All Questions
Tagged with solitons field-theory
42
questions
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100
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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
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0
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30
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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 ...
2
votes
1
answer
133
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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
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0
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61
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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 (...
4
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0
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110
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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'...
0
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2
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126
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$O(3)$ 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 $$\mathcal{L} = (1/4)\partial _{\mu}\phi \cdot \partial ^{\mu}\phi + \nu (1-\phi \cdot \...
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0
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73
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How $\varphi^6$ potential of topological soliton kink and anti-kink are calculated?
how $\varphi^6$ topological soliton kink and anti-kink are calculated ?, what is an anti-kink?
$$L = -\frac{1}{4}F_{\mu\nu}^2− |\varphi|^6 − ieA_\mu(\varphi^\ast\partial_\mu\varphi−\varphi\partial_\mu\...
1
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1
answer
160
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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
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0
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67
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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 ...
2
votes
1
answer
400
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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 ...
1
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1
answer
100
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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 ...
5
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1
answer
469
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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, ...
3
votes
1
answer
306
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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{...
4
votes
1
answer
66
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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, ...
1
vote
1
answer
97
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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:
$$
\mathcal{L} = -\frac{1}{4}...
0
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1
answer
126
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Periodic traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ for a $\phi^4$ model
Consider
\begin{equation}\label{1}
\partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1)
\end{equation}
the $\phi^4$ model.
I know that
$$H(x)=\...
3
votes
1
answer
212
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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 ...
4
votes
1
answer
806
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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: $\...
4
votes
1
answer
268
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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 ...
1
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1
answer
159
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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 ...
2
votes
0
answers
83
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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 ...
1
vote
1
answer
335
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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 ...
7
votes
2
answers
1k
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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
$$n= {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial \mathbf{M}}{\partial x}\times\frac{...
4
votes
1
answer
1k
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What is the intuition for topological currents?
The reason for topological stability of a kink solution in scalar field theory in $1+1$ dimensions is the fact that the finite energy scalar field cannot be continuously deformed into a vacuum.
How ...
3
votes
1
answer
903
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Finding the energy of a solution to the Sine-Gordon equation
I am delving into Quantum-Field Theory, and am stuck trying to work out how to compute the energy of a soliton solution to the Sine-Gordon equation in 1-1 spacetime.
I start with the Lagrangian ...
2
votes
2
answers
231
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Boundary condition for solitons in 1+1 dimensions to have finite energy
Suppose a classical field configuration of a real scalar field $\phi(x,t)$, in $1+1$ dimensions, has the energy $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}...
1
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1
answer
272
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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 $SU(2)$ ...
1
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0
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344
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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}+\...
9
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2
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4k
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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 ...
7
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0
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286
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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, ...