜ 1 i∗ ,
β2 = h∂X Ψ1 , w˜ 2 i∗ ,
ηj = hNˆ (Ψ1 ), w˜ j i∗ ,
and γj = Γj + (hΨ1,t , w˜ j i∗ + hiΨ1 Ht , w˜ j i∗ ) + ηj + o 2
1 log
−1/2
! R−2
,
for j = 1, 2. From (3.6) and (3.11), we obtain 1 1/2 (κ0 + o (1)) as → 0+, Γ = log where κ0 is a positive constant independent of . By (2.9) and (2.18) and the Hölder inequality, we have ! ! 1 −1/2 −2 1 −1/2 −2 R R |αj | = o log , |βj | = o log , ! ! 1 −1/2 −2 1 −1/2 −2 log , γj = Γj + o log , j = 1, 2, R R |ηj | = o
(3.23)
(3.24)
304
T.-C. Lin / Physica D 149 (2001) 293–305
and he−iH P (eiH Ψ1 ), ∂X Ψ0 i∗ = o (R−2 ),
he−iH P (eiH Ψ1 ), ∂Y Ψ0 i∗ = o (R−2 ).
(3.25)
Hence (3.22) implies that q˙x =
1 [(Γ + α2 )γ1 − β1 γ2 ], γ
q˙y =
1 [(Γ + α1 )γ2 − β2 γ1 ], γ
(3.26)
where γ = Γ2 + Γ (α1 + α2 ) + 2 (α1 α2 − β1 β2 ). Thus by (2.9), (3.6) and (3.11) and the mean-valued theorem of harmonic functions, we have Z r0 / C , C = 2πK 2 Rf0 f00 dR ∼ R−1 , |hΨ0 |∇H |2 , w˜ j i| ≤ Γ R Z r0 / π Rf 20 dR ∂y Ht (q, t, ), 2 hiΨ0 Ht , w˜ 1 i = − Γ R Z r0 / π 2 2 Rf 0 dR ∂x Ht (q, t, ). (3.27) hiΨ0 Ht , w˜ 2 i = Γ R By (1.8), (3.3), (3.13), (3.21), (3.25) and (3.27), we obtain Γ1 = −c5 Γ−1 [(c3 + γ c4 ) −2 R−2 η (∂y H (q, t, ) + o (1)) + c ∂y Ht (q, t, )],
(3.28) Γ2 = c5 Γ−1 [(c3 + γ c4 ) −2 R−2 η (∂x H (q, t, ) + o (1)) + c ∂x Ht (q, t, )], R r / where c5 is a positive constant independent of and c = 2 R0 Rf 20 dR > 0. Note that by (2.1) and (3.11), c ∼ 21 r02
as → 0 + .
(3.29)
Therefore by (3.23), (3.24), (3.26), (3.28) and (3.29), we complete the proof of (2.17). References [1] I. Affleck, M. Franz, M.H.S. Amin, Generalized London free energy for high-Tc vortex lattices, Phys. Rev. B 55 (1997) R704. [2] A.J. Berlinsky, A.L. Fetter, M. Franz, C. Kallin, P.I. Soininen, Ginzburg–Landau theory of vortices in d-wave superconductors, Phys. Rev. Lett. 75 (1995) 2200. [3] X. Chen, C.M. Elliott, T. Qi, Shooting method for vortex solutions of a complex-valued Ginzburg–Landau equation, Proc. Roy. Soc. Edinburgh 124A (1994) 1075–1088. [4] D. Chang, C.Y. Mu, B. Rosenstein, C.L. Wu, The effect of anisotropy on vortex lattice structure and flux flow in d-wave superconductors, Chinese J. Phys. 36 (2) (1998) 238–244. [5] D. Chang, C.Y. Mu, B. Rosenstein, C.L. Wu, Static and dynamical anisotropy effects in mixed state of d-wave superconductors, Phys. Rev. B 57 (13) (1998) 7955–7969. [6] Q. Du, Studies of a Ginzburg–Landau model for d-wave superconductors, SIAM J. Appl. Math. 59 (4) (1999) 1225–1250. [7] W. E, Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity, Physica D 77 (1994) 383–404. [8] M. Franz, C. Kallin, P.I. Soininen, A.J. Berlinsky, A.L. Fetter, Vortex state in a d-wave superconductor, Phys. Rev. B 53 (9) (1996) 5795. [9] P.G. de Gennes, Superconductivity of Metals and Alloys, Addison-Wesley, Reading, MA, 1989. [10] P. Hagan, Spiral waves in reaction diffusion equations, SIAM J. Appl. Math. 42 (4) (1982) 762–786. [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer, Berlin, 1981. [12] R.M. Hervé, M. Hervé, Étude qualitative des solutions réelles d’une équation différentielle liée à l’équation de Ginzburg–Landau, Ann. Inst. Henri Poincaré 11 (1994) 427–440. [13] F. John, Partial Differential Equations, 4th Edition, Springer, New York, 1982. [14] J.D. Jorgensen, et al., Structural features that optimize high temperature superconductivity, in: B.W. Veal, B.M. Dabrowski, P.W. Klamut (Eds.), Proceedings of the First Polish US Conference on Recent Developments in High Temperature Superconductivity, Springer, Berlin, 1996, pp. 1–15.
T.-C. Lin / Physica D 149 (2001) 293–305 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
305
G. Kotliar, Resonating valence bonds and d-wave superconductivity, Phys. Rev. B 37 (1988) 3664. F.H. Lin, T.C. Lin, Vortex state of d-wave superconductors in the Ginzburg–Landau energy, SIAM J. Math. Anal. 32 (3) (2000) 493–503. F.H. Lin, J.X. Xin, On the dynamical law of the Ginzburg–Landau vortices on the plane, Comm. Pure Appl. Math. 52 (10) (1999) 1189–1212. T.C. Lin, The stability of the radial solution to the Ginzburg–Landau equation, Comm. PDE 22 (1997) 619–632. T.C. Lin, Spectrum of the linearized operator for the Ginzburg–Landau equation, Electron. J. Diff. Eqns. 2000 (42) (2000) 1–25. K.A. Müller, Possible coexistence of s- and d-wave condensates in copper oxide superconductors, Nature 377 (1995) 133–135. J.C. Neu, Vortices in complex scalar fields, Physica D 43 (1990) 385–406. K. Park, D.A. Huse, The phase transition to a square vortex lattice in type-II superconductors with fourfold anisotropy, Phys. Rev. B 58 (14) (1998) 9427–9432. Y. Ren, J.H. Xu, C.S. Ting, Ginzburg–Landau equations and vortex structure of a dx 2 −y 2 superconductor, Phys. Rev. Lett. 74 (1995) 3680. Y. Ren, J.H. Xu, C.S. Ting, Ginzburg–Landau equations for mixed s + d symmetry superconductors, Phys. Rev. B 53 (5) (1996) 2249. P.I. Soininen, C. Kallin, A.J. Berlinsky, Structure of a vortex line in a dx 2 −y 2 superconductor, Phys. Rev. B 50 (1994) 13883. J. Shiraishi, M. Kohmoto, K. Maki, Vortex lattice transition in d-wave superconductors, Phys. Rev. B 59 (6) (1999) 4497–4503. L. Ting, R. Klein, Viscous Vortical Flows, Lecture Notes in Physics, Vol. 374, Springer, Berlin, 1991. Y. De Wilde, M. Iavarone, U. Welp, V. Metlushko, A.E. Koshelev, I. Aranson, G.W. Crabtree, P.C. Canfield, Scanning tunneling microscopy observation of a square Abrikosov lattice in LuNi2 B2 C, Phys. Rev. Lett. 78 (1997) 4273. J.H. Xu, Y. Ren, C.S. Ting, Structures of single vortex and vortex lattice in a d-wave superconductor, Phys. Rev. B 53 (6) (1996) 2991.