Post on 23-Aug-2021
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
Sine-Gordon, φ4 and φ6 models
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
Demonstration
How to make an experimental setup for sine-Gordon kinks with abelt and pins?
A. Halavanau Topological solitons in scalar field theory
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))
A. Halavanau Topological solitons in scalar field theory
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..
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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)
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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)
A. Halavanau Topological solitons in scalar field theory
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.
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
Beyond 1+1...
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
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]
A. Halavanau Topological solitons in scalar field theory
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
A. Halavanau Topological solitons in scalar field theory
Questions? Comments?
A. Halavanau Topological solitons in scalar field theory