All Questions
Tagged with solitons field-theory
42
questions
0
votes
0
answers
100
views
What is the topology of sine-Gordon equation?
In one pdf on solitons, I am finding the following written
For the sine-Gordon theory, it is much better to think of $\phi$ as a field modulo $2\pi$, i.e. as a function $\phi: R \rightarrow S_{1}$. ...
0
votes
0
answers
30
views
Do topological solitons allow modeling non-degenerate multiple vacua?
I am not well-versed in the research on topological solitons but am interested to make a good sense of its implication.
The highly interesting point in this new talk by Nick Manton was where he is ...
- 489
2
votes
1
answer
133
views
Scale transformation of the scalar field and gauge field
I am reading this paper: "Magnetic monopoles in gauge field theories", by Goddard and Olive. I don't understand some scale transformations that appear in Page 1427.
Start from the energy ...
- 1,421
1
vote
0
answers
61
views
If principle $SU(N)$ bundles on 3-manifolds are trivial, how can there be magnetic monopoles?
Magnetic monopoles are solitons, i.e. field configurations on space (which is 3 dimensional). In pure $SU(N)$ gauge theory, magnetic monopoles can be constructed via 't Hooft's abelian projection (...
- 742
4
votes
0
answers
110
views
Why are non-perturbative solutions important and how to take them into account?
I am guilty of studying physics with an almost complete focus on the mathematical constructions (together with the motivating physical premisses) and ignoring the semantic physical intuition, which I'...
- 1,742
0
votes
2
answers
126
views
$O(3)$ sigma model for lumps
I'm studying the $O(3)$ $\sigma$-model related to lumps through chapter 6 of Manton's book.
There appears that $$\mathcal{L} = (1/4)\partial _{\mu}\phi \cdot \partial ^{\mu}\phi + \nu (1-\phi \cdot \...
user320803
0
votes
0
answers
73
views
How $\varphi^6$ potential of topological soliton kink and anti-kink are calculated?
how $\varphi^6$ topological soliton kink and anti-kink are calculated ?, what is an anti-kink?
$$L = -\frac{1}{4}F_{\mu\nu}^2− |\varphi|^6 − ieA_\mu(\varphi^\ast\partial_\mu\varphi−\varphi\partial_\mu\...
1
vote
1
answer
160
views
A primer on topological solitons in scalar field theories
As the title suggests I want to learn more about topological solitons in scalar field theories. I am searching for a resource which is self-contained, in the sense that it also explains the ...
2
votes
0
answers
67
views
How do you check the stability of a kink solution?
I am reading a nice introductory note by Hugo Laurell (http://uu.diva-portal.org/smash/get/diva2:935529/FULLTEXT01.pdf) but got confused on section 3.2.
He claims the stability of kink by expanding a ...
- 311
2
votes
1
answer
400
views
An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?
The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the ...
- 713
1
vote
1
answer
100
views
What are the two different $\mathbb{S}^n$ in the construction of the homotopy group $\pi_n(\mathbb{S}^n)$ that classifies topological defects?
According to Mukhanov's Physical Foundations of Cosmology,
Homotopy groups give us a useful unifying description of topological defects. Maps of the $n$-dimensional sphere $\mathbb{S}^n$ into a ...
- 26.8k
5
votes
1
answer
469
views
How to Diagonalize Self-Interacting Scalar Hamiltonian for Mass Term from Polyakov Paper?
So, I'm reading through Polyakov's paper from 1974, "Particle Spectrum in Quantum Field Theory." I'm trying to work through all of the steps and properly understand everything. For context, ...
- 133
3
votes
1
answer
306
views
Stability and topological charge of kink (anti-kink) solutions (soliton)
I am reading the book << Gauge theory of elementary particle physics >>. In chapter 15, it presents a model having finite-energy solution.
First, we have a $1+1D$ spacetime model
\begin{...
- 995
4
votes
1
answer
66
views
Must a field approach one of its vacua to have finite energy?
I'm reading these Cornell lectures on solitons (link doesn't work right now, but it just worked yesterday), and I can't seem to prove what I thought would be a simple analysis exercise.
Namely, ...
- 4,780
1
vote
1
answer
97
views
Set of zeroes as coset space
I am currently studying Chapter 6 of Coleman S. - Aspects of Symmetry.
We study a spontaneously broken gauge theory in two spatial dimensions where the Lagrangian reads:
$$
\mathcal{L} = -\frac{1}{4}...
- 449
0
votes
1
answer
126
views
Periodic traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ for a $\phi^4$ model
Consider
\begin{equation}\label{1}
\partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1)
\end{equation}
the $\phi^4$ model.
I know that
$$H(x)=\...
- 103
3
votes
1
answer
212
views
In which representation are monopoles of grand unifying theories classified?
In the context of grand unification theories A.Zee's book states that $SU(5)$ (or $SO(10)$ if $SU(5)$ is considered as outdated as GUT candidate) as GUT and as spontaneously broken non-abelian gauge ...
- 9,778
4
votes
1
answer
806
views
How does the $U(1)$ global symmetry break in the gauged $XY$ model?
I'm studying the particle vortex duality, and I'm confused how we're able to say that in the Coulomb phase, the "hidden" $U(1)$ global magnetic symmetry spontaneously breaks.
gauged XY model: $\...
- 737
4
votes
1
answer
268
views
Is there a difference between topological defects and topological soliton?
Is there a difference between topological defects and topological soliton? Or are these objects the same thing? I ask this because it very common find some papers whose the authors itself refer, for ...
- 387
1
vote
1
answer
159
views
Why do we consider solitons as a composite object?
Can someone explain why do we consider solitons as a composite object? I know that there are dual theories which the role of fundamental and solitonic objects can be mapped to each other, but I can't ...
- 49
2
votes
0
answers
83
views
Topological solitons in general dimension
Let's begin with a simple model of a field theory:
$$
\mathcal{H} = \int ( \nabla \phi ) ^2
$$
where $\phi$ is an angle valued field defined on some space. We suppose for the moment to freeze out ...
- 781
1
vote
1
answer
335
views
Asymptotic behaviour of soliton-antisoliton solution for the Sine Gordon equation
The question isn't about any actual homework, it's rather a (probably simple) intermediate step I've encountered on Rajaraman's Solitons and instantons : an introduction to solitons and instantons in ...
- 150
7
votes
2
answers
1k
views
Non-chiral skyrmion v.s. Left/Right chiral skyrmion
A skyrmion in a 3-dimensional space (or a 3-dimensional spacetime) is detected by a topological index
$$n= {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial \mathbf{M}}{\partial x}\times\frac{...
- 7,848
4
votes
1
answer
1k
views
What is the intuition for topological currents?
The reason for topological stability of a kink solution in scalar field theory in $1+1$ dimensions is the fact that the finite energy scalar field cannot be continuously deformed into a vacuum.
How ...
- 17.8k
3
votes
1
answer
903
views
Finding the energy of a solution to the Sine-Gordon equation
I am delving into Quantum-Field Theory, and am stuck trying to work out how to compute the energy of a soliton solution to the Sine-Gordon equation in 1-1 spacetime.
I start with the Lagrangian ...
- 1,224
2
votes
2
answers
231
views
Boundary condition for solitons in 1+1 dimensions to have finite energy
Suppose a classical field configuration of a real scalar field $\phi(x,t)$, in $1+1$ dimensions, has the energy $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}...
- 26.8k
1
vote
1
answer
272
views
Vacuum Manifold of an $SU(2)$ Theory
I am reading Coleman's book "Aspects of Symmetry", specifically chapter 6 "Classical Lumps and their Quantum Descendants". He gives an Example 5 p. 209 for the topological solutions for an $SU(2)$ ...
- 21
1
vote
0
answers
344
views
How is the velocity of a soliton defined?
The equation of motion of a real scalar field $\phi(x,t)$ in 1+1 dimension in an arbitrary potential $V(\phi)$ is given by $$\frac{\partial^2\phi}{\partial t^2}-\frac{\partial^2\phi}{\partial x^2}+\...
- 26.8k
9
votes
2
answers
4k
views
What is the definition of soliton?
What is the definition of soliton? I've encountered this name in different situations like when the topic discussed is about QFT, fluid dynamics or optics, but I cannot find a general definition. I've ...
- 1,193
7
votes
0
answers
286
views
Non-topological solitons in condensed matter physics
As I know most well-known soliton solutions in condensed matter physics are topological ones: kinks, domain walls etc.
In field theory there are several examples on non-topological solitons: Q-balls, ...
- 546
3
votes
2
answers
502
views
From the viewpoint of field theory and Derrick's theorem, what's the classical field configuration corresponding to particle? Is it a wavepacket?
In the framework of QM, we have known that particle, like electron, cannot be a wavepacket, because if it is a wavepacket then it will become "fatter" due to dispersion and it's impossible.
However ...
- 5,971
2
votes
1
answer
78
views
Question from Terning's book
In Chapter 7 of Terning's book (Modern Supersymmetry), the first example considered is that of an $SO(3)$ gauge theory, a complex scalar in the triplet representation of $SO(3)$ and a potential term:
$...
- 2,095
3
votes
2
answers
328
views
Why can kink not tunnel to the vacuum, making it topologically stable?
Why can the kink
$$\phi(x)=v\tanh\left(\frac{x}{\xi}\right)$$
not tunnel into vacuum $+v$ or $-v$ (with spontaneous symmetry breaking in the vacuum)?
From the boundary condition, $\phi(x)\rightarrow \...
- 49
3
votes
2
answers
342
views
Why is the solution of the $\phi^6$ potential not a soliton?
Consider a theory with a $\phi^6$-scalar potential:
$$
\mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2.
$$
I solved its equation of motion but found that the general form of its ...
- 175
2
votes
1
answer
643
views
Dimension analysis in Derrick theorem
The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe:
What I don't understand from the above statement:
why $e(\mu)$ has minimum for ...
- 651
4
votes
2
answers
1k
views
Derrick’s theorem
Consider a theory in $D$ spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form
$$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- ...
user12906
0
votes
1
answer
550
views
Vortex in D dimensions soliton
let us consider
the two-dimensional configuration shown in Fig. 3.1a. The lengths of the arrows
represent the magnitude of φ, while their directions indicate the orientation in
the $φ_1 -φ_2$ plane. ...
user12906
1
vote
0
answers
668
views
Domain wall and kink solutions from solitions equations
A general solition equation can be obtaion from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$
where $x_0$ is a constant of integration when we drived this ...
user12906
1
vote
1
answer
577
views
Potential in Relativistic Scalar Field Theory
My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576.
I have understand the equation (92.1) but don't understand that how they guessed the ...
user12906
-1
votes
1
answer
297
views
Oscillon and soliton
I want to know the major difference between oscillon and soliton in terms of radiating energy with respect to time and position. And what about their localization?
- 21
4
votes
0
answers
194
views
Asymptotic limit of the two kink solution of the sine-gordon equation
I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as:
$$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\...
user7757
15
votes
1
answer
993
views
Could this model have soliton solutions?
We consider a theory described by the Lagrangian,
$$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$$
The corresponding field equations are,
$$(i\...
- 1,714