jarmo hietarinta- topological solitons in the faddeev-skyrme model

8/3/2019 Jarmo Hietarinta- Topological solitons in the Faddeev-Skyrme model

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Topological solitons in the

Faddeev-Skyrme model

Jarmo Hietarinta

Department of Physics and Astronomy, University of TurkuFIN-20014 Turku, Finland

in collaboration with P. Salo, J. Jykk, J. Palmu, and P. Pakkanen.

NLP6, Gallipoli, 23.6-3.7.2010
http://find/http://goback/


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Faddeevs model

Vortices

Hopfion scattering

Introduction

Numerical results for knots

Knot theory

Introduction

Solitons:

Localized on the line (usually) or in the plane (lumps,

dromions).

Stability due to infinite number of conservation laws.

Amenable to exact analytical treatment; completelyintegrable.

Jarmo Hietarinta Faddeevs model


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Faddeevs model

Vortices

Hopfion scattering

Introduction


Knot theory

Introduction

Solitons: Localized on the line (usually) or in the plane (lumps,

dromions).



Topological solitons:

Localized on the line (kinks), on the plane (vortices) or in

3D-space (Hopfions).

Stability due to topological properties.

Often can only solve numerically.



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Faddeevs model

Vortices

Hopfion scattering

Introduction


Knot theory

Introduction

Solitons: Localized on the line (usually) or in the plane (lumps,

dromions).



Topological solitons:

Localized on the line (kinks), on the plane (vortices) or in

3D-space (Hopfions).

Stability due to topological properties.

Often can only solve numerically.

In this talk: Review of work on Hopfions inFaddeevsmodelJarmo Hietarinta Faddeevs model


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Faddeevs model

Vortices

Hopfion scattering

Introduction


Knot theory

Topology in R3: Hopfions

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


F dd d l I d i


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Faddeevs model

Vortices

Hopfion scattering

Introduction


Knot theory



Asymptotically trivial: n(r) n, when |r| can compactify R3 S3.


F dd d l I t d ti


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Faddeevs model

Vortices

Hopfion scattering

Introduction


Knot theory




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

the sphere S2. Therefore

n : S3 S2.


Faddeevs model Introduction


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Faddeev s model

Vortices

Hopfion scattering

Introduction


Knot theory




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

the sphere S2. Therefore

n : S3 S2.

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

by the homotopy class 3(S2) = Z.




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Faddeev s model

Vortices

Hopfion scattering

Introduction


Knot theory

A concrete Hopfion

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. Note that

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




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Faddeev s model

Vortices

Hopfion scattering

Introduction


Knot theory

A concrete Hopfion

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. Note that

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

Computing the Hopf charge:

Given n : R3 S2 define Fij = abcnainbjnc.Given Fij construct Aj so that Fij = iAj jAi, then

Q =1

162

ijkAiFjk d

3x.




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Faddeev s model

Vortices

Hopfion scattering

Introduction


Knot theory

Possible physical realization




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Faddeev s model

Vortices

Hopfion scattering

Introduction


Knot theory

Dynamics: Faddeevs model

In 1975 Faddeev proposed the Lagrangian (energy)

E =

(in)

2 + gF2ij

d3x, Fij := n in jn.




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Vortices

Hopfion scattering


Knot theory



E =

(in)

2 + gF2ij


Under the scaling r r the integrated kinetic term scales as

and the integrated F2 term as 1.

Nontrivial configurations will attain some fixed size determined

by the dimensional coupling constant g. (Virial theorem)




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Vortices

Hopfion scattering


Knot theory



E =

(in)

2 + gF2ij


Under the scaling r r the integrated kinetic term scales as

and the integrated F2 term as 1.

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.

Similar upper bound has been derived recently by Lin and Yang.




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Vortices

Hopfion scattering


Knot theory

Minimum energy states in Faddeevs 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.




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Vortices

Hopfion scattering


Knot theory

Minimum energy states in Faddeevs 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.

Our work:Full 3D minimization using dissipative dynamics:

nnew = nold n(r)L.No assumptions on symmetry, on the contrary:

Linked unknots of various charges.

J.H. and P. Salo, Phys. Lett. B 451, 60-67 (1999),

Phys. Rev. D 62, 081701(R) (2000).


Faddeevs model

V i

Introduction

N i l l f k


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Vortices

Hopfion scattering


Knot theory

How to visualize vector fields?

There is one fixed direction, n = (0, 0, 1), the north pole.Vortex core is where n = n (the south pole).


Faddeevs model

V ti

Introduction

N i l lt f k t


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Vortices

Hopfion scattering


Knot theory

How to visualize vector fields?

There is one fixed direction, n = (0, 0, 1), the north pole.Vortex core is where n = n (the south pole).

We plot isosurfaces n3 = c with color determined by n1.

Example: Isosurface n3 = 0 for |Q| = 1, 2

Color order and handedness of twist determine Hopf charge.

Inside the torus is the core, where n3 = 1.


Faddeevs model

Vortices

Introduction



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Vortices

Hopfion scattering


Knot theory

Evolution of linked unknots

Total charge = sum of individual charges + linking number


Faddeevs model

Vortices

Introduction



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Vortices

Hopfion scattering


Knot theory

Various final states


Faddeevs modelVortices

IntroductionNumerical results for knots


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Vortices

Hopfion scattering


Knot theory

Evolution 5+ 4 2 trefoil





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Vortices

Hopfion scattering


Knot theory

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

EQ/

(E1

Q3/4)





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Vortices

Hopfion scattering


Knot theory

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, e.g., twisting around it.





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Hopfion scattering Knot theory

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, e.g., twisting around it.

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

which are preimages of line segments.Example: Q = 2 in two ways:





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Computing the charge

For a ribbon define:

twist = linking number of the ribbon core with a ribbon

boundary, locally.

writhe = signed crossover number of the ribbon core with

itself.

linking number = 12 (sum of signed crossings)





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Computing the charge

For a ribbon define:

twist = linking number of the ribbon core with a ribbon

boundary, locally.

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 + writhe

or

linking number of the two ribbon boundaries,

or

linking number of the preimages of any pair of regular points.





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Diagrammatic rule for deformations

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

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

Note that when considering equivalence of ribbon diagrams

type I Reidemeister move is not valid:



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H fi tt i

Topology for vorticesNumerical results 2: Hopfion vortices

P f i di


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Hopfion scattering Process of unwinding

What is different with vortices

Vortices do not allow 1-point compactification of R3 S3.

Instead we have R2 T if periodic in z-direction

or T3 if periodic in all directions.

Topological conserved quantities studied by Pontrjagin in 1941,

They are mainly related to vortex punctures of the periodic box.



Hopfion scattering


Process of unwinding


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Further results on single vortices:

J.H., J. Jykk and P. Salo: JHEP Proc.: PrHEP unesp2002/17

J.H., J. Jykk and P. Salo: Phys. Lett. A 321, 324-329 (2004).

Knotting as usual if tightly wound.



Hopfion scattering




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Further results on single vortices:

J.H., J. Jykk and P. Salo: JHEP Proc.: PrHEP unesp2002/17

J.H., J. Jykk and P. Salo: Phys. Lett. A 321, 324-329 (2004).

Knotting as usual if tightly wound.

If there are several vortices close enough there can be

bunching or in the fully periodic case, Hopfion unwinding.

J. Jykk and J.H., Phys. Rev. D 79, 125027 (2009).



Hopfion scattering




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4 4



Hopfion scattering




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4 4



Hopfion scattering




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4 4



Hopfion scattering

Computational methodResults

Conclusions


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p g

True time evolution

So far: fictitious time giving 1st order dissipative dynamics.

Now: Full dynamics by the Lorentz invariant Lagrangian:

L

=

1

4{

0 (

)

1[(

)

2

(

)(

)]+(1

)}.

The metric is (1, 1, 1, 1) and 0 = 1, with 1 > 0. is the Lagrange multiplier.



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Hopfion scattering


Conclusions


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Computational setting:

Time evolution by 4th order Runge-Kutta.

Spatial derivatives discretized on 53 points,

space typically discretized into a 1503 lattice.

Dirichlet boundary condition with n| = n = (0, 0, 1).Furthermore a thin and mild absorption layer to prevent

radiation from reflecting and re-entering scattering region.

Courant instability was controlled by smoothing.

Platform: SAMRAI for dynamics and Visit for visualization.



Hopfion scattering


Conclusions


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Scattering processes

Initial states constructed by minimization and then Lorentz

boosted numerically.



Hopfion scattering


Conclusions


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Scattering processes

Initial states constructed by minimization and then Lorentz

boosted numerically.

Observations:

Head on scattering:

Various deformations possible, but the Hopf charge is

conserved.

The final state may acquire spinning

With nonzero impact parameter there may be intermediatebound states.



Hopfion scattering


Conclusions


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Conclusions

We have studied of Hopfions in Faddeevs model.



Hopfion scattering


Conclusions


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Conclusions

We have studied of Hopfions in Faddeevs model.

Used dissipative dynamics to find minimum energy states

for a given Hopf charge.

Some results obtained on the taxonomy of the minimumenergy states.

The results follow closely the Vakulenko-Kapitanskii bound

E c|Q|3/4

The trefoil knot is obtained at |Q| = 7 from various initialconfigurations.



Hopfion scattering


Conclusions


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Conclusions

Results on vortices:

Tightly would vortices form knots in the middle.

Closeup vortices form a bunch.

In the fully periodic case the vortices can unwind

completely.

We have also studied Hopfion scattering.



Hopfion scattering


Conclusions


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Conclusions

The videos can be seen at

www.youtube.com/nonlinearUTU

This work is supported by Academy of Finland grants

47188 and 123311.


jarmo hietarinta- topological solitons in the faddeev-skyrme model

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