jarmo hietarinta- faddeev-hopf knots and the two-component ginzburg-landau model

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Faddeev-Hopf knots and the two-componentGinzburg-Landau model

Jarmo Hietarinta

Department of Physics, University of Turku, FIN-20014 Turku, Finland

NorditaQF2007, August 21, 2007



The Faddeev-Skyrme model

Knot theory

The Ginzburg-Landau model

Topology

Faddeev’s model

Numerical results

Topology on the plane

Closed curve on a punctured plane, f : S 1 → R/{(0, 0)} ≈ S 1

Topological characterization using the winding number around

the puncture.

Jarmo Hietarinta Hopfions and GL-model




Knot theory


Topology

Faddeev’s model

Numerical results

Topology on the plane


Topological characterization using the winding number around

the puncture.

• winding number = 0







Knot theory


Topology

Faddeev’s model

Numerical results


Topological characterization of maps f into homotopy classes

according to the winding number,

f : S 1 → S 1, π1(S 1) = Z.





Knot theory


Topology

Faddeev’s model

Numerical results




f : S 1 → S 1, π1(S 1) = Z.

Physical interpretation:Complex field ψ = ρe i θ.

Take a 2D section of R3.

θ : R2 → (0, 2π]




Th F dd Sk d l T l




Knot theory


Topology

Faddeev’s model

Numerical results




f : S 1 → S 1, π1(S 1) = Z.

Physical interpretation:Complex field ψ = ρe i θ.

Take a 2D section of R3.

θ : R2 → (0, 2π]

ψ must be continuous.

At the point where e i θ would be discontinuous (undefined):

we must have ρ = 0, this defines the vortex core (puncture).

3Jarmo Hietarinta Hopfions and GL-model

The Faddeev Skyrme model Topology




Knot theory


Topology

Faddeev’s model

Numerical results

Topology in R3:Hopf charge

• Carrier field: 3D unit vector field n in R3, locally smooth.

• 3D-unit vectors can be represented by points on the

surface of the sphere S 2.


The Faddeev Skyrme model Topology




Knot theory


Topology

Faddeev’s model

Numerical results





• Asymptotically trivial: n(r) → n∞, when |r| → ∞⇒ can compactify R3 → S 3.

Therefore

n : S 3 → S 2.


The Faddeev-Skyrme model Topology




Knot theory


Topology

Faddeev’s model

Numerical results





• Asymptotically trivial: n(r) → n∞, when |r| → ∞⇒ can compactify R3 → S 3.

Therefore

n : S 3 → S 2.

Such functions are characterized by the Hopf charge, i.e.,

by the homotopy class π3(S 2) = Z.





The Faddeev Skyrme model

Knot theory


Topology

Faddeev’s model

Numerical results

Example of vortex ring with Hopf charge 1:

n =4(2x z − y (r 2 − 1))

(1 + r 2)2 ,

4(2y z + x (r 2 − 1))

(1 + r 2)2 , 1 −

8(r 2 − z 2)

(1 + r 2)2

.

where r 2 = x 2 + y 2 + z 2.





The Faddeev Skyrme model

Knot theory


Topology

Faddeev’s model

Numerical results


n =4(2x z − y (r 2 − 1))

(1 + r 2)2 ,

4(2y z + x (r 2 − 1))

(1 + r 2)2 , 1 −

8(r 2 − z 2)

(1 + r 2)2

.

where r 2 = x 2 + y 2 + z 2.

Note that

• n = (0, 0, 1) at infinity (any direction).• n = (0, 0, −1) on the ring x 2 + y 2 = 1, z = 0 (vortex core).





y

Knot theory


p gy

Faddeev’s model

Numerical results


n =4(2x z − y (r 2 − 1))

(1 + r 2)2 ,

4(2y z + x (r 2 − 1))

(1 + r 2)2 , 1 −

8(r 2 − z 2)

(1 + r 2)2

.

where r 2 = x 2 + y 2 + z 2.

Note that


Computing the Hopf charge:

Given n : R3

→ S 2

define F ij = abc n a

∂ i n b

∂ j n c

.





y

Knot theory


p gy

Faddeev’s model

Numerical results


n =4(2x z − y (r 2 − 1))

(1 + r 2)2 ,

4(2y z + x (r 2 − 1))

(1 + r 2)2 , 1 −

8(r 2 − z 2)

(1 + r 2)2

.

where r 2 = x 2 + y 2 + z 2.

Note that



Given n : R3

→ S 2


∂ i n b

∂ j n c

.Given F ij construct A j so that F ij = ∂ i A j − ∂ j Ai ,





Knot theory


Faddeev’s model

Numerical results


n =4(2x z − y (r 2 − 1))

(1 + r 2)2 ,

4(2y z + x (r 2 − 1))

(1 + r 2)2 , 1 −

8(r 2 − z 2)

(1 + r 2)2

.

where r 2 = x 2 + y 2 + z 2.

Note that



Given n :R

3

→ S 2


∂ i n b

∂ j n c

.Given F ij construct A j so that F ij = ∂ i A j − ∂ j Ai , then

Q =1

16π2

ijk Ai F jk d 3x .





Knot theory


Faddeev’s model

Numerical results

Possible physical realization



K h

Topology

F dd ’ d l



Knot theory


Faddeev’s model

Numerical results

Faddeev’s model

In 1975 Faddeev proposed the Lagrangian (energy)

E =

(∂ i n)2 + g F 2ij

d 3x , F ij := n · ∂ in × ∂ jn.



K t th

Topology

F dd ’ d l



Knot theory


Faddeev’s model

Numerical results

Faddeev’s model


E =

(∂ i n)2 + g F 2ij

d 3x , F ij := n · ∂ in × ∂ jn.

Under the scaling r → λr the integrated kinetic term scales as

λ and the integrated F 2 term as λ−1.



Knot theory

Topology

Faddeev’s model



Knot theory


Faddeev s model

Numerical results

Faddeev’s model


E =

(∂ i n)2 + g F 2ij

d 3x , F ij := n · ∂ in × ∂ jn.



Therefore nontrivial configurations will attain some fixed size

determined by the dimensional coupling constant g . (Virial

theorem)



Knot theory

Topology

Faddeev’s model



Knot theory


Faddeev s model

Numerical results

Faddeev’s model


E =

(∂ i n)2 + g F 2ij

d 3x , F ij := n · ∂ in × ∂ jn.



Therefore nontrivial configurations will attain some fixed size

determined by the dimensional coupling constant g . (Virial

theorem)

Vakulenko and Kapitanskii (1979): a lower limit for the energy,

E ≥ c |Q |34 ,

where c is some constant, and Q the Hopf charge.


The Faddeev-Skyrme modelKnot theory

TopologyFaddeev’s model



Knot theory


Faddeev s model

Numerical results

Numerical studies of Faddeev’s model

What is the minimum energy state for a given Hopf charge?

Studied in 1997-2004 by Gladikowski and Hellmund, Faddeev

and Niemi, Battye and Sutcliffe, and Hietarinta and Salo.






Knot theory


Faddeev s model

Numerical results





Our work:

Full 3D minimization without restrictive symmetry assumptions.

Linked unknots of various charges.Later (with Jäykkä) also knotting of twisted Hopf-vortices.






Knot theory


Faddeev s model

Numerical results





Our work:

Full 3D minimization without restrictive symmetry assumptions.

Linked unknots of various charges.Later (with Jäykkä) also knotting of twisted Hopf-vortices.

More technically:• Discretized on a cubic lattice, size typically 2403.

• Discretized the Lagrangian: ∂ i n on links, F ij on plaquettes.• Computed the gradient n(r)L symbolically.• Used dissipative dynamics: nnew = nold − δn(r)L.

Program parallelizes well, have used Cray T3E, SGI Origin

2000, IBM SP, etc.Jarmo Hietarinta Hopfions and GL-model





y

The Ginzburg-Landau model Numerical results

How to visualize vector fields?

Cannot draw vectors at every point and flow lines do not makesense.






y




n is a point on the sphere S 2.

There is one fixed direction, n∞ = (0, 0, 1), the north pole.

All other points are defined by latitude and longitude.

Vortex core is where n = −n∞ (the south pole).













Latitude is invariant under global gauge rotations that keep the

north pole fixed, therefore we plot equilatitude surfaces

i.e., tubes (around the core) defined by {x : n(x) · n∞ = c }.













Latitude is invariant under global gauge rotations that keep the

north pole fixed, therefore we plot equilatitude surfaces

i.e., tubes (around the core) defined by {x : n(x) · n∞ = c }.Longitudes are represented by colors on the equilatitude

surface. (Under a global gauge rotation only colors change).

Paint the tubes using latitudes.



Th Gi b L d d l


N i l l




Isosurface n 3 = 0 (equator) for |Q | = 1, 2

Color order and handedness of twist determine Hopf charge.Inside the torus is the core, where n 3 = −1.

These figures were made using the program funcs developed by J. Ruokolainen at

CSC, Espoo, Finland



Th Gi b L d d l


N i l lt




Results for linked unknots of charge 1+1



The Ginzburg Landau model


Numerical results




Energy evolution in minimization





Numerical results




Vakulenko bound

0 1 2 3 4 5 6 7

Q

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

E Q /

( E 1

Q 3 / 4 )





Numerical results




Different and improved final states





Numerical results



The Ginzburg Landau model Numerical results

Deformation 5 + 4 − 2 → trefoil





Numerical results



The Ginzburg Landau model Numerical results

(1, 5) evolution




Linking numberFramed links and ribbon knots

Ribbon deformations



g

Linking number addition rules for the Hopf charge

Total charge = charges of individual unknots + linking number.

Linking number depends on the relative directionof the unknots.





Ribbon deformations



g




Assign direction as follows: if the unknot is right-handed then

the direction is the same as color direction, if left-handed then

opposite.





Ribbon deformations






Assign direction as follows: if the unknot is right-handed then

the direction is the same as color direction, if left-handed then

opposite.Then linking number is obtained from the following figure:





Ribbon deformations



Framed links and ribbon knots

The proper knot theoretical setting is to use framed links.

Framing attached to a curve adds local information near the

curve, like twisting around it.





Ribbon deformations







One way to describe framed links is to use directed ribbons,

which are preimages of line segments.





Ribbon deformations









We could use equilatitude line segments , then increasing

latitude and longitude give two directions, their cross product

the ribbon direction.





Ribbon deformations









We could use equilatitude line segments , then increasing

latitude and longitude give two directions, their cross product

the ribbon direction.

Another choice would be to use equilongitude line near the

south pole (=core)





Ribbon deformations



Example: ribbon view of Q = −1 unknot

Here we have plotted the preimages of four nearby points on

the tubular preimage of the equator.

These figures were made using OpenDX





Ribbon deformations



Computing the charge

For a ribbon define:

• twist = linking number of the ribbon core with a ribbon

boundary.

• writhe = signed crossover number of the ribbon core with

itself.

• linking number = 12 (sum of signed crossings)





Ribbon deformations



Computing the charge

For a ribbon define:

• twist = linking number of the ribbon core with a ribbon

boundary.

• writhe = signed crossover number of the ribbon core with

itself.

• linking number = 12 (sum of signed crossings)

The Hopf charge can be determined either by

twist + writheor

linking number of the two ribbon boundaries.





Ribbon deformations



Charge from the ribbon view, Q = −1

Sign convention for crossings allows computing the charge.

In this case linking number of ribbon boundaries = −1.





Ribbon deformations

Ch f h ibb i Q 1






On the right the ribbon has been turned vertical and is

viewed from above: a twist in the ribbon becomes a crossing.





Ribbon deformations

Ch f th ibb i Q 1






On the right the ribbon has been turned vertical and is

viewed from above: a twist in the ribbon becomes a crossing.

Note that when considering equivalence of ribbon diagrams

type I Reidemeister move is not valid:





Ribbon deformations

Ribb i Q 2



Ribbon view, Q = −2

Two ways to get charge −2: twice around small vs. large circle.

The first one has twist = −1, writhe = −1,the second twist = −2, writhe = 0.

Both have boundary linking number = −2.





Ribbon deformations

E l f ibb d f ti d i i i i ti



Example of ribbon deformation during minimization





Ribbon deformations

Close up of the deformation process



Close-up of the deformation process





Ribbon deformations

Diagrammatic rule for deformations



Diagrammatic rule for deformations

Knot deformations correspond to ribbon deformations, e.g.,

crossing and breaking, but the Hopf charge will be conserved.





Ribbon deformations

Ribbon connection rules



Ribbon connection rules

• Total Hopf charge = charges of individual unknots + linkingnumber.

• Unknots: Connecting a ribbon with a clockwise a full twist

(on the end at the right hand) yields charge +1.

• Linking number depends on the relative direction

associated with the unknots, as before:




The modelGL and FS?

Preliminary results





Two electromagnetically coupled, oppositely charged Bose

condensates

L = 2

2m 1 + i 2e c

AΨ12

+ 2

2m 2 − i 2e c

AΨ22

+V

Ψ1, Ψ2

+ 1

2µ0

B 2,

Ψ1 and Ψ2: are order parameters for the condensates, A is the electromagnetic vector potential,

B = 1c × A,V is the potential.



Knot theory


The model

GL and FS?

Preliminary results



Faddeev-Skyrme model is hidden here!



Knot theory


The model

GL and FS?

Preliminary results

Change of variables



Change of variables

Introduce new variables ρ, χα by

Ψα =

2m αρχα,

where χ is normalized as

|χ1|2 + |χ2|2 = 1,

and therefore

ρ2 = 12

|Ψ1|

2

m 1+ |Ψ2|

2

m 2

.



Knot theory


The model

GL and FS?

Preliminary results

Change of variables



Change of variables

Introduce new variables ρ, χα by

Ψα =

2m αρχα,

where χ is normalized as

|χ1|2 + |χ2|2 = 1,

and therefore

ρ2 = 12

|Ψ1|

2

m 1+ |Ψ2|

2

m 2

.

In terms of the new fields the GL-model can be written as

L = 2ρ2

+ i 2e c

A

χ1

2+ − i 2e

c A

χ2

2

+ 2

ρ2

+ V

χ1, χ2, ρ2

+ 12µ0

B 2.



Knot theory


The model

GL and FS?

Preliminary results

Next define the unit vector field n by



y

n T = χ∗1 χ2σχ1

χ∗2 =

χ1χ2+χ∗1χ∗

2

i (χ1χ2−χ∗1χ∗

2)

|χ1|2−|χ2|2 ,

where σ are the Pauli matrices.



Knot theory


The model

GL and FS?

Preliminary results

Next define the unit vector field n by



y

n T = χ∗1 χ2σχ1

χ∗2 =

χ1χ2+χ∗1χ∗

2

i (χ1χ2−χ∗1χ∗

2)

|χ1|2−|χ2|2 ,

where σ are the Pauli matrices.

The Lagrangian is invariant under the gauge transformation

Ψ1 → e −i 2e θ(x )Ψ1, Ψ2 → e i

2e θ(x )Ψ2, Aµ → Aµ + c ∂ µθ(x ),

and the corresponding Noether current is

J k = i e m 1 Ψ∗

1∂ k Ψ1 − Ψ1∂ k Ψ∗1 − i e

m 2 Ψ∗2∂ k Ψ2 − Ψ2∂ k Ψ

∗2 − 8e 2ρ2

c Ak

= 2e ρ2i

χ∗1∂ k χ1 − χ1∂ k χ∗1 − χ∗2∂ k χ2 + χ2∂ k χ∗2

− 8e 2ρ2

c Ak

= 4e ρ2

12 j k − 2e

c Ak

,

Later on we also use the gauge invariant vector field C = 1e ρ2

J .



Knot theory


The model

GL and FS?

Preliminary results



Using the above results we can write the original Lagrangian in

the form

L =2ρ2

4 ∂ k n l ∂ k n l + 2

ρ2

+ 2ρ2

16 C 2 + V

ρ, n k

+ 2

128µ0e 2 klm

n · ∂ k n × ∂ l n + ∂ k C l

2

,

The dynamical fields are now ρ, n and C .

If ρ = constant and C = 0, the GL model reduces to the FS

model.



Knot theory


The model

GL and FS?

Preliminary results



Using the above results we can write the original Lagrangian in

the form

L =2ρ2

4 ∂ k n l ∂ k n l + 2

ρ2

+ 2ρ2

16 C 2 + V

ρ, n k

+ 2

128µ0e 2

klm

n · ∂ k n × ∂ l n + ∂ k C l

2

,

The dynamical fields are now ρ, n and C .

If ρ = constant and C = 0, the GL model reduces to the FS

model.

What happens when ρ, C take their proper dynamical roles?



Knot theory


The model

GL and FS?

Preliminary results

The potential



e pote t a

From physical arguments the following special cases is relevant

(Babaev, 2002)

V 1 (Ψ1, Ψ2) = λ

|Ψ1|2 − 12

+

|Ψ2|2 − 12

,

which breaks O (3) to O (2), and

V 2 (Ψ1, Ψ2) = λ

|Ψ1|2 − 1

2 +

|Ψ2|2 − 1

2

+c Ψ1Ψ∗

2 − Ψ2Ψ∗1

+ a 0,

which breaks O (3) completely.

In our computations we have also used

V 3 (Ψ1, Ψ2) = 14 λ

|Ψ1|2 + |Ψ2|2 − ρ20

2

+ 2γ

|Ψ1|2 + |Ψ2|2

−1

+a 0.



Knot theory


The model

GL and FS?

Preliminary results

The initial state



Constructed following Aratyn et al. (1999).

We use toroidal coordinates

η, ξ , ϕ

of R3 defined by

x 1 = (sinh(η) cos(ϕ))/q , x 2 = (sinh(η) sin(ϕ))/q , x 3 = sin(ξ)/q ,

where q = cosh(η) − cos(ξ).



Knot theory


The model

GL and FS?

Preliminary results

The initial state





η, ξ , ϕ

of R3 defined by


where q = cosh(η) − cos(ξ). Then we define

χ1 := g (η)e ip ξ, χ2 :=

1 − g

η2

e iq ϕ

Here g is a smooth monotonous function.



Knot theory


The model

GL and FS?

Preliminary results

The initial state





η, ξ , ϕ

of R3 defined by


where q = cosh(η) − cos(ξ). Then we define

χ1 := g (η)e ip ξ, χ2 :=

1 − g

η2

e iq ϕ

Here g is a smooth monotonous function.

The χi above are continuous, if

• g (∞) = 0 (at the core) and,

• g (0) = ±1 (at the z -axis and at infinity).

The corresponding n has Hopf charge Q = pq .



Knot theory


The model

GL and FS?

Preliminary results

Discretization



The system has been discretized on a cubic rectangular lattice

with periodic boundary conditions, using methods of lattice fieldtheory (Wilson 1974, Damgaard 1988).

Condition on discretization: gauge invariance is preserved.



Knot theory


The model

GL and FS?

Preliminary results

Discretization



The system has been discretized on a cubic rectangular lattice

with periodic boundary conditions, using methods of lattice fieldtheory (Wilson 1974, Damgaard 1988).

Condition on discretization: gauge invariance is preserved.

For A the gauge transformation is

A1|s ,u ,v → A1|s ,u ,v + c a (θs +1,u ,v − θs ,u ,v ),

A2|s ,u ,v → A2|s ,u ,v + c a (θs ,u +1,v − θs ,u ,v ),

A3|s ,u ,v → A3|s ,u ,v + c a (θs ,u ,v +1 − θs ,u ,v ),

where s , u , v are the lattice coordinates.

Thus, Ak should be considered as living on the link between

two lattice points parallel to the coordinate axis k .



Knot theory


The model

GL and FS?

Preliminary results

For the kinetic term of Ψα (κ := 2e c ) this leads to








e iF

12|suv + e −iF

12|suv + e iF

23|suv + e −iF

23|suv + e iF

31|suv + e −iF

31|suv − 6,

where for example


Energy was miminized using the steepest descent method.The gradients were calculated symbolically from the discretized

Lagrangian.

In practice we use the cubic grids of sizes of 1203 . . . 9 6 03.




The model


For the FS-model there were (local) minimum energy states.






The model





Here apparently not.

If minimize both fields ψ and A simultaneously and

independently arrive at squeezed singular states .

Possible explanation: The degree 4 term is

2 n · ∂ k n × ∂ l n + ∂ k C l − ∂ l C k 2

If C adjusts so that this term vanishes, then scaling to zero size

is not prevented.




The model





Here apparently not.

If minimize both fields ψ and A simultaneously and

independently arrive at squeezed singular states .

Possible explanation: The degree 4 term is

2 n · ∂ k n × ∂ l n + ∂ k C l − ∂ l C k 2

If C adjusts so that this term vanishes, then scaling to zero size

is not prevented.

Once the tube has been squeezed to the thickness of 1 lattice

space (where also ρ = 0), topology breaks

n : ↑↑↑↑↓↑↑↑↑ −→ ↑↑↑↑↑↑↑↑↑

and the state degenerates to vacuum.




The model




0

100

200

300

400

E n e r g y

( a r b . u n i t s )

0 2000 4000 6000 8000

Iterations

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

n . n

a n

d

ρ 2




The model




Dynamical stability still possible.




The model





More recent computer runs:

1 iterate ψ one step

2 iterate A until Maxwell equation sufficiently good

3 repeat 1 and 2 until also ψ equation satisfied

Practical problem: The intermediate A iterations take time.




The model





More recent computer runs:

1 iterate ψ one step

2 iterate A until Maxwell equation sufficiently good

3 repeat 1 and 2 until also ψ equation satisfied

Practical problem: The intermediate A iterations take time.

Stability plausible, especially with strong potentials.




The model


λ = 1, g f = 2






The model


λ = 1, g f = 0.01






The model


λ = 1000, g f = 2






The model


Conclusions



The GL-model contains FS as a submodel, the vector field ncan be extracted from ψi .




The model


Conclusions




The extra degrees of freedom change the behavior

(in comparison to FS) as follows:




The model


Conclusions






For a given Hopf charge the energy is not bounded from 0,

i.e, Hopfions are globally unstable, since a deformation path

exists to singular configurations with zero energy.




The model


Conclusions






For a given Hopf charge the energy is not bounded from 0,

i.e, Hopfions are globally unstable, since a deformation path

exists to singular configurations with zero energy.

Dynamical stability possible, especially with strong potentials.




The model


References

J Hi t i t d P S l F dd H f k t d i f li k d



J. Hietarinta and P. Salo: Faddeev-Hopf knots: dynamics of linked

unknots , Phys. Lett. B 451, 60-67 (1999).J. Hietarinta and P. Salo: Ground state in the Faddeev-Skyrme model ,Phys. Rev. D 62, 081701(R) (2000).

J. Hietarinta, J. Jäykkä and P. Salo: Dynamics of vortices and knots in Faddeev’s model , JHEP Proceedings: PrHEP unesp2002/17http://jhep.sissa.it/archive/prhep/preproceeding/008/017/sp-proc.pdf

J. Hietarinta, J. Jäykkä and P. Salo: Relaxation of twisted vortices in the Faddeev-Skyrme model , Phys. Lett. A 321, 324-329 (2004).

J. Hietarinta, J. Jäykkä and P. Salo: Investigation of the stability of

Hopfions in the two-component Ginzburg-Landau model ,cond-mat/0608424

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jarmo hietarinta- faddeev-hopf knots and the two-component ginzburg-landau model

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