Jarmo Hietarinta- Ribbon knots in the Faddeev-Skyrme model

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Ribbon knotsin the Faddeev-Skyrme model

Jarmo HietarintaUniversity of Turku, Turku, Finland

in collaboration with Petri Salo and Juha Jaykka

[email protected]

LMS Durham Symposium: Topological Solitons and their Applications

August 2004

Ribbon knots in the Faddeev-Skyrme model – p.1/3



The setting

Carrier field: 3D unit vector field n in R3




The setting


Locally smooth




The setting


Locally smooth

Asymptotically trivial: n(r) → n∞, when |r| → ∞




The setting


Locally smooth

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




The setting


Locally smooth

Asymptotically trivial:n

(r

) →n∞, when |

r

| → ∞⇒ can compactify R3 → S 3.

⇒ n : S 3 → S 2.




The setting


Locally smooth

Asymptotically trivial:n

(r

) →n∞, when |

r

| → ∞⇒ can compactify R3 → S 3.

⇒ n : S 3 → S 2.

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

homotopy class π3(S 2) = Z.




Example of vortex ring with Hopf charge 1:

n =

4(2xz − y(r2 − 1))

(1 + r2)2,

4(2yz + x(r2 − 1))

(1 + r2)2, 1−

8(r2 − z2)

(1 + r2)2

.

where r2 = x2 + y2 + z2.





n =

4(2xz − y(r2 − 1))

(1 + r2)2,

4(2yz + x(r2 − 1))

(1 + r2)2, 1−

8(r2 − z2)

(1 + r2)2

.

where r2 = x2 + y2 + z2.

Note that•n = (0, 0, 1) at infinity (any direction).•n = (0, 0,−1) on the ring x2 + y2 = 1, z = 0 (vortex core).





n =

4(2xz − y(r2 − 1))

(1 + r2)2,

4(2yz + x(r2 − 1))

(1 + r2)2, 1−

8(r2 − z2)

(1 + r2)2

.

where r2 = x2 + y2 + z2.


Computing the Hopf charge:

Givenn

:R3

→ S 2

define F ij = abcna

∂ inb

∂ jnc

.





n =

4(2xz − y(r2 − 1))

(1 + r2)2,

4(2yz + x(r2 − 1))

(1 + r2)2, 1−

8(r2 − z2)

(1 + r2)2

.

where r2 = x2 + y2 + z2.



Givenn

:R3

→ S 2

define F ij = abcna

∂ inb

∂ jnc

.Given F ij construct A j so that F ij = ∂ iA j − ∂ jAi,





n =

4(2xz − y(r2 − 1))

(1 + r2)2,

4(2yz + x(r2 − 1))

(1 + r2)2, 1−

8(r2 − z2)

(1 + r2)2

.

where r2 = x2 + y2 + z2.



Givenn

:R3

→ S 2

define F ij = abcna

∂ inb

∂ jnc

.Given F ij construct A j so that F ij = ∂ iA j − ∂ jAi,then

Q =1

16π2 ijkAiF jk d3x.




Possible physical realization




Faddeev’s model

In 1975 Faddeev proposed the Lagrangian (energy)

E =

(∂ in)2 + g F 2ij

d3x, F ij := n · ∂ in× ∂ jn.




Faddeev’s model


E =

(∂ in)2 + g F 2ij

d3x, F ij := n · ∂ in× ∂ jn.

Under the scaling r → λr the integrated kinetic term scales as λand the integrated F 2 term as λ−1.




Faddeev’s model


E =

(∂ in)2 + g F 2ij

d3x, F ij := n · ∂ in× ∂ jn.


Therefore nontrivial configurations will attain some fixed size

determined by the dimensional coupling constant g. (Virialtheorem)




Faddeev’s model


E =

(∂ in)2 + g F 2ij

d3x, F ij := n · ∂ in× ∂ jn.


Therefore nontrivial configurations will attain some fixed size

determined by the dimensional coupling constant g. (Virialtheorem)

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

E ≥ c |Q|34 ,

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




Other models

Other field theoretical models having configurations withnonzero Hopf charge include:

de Vega (1978), Higgs-models (Abelian and SU(2)).

Nicole (1978), L = −−1

4(∂ µna)23/2

Kundu and Rubakov (1982), S 2 nonlinear σ-model.

Aratyn, Ferreira, Zimerman (1999) L = −

F 2µν 3/4


i i f ’



Numerical studies of Faddeev’s model

Question: What are the minimun energy states for various Hopfcharges?


N i l di f F dd ’ d l





Gladikowski and Hellmund (1997): Charges 1 and 2, also

various other models, cylindrical ansatz.


N i l t di f F dd ’ d l







Faddeev and Niemi (1997): Charges 1 and 2, cylindricalansatz, speculations on trefoil knots.










Battye and Sutcliffe (1998): Charges 1-8, ring initial states,deformation to trefoil seen.










Battye and Sutcliffe (1998): Charges 1-8, ring initial states,deformation to trefoil seen.

Hietarinta and Salo

(1999): Linked initial configurations, deformation to trefoilseen.(2000): More on the lowest energy states, agreement withVK bound.(2004): Knotting of twisted vortices (w/Jäykkä).





Our work



Our work

Full three dimensional minimization.


Our work



Our work


Use n rather than w := n1+in21+n3

, because singularities in w

are hard to handle numerically.

(w = ∞ at the vortex core or at spatial infinity)However, with n have to renormalize to n2 = 1 after each

iteration step.


Our work



Our work






iteration step.

Discretize so that code parallelization simple.


Our work



Our work






iteration step.


More complicated initial states generated, in particularlinked unknots.


Our work



Our work






iteration step.


More complicated initial states generated, in particularlinked unknots.

Special emphasis on visualization.


Computational



Computational

Discretized on a cubic lattice, size typically 2403.


Computational



Computational


Discretized the Lagrangian (energy):∂ in on links, F ij on plaquettes.


Computational



Computational



Computed the gradient n(r)L symbolically.


Computational



Computational




Used dissipative dynamics: nnew = nold − δn(r)L.


Computational



Computational




Used dissipative dynamics: nnew = nold − δn(r)L.

Program parallelizes well, have used Cray T3E, SGI Origin2000, IBM SP


How to visualize vector fields?




Cannot draw vectors at every point.

Flow lines do not make sense, because of global gaugeinvariance.

(In fact our vectors could live in another space, e.g., spin.)
















Vector are represented by points on the sphere. We have onefixed direction, n at infinity, define it as the north pole, i.e,n∞ = (0, 0, 1). All other directions are defined by latitude andlongitude.









Latitude is invariant under global gauge rotations that keep thenorth pole fixed, therefore plot equilatitude surfaces (i.e., pointsat which n3 has a fixed value).









Latitude is invariant under global gauge rotations that keep thenorth pole fixed, therefore plot equilatitude surfaces (i.e., pointsat which n3 has a fixed value).

Longitudes are represented by colors on the equilatitudesurface. (Under a global gauge rotation only colors change).


Isosurface n3 = 0 (equator) for |Q| = 1, 2



3 ( q ) |Q| ,

Color order and handedness of twist determine Hopf charge.Inside the torus is the core, where n3 = −1.

These figures were made using the program funcs developed by

J. Ruokolainen at CSC, Espoo, Finland


Results for linked unknots of charge 1+1



g


Deformation 5 + 4 − 2 → trefoil




Energy evolution in minimization



0 20000 40000 60000 80000

Number of iterations

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

T o t a l e

n e r g y

2+2+2

1+1+2

2+2−2

1−1+2 1−1−2

2−2+2

2−2−2

1+1−2


Vakulenko bound



0 1 2 3 4 5 6 7 8

Q

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

E Q /

( E 1

Q 3 / 4 )

Filled circles give the best result (global minima) we have forgiven Hopf charge; open squares are the results of Battye and

Sutcliffe.


Different and improved final states




Knot theory



The proper knot theoretical setting is to use framed links.

Framing attached to a curve adds local information near thecurve, like twisting around it.




Knot theory





One way to describe framed liks is to use directed ribbons,which are preimages of line segments.

We could use equilatitude line segments, then increasing

latitude and longitude give two directions, their cross product theribbon direction.


Knot theory





One way to describe framed liks is to use directed ribbons,which are preimages of line segments.

We could use equilatitude line segments, then increasing

latitude and longitude give two directions, their cross product theribbon direction.

In practice we often choose 4 points on equilongitude line near

the south pole (=core)


Example: ribbon view of Q = −1 unknot



Four nearby points on the equator.These figures were made using OpenDX


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)








linking number = 12 (sum of signed crossings)

The Hopf charge can be determined either by

twist + writheorlinking number of the two ribbon boundaries.


Charge from the ribbon view, Q = −1



Sign convention for crossings allows computing the charge.

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







On the right the ribbon has been turned vertical and isviewed from above: a twist in the ribbon becomes a crossing.







On the right the ribbon has been turned vertical and isviewed 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 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.


Example of ribbon deformation




Close-up of the deformation process




Diagrammatic rule



Knot deformations correspond to ribbon deformations, e.g.,crossing and breaking, but the Hopf charge will be conserved.

Ribbon deformation can be seen if we look the sameconfiguration at different latitudes (the first example before), orat same latitude at different times (animations).


Deformation rule for physical ribbons
























Ribbon connection rules



Total Hopf charge = charges of individual unknots + linkingnumber.






Unknots: Twisting the end on the right hand clockwise a fulltwist yields charge +1.






Unknots: Twisting the end on the right hand clockwise a fulltwist yields charge +1.

Linking number depends on the relative direction associatedwith the unknots:


Physically relevant extensions



Faddeev’s model is hidden in more physical models:

Faddeev and Niemi (2000), electrically conducting plasmas

Cho et al (2001), QCD

Cho (2001), Weinberg-Salam model.

Babaev, Faddeev, Niemi (2002) =⇒

Cho (2002), Bose-Einstein condensates

Babaev (2002), triplet superconductors


Example: Charged two-boson condensate



Change of variables: Ψα =√

2mα ρχα, χ = (χ1, χ∗

2), |χ|2 = 1,

na = (χ, σaχ). Eliminate all χ dependence in favor of n. [Translate

A

→ cC

+ q(χ,∂χ)]


References



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)

Relevant video animations can be seen at

http://users.utu.fi/hietarin/knots/index.html


Jarmo Hietarinta- Ribbon knots in the Faddeev-Skyrme model

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