Topological solitons in scalar field theoryHave bounce windows structure Fractal structure Bounce...

$: Topological solitons in scalar field theoryHave bounce windows structure Fractal structure Bounce windows and fractal structure refers to the energy exchange mechanism between translational$
Topological solitons in scalar field theory

A. Halavanau

Department of Physics, Northern Illinois University

2014

A. Halavanau Topological solitons in scalar field theory

Topological solitons in scalar field theory

Introduction

Sine-Gordon, φ4 and φ6 models

Collisions (Scattering)

Boundary collisions (φ4 example)

Demonstration

Skyrme models (hopfions, baby Skyrmions, Skyrme crystals)

Conclusions

A. HalavanauDepartment ofPhysics, NIU

APS March Meeting, 2014


Solitons in nonlinear physics

Soliton: This is a solution of a nonlinear partial differentialequation which represents a solitary travelling wave, which:

Is localized in space

Has a constant shape

Does not obey the superposition principle.

Examples:

Optical fibres, rogue waves in ocean - NLSE

Josephson junctions - sine-Gordon model

Lattice QCD - caloron solutions

Superconductivity - Abrikosov-Nielsen-Olesen model


Topology

Topological charge

Q =1

2

∞∫−∞

dx∂φ

∂x

Non-toplogical solitons

KdV solutions

Lump-solution in various polynomial models

Oscillons

Topological solitons

Kinks, domain walls, vortices in various models

Skyrmions

Hopfions


Sine-Gordon, φ4 and φ6 models


sine-Gordon model

Field equations

∂2v

∂t ∂t− ∂2v

∂x ∂x+ sin(v) = 0

v(x , t) = 4 arctan e(−√−1+k2t+kx)

A simple mechanical example: the chain of coupled pendula


sine-Gordon model

Hamiltonian and equations of motion

H =N∑n

I

2

(dθndt

)2

+C

2(θn − θn−1)2 + mgl(1− cos θn)

Id2θndt2

− C (θn+1 + θn−1 − 2θn) + mgl sin θn = 0

- 6 - 4 - 2 2 4 6z

1

2

3

4

5

6

4 tan - 1 I ± ã z M

- 10 - 5 5 10z

1

2

3

4

5

6

4 tan - 1 I ± ã z M


Demonstration

How to make an experimental setup for sine-Gordon kinks with abelt and pins?


φ4 potential model

- 4 - 2 2 4

- 1.0

- 0.5

0.5

1.0

- 1.5 - 1.0 - 0.5 0.5 1.0 1.5

0.5

1.0

1.5

2.0

Lagrangian and equation of motion

L =1

2∂µφ∂

µφ− 1

2

(φ2 − 1

)2

∂ttφ− ∂xxφ+ 2φ(φ2 − 1) = 0

Static solution (kink/antikink)

φ = tanh(A(±x + x0))


Modes of φ4 kink

Linear oscillations on the background

φ = φK + δφ

where δφ can be expanded in a set:

δφ =∞∑n=0

Cn(t)ηn(x)

Eigenfunctions

η0(x) =1

cosh2 x

η1(x) =sinh x

cosh2 x

ηk(x) = e ikx(3 tanh2 x − 3ik tanh x) − e ikx(1 + k2) + C.C..


φ4 kink-antikink collisions

Properties

Annihilate or repel

Produce oscillon state after annihilation

Have bounce windows structure

Fractal structure

Bounce windows and fractal structure refers to the energyexchange mechanism between translational and internal modes


Bounce windows structure

Time, sec

Velocity

100

80

60

40

20

0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.50.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30

Figure: Bounce window structure of KK̄ collision in a usual φ4 case


φ6 potential model

- 10 - 5 5 10

- 1.0

- 0.5

0.5

1.0

- 1.5 - 1.0 - 0.5 0.5 1.0 1.5

0.05

0.10

0.15

0.20

Lagrangian and equation of motion

L =1

2∂µφ∂

µφ− 1

2φ2(φ2 − 1

)2

∂ttφ− ∂xxφ+ 2φ3(φ2 − 1) + φ(φ2 − 1) = 0

Static solution (kink/antikink)

φ = ±√

1± tanh x

2


Mysterious φ6

Quick glance on φ6 window structure

No internal mode. Energy exchange is due to continuous spectrum

Missing window is probably an interference effect (no theory yet)

Collective coordinate model is under construction (no theory yet)


Coupled φ4 model

Lagrangian of the coupled two-component model can be written as:

L =1

2[(∂tφ1)2 − (∂xφ1)2 − (φ2

1 − 1)2]

+1

2[(∂tφ2)2 − (∂xφ2)2 − (φ2

2 − 1)2] + κφ21φ

22

Field equations:{∂2t φ1 − ∂2

xφ1 + 2φ1(φ21 − 1)− 2κφ1φ

22 = 0

∂2t φ2 − ∂2

xφ2 + 2φ2(φ22 − 1)− 2κφ2φ

21 = 0

Wess-Zumino model with two coupled Majorana spinor fieldsMontonen-Sarker-Trullinger-Bishop model with two scalar coupled fields


Static configuration (numerical)

-5 5x

1.5

1.0

0.5

0.5

1.0

1.5

-5 5x

1.15

1.20

1.25

1.30

φ

φ

Figure: Static configuration for κ = 0.5

A. Halavanau, T. Romanczukiewicz, Ya. Shnir - Resonance structures incoupled two-component φ4 model, Physical Review D 86, 085027


Different collision channels

Topological charge

Qi =1

2

√1− κ

∞∫−∞

dx∂φi∂x

, i = 1, 2

Interesting channels

(−1, 0) + (1, 0)→

(0, 0)(0,−1) + (0, 1)(−1, 0) + (1, 0)

(−1, 1) + (−1, 1)→{

(0, 0)(−1, 1) + (−1, 1)


Topological flipping and double kink configuration

0

10

20

30

40

50

60

70

0.41 0.415 0.42 0.425 0.43 0.435 0.44 0.445 0.45 0.455

Tim

e,se

c

Velocity

-1.5

-1

-0.5

0

0.5

1

1.5

0

10

20

30

40

50

60

70

0.2 0.25 0.3 0.35 0.4

Tim

e,se

c

Velocity

-1.5

-1

-0.5

0

0.5

1

1.5

Kink-lump collision (left) and double kink collision (right). Numericalerrors prove FDM is not sufficient for the problem and spectral methodsare to be used.


Boundary collisions - Lagrangian mechanics

Applying boundary conditions on the equations of motion

following the principle of stationary action, one can obtain the equationsof motion for the bulk and the boundary(assuming M(φ, φt) = B(φ)− A(φ, φt)):(

∂L

∂φ− d

dt

∂L

∂φt− d

dt

∂L

∂φx

)= 0

and for the boundary point x = xb(∂L

∂φx− ∂M

∂φ+

d

dt

∂L

∂φt

)= 0

φx(0, t) = Hb = const from where x0 = xb + cosh−1 (√

1H ) and few

modes survive from collective spectrum:

η(x) = e ik(x−x0)(−1− k2 − 3ik tanh (x − x0) + 3 tanh (x − x0)2

)A. Halavanau Topological solitons in scalar field theory

Neumann boundary (applied magnetic field)

by courtesy of T. Romanczukiewicz


Beyond 1+1...


Higher dimensions

Derrick’s theorem shows that stationary localized solutions to anonlinear wave equation or nonlinear Klein–Gordon equation indimensions three and higher are unstable.

Skyrme model

E =1

32π2√

2

∫ (∂iφ∂iφ+

1

2(∂iφ∂iφ× ∂iφ∂iφ)(∂iφ∂iφ× ∂iφ∂iφ)

)The first term in the energy is that of the usual O(3) sigma model andthe second is a Skyrme term, required to provide a balance under scalingand hence allow solitons with a finite non-zero size.

Skyrmions, baby Skyrmions

Hopfions

Skyrme crystals


Isorotating baby Skyrmions

Lagrangian and topological charge

L =1

2∂µ~φ · ∂µ~φ−

1

4(∂µ~φ× ∂ν~φ)2 − U(~φ)

B =1

4π

∫~φ · ∂1

~φ× ∂2~φ d2x

Different potential choices

U(~φ) = µ2[1− φ3]

U(~φ) = µ2[1− φ23]

U(~φ) = µ2[1− φ23][1− φ2

1]

Rotation invariance

(φ1 + iφ2)→ (φ1 + iφ2)e iωt


Isorotating baby Skyrmions

A. Halavanau, Yakov Shnir - Isorotating Baby Skyrmions, Phys.Rev. D 88, 085028 (2013)

-4 -3 -2 -1 0 1 2 3 4-4-3

-2-1

01

23

4

0123456789

x

y

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2-1.5

-1-0.5

0 0.5

1 1.5

2

0

5

10

15

20

25

x

y

-4 -3 -2 -1 0 1 2 3 4 -4-3

-2-1

0 1

2 3

4

0

1

2

3

4

5

6

x

y

-4 -3 -2 -1 0 1 2 3 4-4-3

-2-1

0 1

2 3

4

0 1 2 3 4 5 6 7 8 9

x

y


Faddeev-Skyrme model

Lagrangian and mapping

L =1

32π2

(∂µφ

a∂µφa − κ

4(εabcφ

a∂µφb∂νφ

c)2 + µ2[1− (φ3)2])

R3 → C 2 mapping

W =p(Z1,Z0)

q(Z1,Z0)=

Zα1 Zβ0

Z a1 + Zb

0

Topological charge (Hopf charge)

Q = αb + βa


Links, knots and seals in Faddeev-Skyrme model

Massive and isospinning hopfions (work in progress). arXiv:1301.2923[hep-th], J. Jaykka, Ya. Shnir, D. Harland and M. SpeightA. Acus, A. Halavanau, E. Norvaisas and Ya. Shnir - Hopfion canonicalquantization, Physics Letters B 711 (2012) [hep-th]


Collaborations and resources

Existing collaborations

University of Oldenburg, with Ya. Shnir

Uniwersytet Jagiellonski, with T. Romanczukiewicz

Durham University, with P. E. Dorey and P. Sutcliffe

Vilnus University, with A. Acus

Stockholm University, with J. Jaykka

Resources

Mathematica 8

SKIF supercomputer, Belarusian State University

Condor computing cluster, Durham University

Over 2000 hours of computing time


Questions? Comments?


Topological solitons in scalar field theoryHave bounce windows structure Fractal structure Bounce...

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