Aspect on vortex structure in noncentrosymmetric superconductors

Annals of Physics 325 (2010) 1316–1325

Contents lists available at ScienceDirect

Annals of Physics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a o p

Aspect on vortex structure in noncentrosymmetric superconductors Tao Xu College of Electric and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 20 February 2010 Accepted 15 March 2010 Available online 13 April 2010

Helicity of vortices is studied in a single order parameter Ginzburg– Landau model with n × B term. This unusual term changes the superconductor current, but does not change the quantum of magnetic flux, and the cross helicity of the probability density current in noncentrosymmetric superconductors. In this paper I compute the kinetic helicity and magnetic helicity of a superconductor and introduce a new helicity, i.e. a new topological invariant, to describe vortex phase. © 2010 Elsevier Inc. All rights reserved.

Keywords: Ginzburg–Landau model Vortices Helicity

1. Introduction In most superconducting materials, there is an inversion center in the crystal which guarantees the parity conservation [1,2]. According to the parity symmetry of their pairing wave function [3], the superconductive fluids can be labeled as s-wave, p-wave, and d-wave super-fluids, respectively, and each has distinct thermodynamic and transport properties. However, when a superconductor lacks a crystal inversion center [4–6], the above-described rule (parity conservation) is violated due to the asymmetric spin–orbit coupling, and the pairing symmetry becomes nontrivial [7,8]. Theoretical studies based on the addition of a spin–orbital interaction in the Hamiltonian predict the Cooper pair to be a mixed state of singlets and triplets in pseudospin space. Recently, superconductivity has been discovered in a number of compounds lacking inversion symmetry, such as CePt3Si [4], UIr [9], CeRhSi3 [10], and Y2C3 [11], CeIrSi3 [5], Li2(Pd1 − z,Ptx)3B [12], KOs2O6 [13], and many others. The broken parity symmetry implies that these materials all allow a Rashba spin–orbit coupling [14–17]. The complete theory of the behavior of superconductor in a magnetic field is very complex, the Ginzburg–Landau theory for superconductivity represents one of the most useful tools available for the theoretical description of superconductivity. The total free energy [18] of a superconducting body is ( 2

F = Fno + ∫ ajψj +

) 2 β 4 1 1 B 2 2 jψj + jDψj + jDz ψj + ζn·B × ½ψðDψÞ* + ψ* ðDψÞ + dτ ð1Þ 2 2m 2m 8π

E-mail address: [email protected]. 0003-4916/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.03.003

T. Xu / Annals of Physics 325 (2010) 1316–1325

1317

where Fno is the free energy of the body in the normal state in the absence of the magnetic field, a = a0(T – Tc), D = (Dx, Dy), Di = –i∂i – 2eAi/c, c is velocity of light, n is the unit vector oriented along the c axis, m is the boson mass, and B = ∇ × A. Eq. (1) applies to all possible pairing symmetries with single complex order parameter, as discussed also by Samokhin [6]. Varying the integral with respect to ψ⁎, putting δF = 0, we obtain the Ginzburg–Landau (GL) equation 2

aψ + βjψj ψ +

1 2 D ψ + 2ζn × B·Dψ = 0: 2m

ð2Þ

Similarly, varying the integral with respect to A gives Maxwell's equation    2 c ie 4e −i −i 2e 2 2 2 ∇ × B = − ðψ*∇ψ−ψ∇ψ* Þ− ψ*∇ψ− ψ∇ψ*− Ajψj jψj A−2ζc∇ × n × + 4eζjψj ðn × BÞ; 4π m 2 2 c mc

ð3Þ or c ie 4e2 2 2 ∇ × ðB−4πMÞ = − ðψ*∇ψ−ψ∇ψ* Þ− jψj A + 4eζjψj ðn × BÞ; 4π m mc

ð4Þ

where M is the magnetization due to the ζ term " # 2 ζm ie 4e 2 − ðψ*∇ψ−ψ∇ψ* Þ− jψj A : M= e m mc

ð5Þ

The superconductivity current density Js is Js ie 4e2 ðψ*∇ψ−ψ∇ψ* Þ− =− A + 4eζðn × BÞ: mψ*ψ ρ mc

ð6Þ

where ρ = |ψ|2 is the superconducting electron density. The lack of inversion symmetry allows for the existence of the term proportional to ζ. This term induces a spatially modulated solution in a uniform magnetic field. It is obvious that the GL equation for the order parameter ψ(R) = e− iq·Rψ0(R) in the helical vortex phase, where ψ0(R) is the order parameter of the zero-ζ GL equation, and q = −2mζn × B:

ð7Þ

Then Eq. (6) can be written as 2

Js 4e 2e =− A + 2Vp − q; m ρ mc

ð8Þ

where the velocity field Vp is defined in terms of the probability current Vp =

−ie ðψ*∇ψ−ψ∇ψ* Þ: 2mψ*ψ

ð9Þ

Within the body (beyond the range of penetration of the field) the current density Js = 0; the vector potential is not zero, but only its curl, i.e. the magnetic induction B. We take any closed contour and passing through the torus far from the surface, so condition (6) is satisfied, namely the slowness of the spatial variation of the phase and the potential A. The circulation of the vector A along the contour is equal to the flux of magnetitic induction through a surface spanning the contour, i.e. ∮AdI = ∫B·ds;

ð10Þ

1318

T. Xu / Annals of Physics 325 (2010) 1316–1325

On the other hand, equating Eq. (6) to zero and integrating it along the contour, we have ∮AdI =

mc ζmc ∮ðn × BÞdI: ∮Vp dI + e 2e2

ð11Þ

The second term of the right hand side of Eq. (11) vanishes identically. So, the q term does not contribute to the integral here. Then ∮AdI =

mc ∮Vp dI: 2e2

ð12Þ

The order parameter ψ function is usually written as ψ = ‖ψ‖eiφ, φ is the phase factor. Then velocity Vp is just a gradient of the phase factor, i.e. Vp =

e ▽φ: m

ð13Þ

It leads to a trivial curl-free result ð14Þ

▽×Vp = 0:

Therefore, the flow is strictly irrotational in the bulk. Feynmann found [19] that this statement has to be modified. He points out that the curl of velocity can be non-zero at a singular line, the core of quantum vortex line. So vorticity may live only on the lines of singularity of the phase. Based on the phase singularity theory, we not only obtain the correct result of curl of velocity field Vp, but also get the precise expression of kinetic helicity of vortex lines. The organization of the rest of the paper is as follows. In the section ‘Quantum of magnetic flux’, a detailed calculation of quantum of magnetic flux is given. A computation of kinetic helicity of vortex is performed in the section ‘Kinetic helicity of vortices’. In the section ‘Helicity of the superconductive current in mixed state’, the problem of the cross helicity and magnetic helicity is considered. The conclusions are presented in the ‘Conclusion’ section. 2. Quantum of magnetic flux We denote the order parameter ψ as 1

2

ψ = ψ + iψ :

ð15Þ

The velocity (Eq. (9)) is written as VP =

e ab ψa ψb  ∇ ; m ‖ψ‖ ‖ψ‖

a; b = 1; 2 :

ð16Þ

The vorticity of the probability current is Ω = ∇ × Vp, which can be written in terms of the wave function: i

Ω =

e 1 ijk ψa ψb  ab ∂j ∂ :i; j; = 1; 2; 3: m2 ‖ψ‖ k ‖ψ‖

ð17Þ

We use the relation a

∂b

a

a

b

ψ ∂ ψ ψ ψ = b − ; ‖ψ‖ ‖ψ‖ ‖ψ‖3

2

∂a ∂a ln ‖ψ‖ = 2πδ ðψÞ;

ð18Þ

T. Xu / Annals of Physics 325 (2010) 1316–1325

1319

The vorticity becomes i

Ω =

  2πe 2 i ψ δ ðψÞD ; m x

ð19Þ

where [20,21] Di

  ψ 1 ijk ab a b =   ∂j ψ ∂k ψ ; x 2

i; j; k = 1; 2; 3; a; b = 1; 2 :

ð20Þ

Eq. (20) tells us that the vorticity field Ωi = 0 i Ω ≠0

only if ψ≠0 only if ψ = 0:

ð21Þ

Under the regular condition   ψ ≠0; D x

ð22Þ

the general solution of 1

ψ

  1 2 3 t; x ; x ; x = 0;

2

ψ

  1 2 3 t; x ; x ; x = 0

ð23Þ

is just the vortex line. The k-th vortex line Lk can be expressed by line parameter s: 1

1

2

2

3

3

xk = xk ðt; sÞ; xk = xk ðt; sÞ; x = xk ðt; sÞ:

ð24Þ

The δ-function theory tells us N

2

δ ðψÞ = ∑ βk ∫ k=1

Lk

‖

δ3 ðxðsÞÞ ds;   ψ D X Mk

‖

ð25Þ

where

D

  ψ 1 ij ab ∂ψa ∂ψb =   ; X 2 ∂X i ∂X j

i; j = 1; 2; a; b = 1; 2 ;

ð26Þ

and Mk is the k-th planar element transverse to the vortex line Lk with local coordinates (X1, X2). Local coordinates (X1, X2) are the vortex plane. The positive integer number βk is the Hopf index, which means that when x covers the zero point once, the wave function covers the corresponding region in wave function space βk times. In Moffatt's paper, βk is also called the winding number traced from Gauss. Usually, βk is 1. In superconductor, vorticity can also describe vortex reconnection [22]. Under some condition, vortices can link in different superconductive planes. If two vortices are linked with β = 1, then the total Hopf index of linked vortex is 2. The direction of vector vortex line is given by   ψ dx X =  : ψ ds D X i

D

i

ð27Þ

1320

T. Xu / Annals of Physics 325 (2010) 1316–1325

Then from Eqs. (25) and (27), the vorticity field Ω can be written as i

Ω =

N e dxi 3 2π ∑ βk ηk ∫Lk δ ðx−xk ðsÞÞds; m ds k=1

ð28Þ

  ψ = F1. It is the Brouwer degree of ψ mapping, which characterizes the direction of where ηk = sgn D X the vortex line. Hence ∫Mk Ωi dσi =

2πe N1 2πe ∑β η = N; m k=1 k k m

N1 = 1; 2; 3…

ð29Þ

N

where N = ∑k 1= 1 βk ηk . It is obvious that the vortex line can be classified by the Brouwer degree and Hopf index. Then using Eq. (29) we write Eq. (12) as ∮AdI =

cπ N: e

ð30Þ

It is the quantization of the magnetic flux, which also can be written as ∮AdI = ϕ0 N

ð31Þ

−7 Gcm2 , is the quantum of magnetic flux. where ϕ0 = cπ e = 2 × 10 In helical phase or flux-line liquids, superconductive current Js ≠ 0 usually. We take any closed contour, the circulation of the vector A along the contour is equal to the flux of magnetitic induction through a surface spanning the contour, using Eq. (6), we get

∮AdI +

mc Js c πc N; ∮ dI− ∮qdI = 2e e 4e2 ρ

N = 1; 2; 3; …

ð32Þ

It is clear that ∮AdI = ϕ cannot be quantized now. For a closed integral, the q term contributes nothing to the result. Then Eq. (32) can be written as ∮AdI +

mc Js πc N; ∮ dI = e 4e2 ρ

N = 1; 2; 3; …

ð33Þ

If ∮ Jρs dI≠0, we denote mc Js πc γ: ∮ dI = e 4e2 ρ

ð34Þ

Then we get the universal expression of magnetic flux ∮AdI =

πc ðN−γÞ; e

N = 1; 2; 3; …

ð35Þ

In the center of vortex, the magnetic field has its maximum, and it decreases toward the outside. The decrease of B is controlled by the penetration depth. In the vortex core, Js = 0 and q term contributes nothing to the closed integral. So, each vortex can contain only an integer multiple of flux quantum, i.e. ϕ0βkηk. 3. Kinetic helicity of vortices The vortices govern the electromagnetic response of type-II superconductors and have been extensively studied, both experimentally and theoretically [23]. A vortex is formed from a core of radius rc ∼ ξ, where ξ is the superconducting coherence length, and it is surrounded by a supercurrent that

T. Xu / Annals of Physics 325 (2010) 1316–1325

1321

screens the magnetic field on a length of order δ, the penetration depth. The vortices are strong inhomogeneities of the superconducting condensate, and they scatter the quasiparticles in several different ways. In particular, the vortices can capture Bogoliubov excitations into low-energy localized states. Thermal fluctuations are significant over a wide range of the (H, T) phase diagram, causing the lattice of stiff vortex lines to melt well below Tc. The nature of this molten state has remained a subject of intense debate. Linking numbers are the simplest topological relation between two closed curves; this number is zero for two un-linked curves. In this section, we will discuss the linking numbers of vortices in noncentrosymmetric superconductors. The kinetic helicity Γk of vortex is Γk = ∫ Vp ⋅ Ωd3x. From Eq. (29), we can obtain

Γk =

2πe2 N i ∑ βk ηk ∫Lk Vpi dx : m2 k = 1

ð36Þ

When these vortex line are closed curves, i.e. a family of knots ξk(k = 1, 2, …N), Eq. (36) becomes Γk =

4π2 e2 N i ∑ βk ηk ∮ξk Vpi dx : m2 k = 1

ð37Þ

In order to compute it, we define Gauss mapping: ˜ : S1 × S1 → S2 ; n

ð38Þ

where ñ is a unit vector ˜ ðx; yÞ = n

y−x ; ‖y−x‖

ð39Þ

x and y are two points respectively on the knotted vortex lines ξj and ξk. When x and y are the same point on the same vortex line ζ, ñ is just the unit tangent vector. When x and y cover the corresponding vortex lines ξj and ξk, ñ becomes the section of sphere bundle S2. Similarly as the above section, we can define two two-dimensional unit vectors e˜ = e˜(x, y). e˜, ñ are normal to each other, i.e. ˜ = e˜2 : n ˜ =0 e˜1 : e˜2 = e˜2 : n

ð40Þ

˜:n ˜ = 1: e˜1 : e˜ 1 = e˜ 2 : e˜ 2 = n In fact, the velocity Vp can be expressed as Vpi =

e ab a b  e ∂i e ; 2m

a; b = 1; 2 :

ð41Þ

Substitute it into Eq. (37), one can obtain a new expression of kinetic helicity 2

Γk =

e π N ab a b i ∑ βk ηk ∮ξk  e ðx; yÞ∂i e ðx; yÞdx : m2 k = 1

ð42Þ

It can be written as 2

Γk =

e π N ab a b i j ∑ βk ηk ∮ξk ∮ξl  ∂i e ðx; yÞ∂j e ðx; yÞdx ∧dx : m2 k = 1

ð43Þ

1322

T. Xu / Annals of Physics 325 (2010) 1316–1325

There are three cases: (1) ξk, ξl are different vortex lines, x, y are different points; (2) ξk, ξl are the same vortex lines, x, y are different points; (3). ξk, ξl are the same vortex lines, x, y are the same points. So the Eq. (43) can be written as

Γk =

f

N 4π2 e2 1 ab a b i i ∑ β η ∮ ∮  ∂i e ðx; yÞ∂j e ðx; yÞdx ∧dx 2 4π k = 1;ðx≠yÞ k k ξk ξk m

+

N 1 ab a b i j ∑ β η ∮  ∂i e ðx; yÞ∂j e ðx; yÞdx ∧dx 4π k = 1ðx = yÞ k k ξk

+

1 N ab a b i j ∑ β η ∮ ∮  ∂i e ðx; yÞ∂j e ðx; yÞdx ∧dx : 4π k;j = 1 k k ξk ξl

ð44Þ

g

The first term is just the writhing number [24] wr(ξk) of the vortex line ξk. The second term is the twisting number Tw(ξk) of the vortex line ξk. From White's formula [25], the self linking number S(ξk) of the vortex line ξk is: Sðξk Þ = wr ðξk Þ + Tw ðξk Þ:

ð45Þ

The third term is the Gauss linking number L(ξk, ξl) of vortex lines ξk and ξl, i.e. Lðξk ; ξl Þ =

1 N ab a b i j ∑ β η ∮ ∮  ∂i e ðx; yÞ∂j e ðx; yÞdx ∧dx 4π l = 1 k k ξk ξl

k≠l :

ð46Þ

Then we obtain " # N 4π2 e2 N ∑ βk ηk Sðξk Þ + ∑ βk ηk Lðξk ; ξl Þ : Γk = m2 k = 1 k;l = 1

ð47Þ

This result is correct not only in the quantum case [26] but also for the classical fluid [27]. If there are N filaments with a strength χk(k = 1, 2, ⋯N) whose self knottedness degree, i.e. βk = 1 in classical fluid, the N N kinetic helicity equals 4π2 ∑k,l= 1ηkL(ξk, ξl) = ∑k,l= 1χkχlηkηlαkl (αkl = 1 if two vortex lines ξk, ξl are linked, αkl = 0 if ξk, ξl are not singly linked). There are strong experimental [28–30] indications that the conventional Abrikosov flux lattice or Josephson flux lattice is melted in superconductors over much of (H, T) phase diagram. Using Eqs. (29) and (47), we obtain Γk =

4π2 e2 ℏ2 N1 N2 ∑ ∑ βk ηk βj ηj ; m2 k = 1 j = 1

N1 ; N2 = 1; 2; 3…

ð48Þ

From Eq. (8), we know that the q term is the distinct difference of vector field in noncentrosymmetric superconductors with centrosymmetric superconductors. In order to study its topology, we define a velocity vector 0

V = Vp −

2e q: m

ð49Þ

The new kinetic helicity Γk′ of V′ is Γk′ = ∫V′ ⋅ ∇ × V′d3x, using Eq.(29,46), we can easily obtain an important result

Γk′ =

" # N 4π2 e2 N ð Þ ð Þ + ∑ ∑ β η S ξ β η L ξ ; ξ k k k k k k l : m2 k = 1 k;l = 1

ð50Þ

T. Xu / Annals of Physics 325 (2010) 1316–1325

1323

If the vortex line is closed, the result is the same as Γk because the closed integral of the q term is zero. But if the vortex line is not closed, the result is different from kinetic helicity Γk because the integral of the q term is not zero. Moreover, when the vortex lines are curved in the helical vortex phase, but not closed, we can obtain Γ′ =

4π2 e2 N ∑ βk ηk Sðξk Þ; m2 k = 1

ð51Þ

which is a new result in the helical vortex phase. 4. Helicity of the superconductive current in mixed state In this section, we will consider other topological aspects of noncentrosymmetric superconductors with q term. We define Jρs = Vs . Then from Eq. (8), we know Vs = −

4e2 2e A + 2Vp − q: m mc

ð52Þ

The cross helicity of superconductive current is Γs = ∫Vs ·Bdτ: In the interior of the superconductor beyond penetration depth, the cross helicity or magnetic helicity equals zero for magnetic induction B = 0. However, the result is not correct in the region near the superconductive surface. In the noncentrosymmetric superconductors, the closed integral from n × B term does not contribute to the cross helicity of a superconductor. It is natural that the cross helicity of a superconductor with q term and without q term is exactly the same. The cross helicity of a superconductive density current Vs has two parts, one from the probability density current and the other from vector potential A. The total cross helicity Γp of probability density current is Γc = ∫Vp ·Bdτ:

ð53Þ

If the vector potential does not pass through the circle of the probability density current, then the cross helicity is zero. However, if the magnetic induction does pass, the result will be different. We assume that ∂Vp ∂t

= −Vp ·∇Vp −∇P + Js × B;

ð54Þ

where P is pressure. We get expression of cross helicity about Vp   Vp ·Vp ∂Γc B + PB ⋅dS: = −∮ Vp ·BVp − 2 ∂t If there are no fluids or magnetic induction flowing out of the integral surface, we get means that the cross helicity of vortices is a conservation quantity. The magnetic helicity Γm of superconductor is 3

Γm = ∫A⋅Bd x;

ð55Þ

∂Γc ∂t

= 0, which

ð56Þ

which does not equal zero in a mixed state. After the penetration of magnetic flux into the superconducting sample, the transport current can also flow within the interior of the superconductor. In this case there exists an important interaction between the transport current and the vortices. The transport current, also passes through the vortices, i.e., through a region where a magnetic field exists. The vortex motion across

1324

T. Xu / Annals of Physics 325 (2010) 1316–1325

the superconductor causes dissipation, i.e., electric energy is changed into heat. Using Maxwell's equations, we obtain   ∂Γm ∂A + ∇⋅ 2κB + A × = −2ηJ·B; ∂t ∂t

ð57Þ

where κ is the electric potential, and η is resistance. If η = 0, the magnetic helicity Γm of superconductor is conservative. Substituting B by n × B in the expression of cross helicity (Eq. (53)) and magnetic helicity (Eq. (56)), we obtain the cross-helicity-like term including n × B term, and find Γq = ∫Vs ·ðn × BÞdτ:

ð58Þ

It is just the term of n × B in GL free energy. In superconductors, a new topological invariant can be defined as 3

ΓΩ = ∫A·Ωd x;

ð59Þ

where Ω is vorticity of the probability current. It is a new helicity for the superconductor. Using Eq. (31), we easily get   2π c N1 N2 ∑ ∑ βi ηi βj ηj −γj : m j=1 j=1 2

ð60Þ   πc where 2πe m βi ηi is the velocity flux of i-th vortex, e βj ηj −γj is the magnetic flux passing through the surface spanned by i-th curved vortex line. It is useful to visually describe vortex phase in superconductor. ΓΩ =

5. Conclusion If the regular condition (22) is not satisfied, i.e.   ψ = 0; D x

ð61Þ

then the vortex is unstable, and the vortex line will break down. There are three kinds of bifurcation possibilities [20], and corresponding three kinds of length approximation relations at the neighbor of the 1 bifurcation point, i.e. l∝ðϑ−ϑ* Þ2 , l ∝ ϑ – ϑ⁎, l = const, where l is the distance of two bifurcated vortices, ϑ is time interval. In this paper, we have considered the role of magnetic fields in the noncentrosymmetric superconductor, such as CePt3Si, UIr, CeRhSi3 and so on. In this modified GL model, a new n × B term, i.e. the q term is added to the classical GL model. This unusual term changes the superconductor current, but does not change the quantum of magnetic flux. After the penetration of magnetic flux into the superconducting sample, the transport current passes through the vortices, i.e., through region where a magnetic field exists. There exists an important interaction between the transport current and the vortices. The cross helicity and magnetic helicity are two important topological quantities to describe the interaction between current and the vortices. The kinetic helicity is the most important quantity to describe the topological relation between vortices. These helicities of a superconductor are calculated in detail. Moreover, a new helicity, i.e. a new topological invariant, is introduced to describe vortex phase. The cross helicity in a superconductor can be changed by boundary helicity injection, which will be verified by an experiment in the future. Acknowledgments This work has been supported by the National Natural Science Foundation of China (NSFC-10805022).

T. Xu / Annals of Physics 325 (2010) 1316–1325

1325

References [1] P.W. Anderson, E.I. Blount, Phys. Rev. Lett. 14 (1965) 217. [2] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. [3] G.E. Volovik, L.P. Gor'kov, Pis'ma Zh. Eksp. Teor. Fiz. 39 (1984) 550; JETP Lett. 39 (1984) 675. [4] E. Bauer, G. Hilscher, H. Michor, C. Paul, E.W. Scheidt, A. Gribanov, Y. Seropegin, H. Noel, M. Sigrist, P. Rogl, Phys. Rev. Lett. 92 (2004) 027003. [5] I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. Thamizhavel, E. Yamamoto, T.D. Matsuda, Y. Haga, T. Takeuchi, R. Settai, Y. Onuki, J. Phys. Soc. Jpn 75 (2006) 043703. [6] K.V. Samokhin, E.S. Zijlstra, S.K. Bose, Phys. Rev. B 69 (2004) 094514; Phys. Rev. Lett. 94 (2005) 027004. [7] M. Nishiyama, Y. Inada, Guo-qing Zheng, Phys. Rev. Lett. 98 (2007) 047002. [8] L.P. Gorkov, E.I. Rashba, Phys. Rev. Lett. 87 (2001) 037004. [9] T. Akazawa, H. Hidaka, T. Fujiwara, T.C. Kobayashi, E. Yamamoto, Y. Haga, R. Settai, Y. Onuki, J. Phys. Condens. Matter 16 (2004) L29. [10] N. Kimura, K. Ito, K. Saitoh, Y. Umeda, H. Aoki, T. Terashima, Phys. Rev. Lett. 95 (2005) 247004. [11] G. Amano, S. Akutagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, J. Phys. Soc. Jpn 73 (2004) 530. [12] K. Togano, P. Badica, Y. Nakamori, S. Orimo, H. Takeya, K. Hirata, Phys. Rev. Lett. 93 (2004) 247004; P. Badica, T. Kondo, K. Togano, J. Phys. Soc. Jpn 74 (2005) 1014. [13] G. Schuck, S.M. Kazakov, K. Rogacki, N.D. Zhigadlo, J. Karpinski, Phys. Rev. B 73 (2006) 144506. [14] Lu. Chi-Ken, Sungkit Yip, Phys. Rev. B 77 (2008) 054515. [15] Ol'ga Dimitrova, M.V. Feigel'man, Phys. Rev. B 76 (2007) 014522. [16] V.M. Edelstein, Sov. Phys. JETP 68 (1989) 1244. [17] T. Yasuda, et al., J. Phys. Soc. Jpn 73 (2004) 1657. [18] R.P. Kaur, D.F. Agterberg, M. Sigrist, Phys. Rev. Lett. 94 (2005) 137002. [19] R.P. Feynmann, in: C.G. Gorter (Ed.), Progress in Low Temperature Physics, Vol. 1, North-Holland, Amsterdam, 1955, p. 17. [20] T. Xu, Ann. Phys. 321 (2006) 2017; Phys. Rev. E 72 (2005) 036303. [21] G.F. Mazenko, Phys. Rev. Lett. 78 (1997) 401; F. Liu, G.F. Mazenko, Phys. Rev. B 46 (1992) 5963. [22] G. Blatter, M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [23] R.P. Huebener, N. Schopohl, G.E. Volovik (Eds.), Vortices in Unconventional Superconductors and Superfluids, Springer, Berlin, 2002. [24] E. Witten, Comm. Math. Phys. 121 (1989) 351; A.M. Polyakov, Mod. Phys. Lett. A3 (1988) 325. [25] D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, CA, 1976. [26] G.E. Volovik, V.P. Mineev, Zh. Eksp. Teor. Fiz. 72 (1977) 2256. [27] H.K. Moffatt, J. Fluid Mech. 35 (1969) 117. [28] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak, D.J. Bishop, Phys. Rev. Lett. 61 (1988) 1666. [29] R.B. van Dover, L.F. Schneemeyer, E.M. Gyorgy, J.V. Waszczak, Phys. Rev. B 39 (1989) 4800. [30] P.L. Gammel, D.J. Bishop, G.J. Dolan, J.R. Kwo, C.A. Murray, L.F. Schneemeyer, J.V. Waszczak, Phys. Rev. Lett. 59 (1987) 2592.