Jarmo Hietarinta- Localized objects in the Faddeev-Skyrme model

8/3/2019 Jarmo Hietarinta- Localized objects in the Faddeev-Skyrme model

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Localized objects in theFaddeev-Skyrme model

Jarmo Hietarinta

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

IWNMMP, Beijing, China, July 15-21,2009


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Faddeevs model

Knot theory

Vortices

Topological solitons

The model

Numerical results 1: Knots

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; completely

integrable.

Jarmo Hietarinta Faddeevs model


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Faddeevs model

Knot theory

Vortices


The model


Introduction

Solitons:


dromions).



integrable.

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

Knot theory

Vortices


The model


Introduction

Solitons:


dromions).



integrable.

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 in Faddeevs model,

done in collaboration with P. Salo and J. Jykk.Jarmo Hietarinta Faddeevs model


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Faddeevs model

Knot theory

Vortices


The model


Topology in R3: Hopfions

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 S2.

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


F dd d l T l i l li


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Faddeevs model

Knot theory

Vortices


The model


Topology in R3: Hopfions

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 S2.

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

Therefore

n : S3 S2.

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

by the homotopy class 3(S2) = Z.


F dd d l T l i l lit


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Faddeevs model

Knot theory

Vortices


The model


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.


Faddeevs model Topological solitons


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Faddeev s model

Knot theory

Vortices


The model


A concrete Hopfion


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 thatn = (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

Knot theory

Vortices


The model


A concrete Hopfion


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 thatn = (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 = abcn

ainbjn

c.

Given Fij construct Aj so that Fij = iAj jAi, then

Q =1

162

ijkAiFjk d3x.




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Faddeev s model

Knot theory

Vortices


The model


Possible physical realization




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Faddeev s model

Knot theory

Vortices


The model


Faddeevs model

In 1975 Faddeev proposed the Lagrangian (energy)

E =

(in)

2 + gF2ij

d3x, Fij := n in jn.




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Faddeev s model

Knot theory

Vortices


The model


Faddeevs model


E =

(in)

2 + gF2ij


Under the scaling r r the integrated kinetic term scales as and the integrated F2 term as 1.

Therefore nontrivial configurations will attain some fixed size

determined by the dimensional coupling constant g. (Virial

theorem)




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Knot theory

Vortices

p g

The model


Faddeevs model


E =

(in)

2 + gF2ij


Under the scaling r r the integrated kinetic term scales as and the integrated F2 term as 1.

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.

Similar upper bound has been derived recently by Lin and Yang.Jarmo Hietarinta Faddeevs model



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Knot theory

Vortices

p g

The model


Numerical studies of 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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Knot theory

Vortices

The model






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.




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Knot theory

Vortices

The model






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. Hietarinta and P. Salo: Faddeev-Hopf knots: dynamics oflinked 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).


Faddeevs model

K h


Th d l


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Knot theory

Vortices

The model


How to visualize vector fields?

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


Faddeevs model

K t th


Th d l


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Knot theory

Vortices

The model




n is a point on the sphere S2.

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).


Faddeevs model

Knot theory


The model


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Knot theory

Vortices

The model





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

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


Faddeevs model

Knot theory


The model


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Knot theory

Vortices

The model





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

(e.g., 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):

we paint the surfaces using longitudes.


Faddeevs modelKnot theory

Topological solitonsThe model


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Knot theory

Vortices

The model


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

Color order and handedness of twist determine Hopf charge.

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





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Knot theory

Vortices

The model


Evolution of linked unknots

Total charge = sum of individual charges + linking number





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Knot theory

Vortices

The model


Energy evolution in minimization





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y

Vortices Numerical results 1: Knots

Evolution (1, 5) 1 + 2 + 2





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y


Evolution 5 + 4 2 trefoil





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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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Different and improved final states



V i

Framed links and ribbon knots

Ribbon deformations


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VorticesRibbon deformations


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.



V ti


Ribbon deformations


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Vorticesbbo de o at o s


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.



Vortices


Ribbon deformations


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Vortices

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)



Vortices


Ribbon deformations


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Vortices

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

orlinking number of the two ribbon boundaries,

or

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



Vortices


Ribbon deformations


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Vortices

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.



Vortices


Ribbon deformations


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Vortices

Example of ribbon deformation during minimization



Vortices


Ribbon deformations


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Vortices

Close-up of the deformation process



Vortices


Ribbon deformations


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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:



Vortices

Topology for vorticesNumerical results 2: Hopfion vortices

Process of unwinding


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g

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.



Vortices




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Single twisted vortex

J. Hietarinta, J. Jykk and P. Salo: Dynamics of vortices and

knots in Faddeevs model, JHEP Proc.: PrHEP unesp2002/17

http://pos.sissa.it//archive/conferences/008/017/un

J. Hietarinta, J. Jykk and P. Salo: Relaxation of twisted

vortices in the Faddeev-Skyrme model, Phys. Lett. A 321,

324-329 (2004).

Knotting as usual if tightly wound:



Vortices




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Vortex bunches and unwinding

If the vortices are close enough there can be bunching

or in the fully periodic case, Hopfion unwinding.

J. Jykk and J. Hietarinta: Unwinding in Hopfion vortex

bunches, Phys. Rev. D 79, 125027 (2009).



Vortices




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In the following pictures the lines corresponding to two

preimages have been plotted, one in blue and one in red.



Vortices




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In the following pictures the lines corresponding to two

preimages have been plotted, one in blue and one in red.

First a 3 3 section of the fully periodic case.



Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices




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Vortices



4 4


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4 4



Vortices



4 4


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4 4



Vortices



4 4


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4 4


Faddeevs model

Knot theory

Vortices

Topology for vortices

Numerical results 2: Hopfion vortices


Conclusion


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Conclusion

We have studied the minimum energy states and the

deformation dynamics of Hopfions in Faddeevs model.


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion



The results follow closely the Vakulenko-Kapitanskii bound


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion




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

configurations.


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion





configurations. Tightly would vortices form knots in the middle.


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion






Closeup vortices form a bunch.


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion







In the fully periodic case the vortices can unwind

completely.


Faddeevs model

Knot theory

Vortices




Conclusion


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Conclusion







In the fully periodic case the vortices can unwind

completely.

See also:

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


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Jarmo Hietarinta- Localized objects in the Faddeev-Skyrme model