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An introduction to kinks in $\varphi^4$-theory

by Mariya Lizunova, Jasper van Wezel

This is not the current version.

Submission summary

As Contributors: Jasper van Wezel
Arxiv Link: https://arxiv.org/abs/2009.00355v1 (pdf)
Date submitted: 2020-09-02 07:53
Submitted by: van Wezel, Jasper
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
Approach: Theoretical

Abstract

As a low-energy effective model emerging in disparate fields throughout all of physics, the ubiquitous $\varphi^4$-theory is one of the central models of modern theoretical physics. Its topological defects, or kinks, describe stable, particle-like excitations that play a central role in processes ranging from cosmology to particle physics and condensed matter theory. In these lecture notes, we introduce the description of kinks in $\varphi^4$-theory and the various physical processes that govern their dynamics. The notes are aimed at advanced undergraduate students, and emphasis is placed on stimulating qualitative insight into the rich phenomenology encountered in kink dynamics. The appendices contain more detailed derivations, and allow enquiring students to also obtain a quantitative understanding. Topics covered include the topological classification of stable solutions, kink collisions, the formation of bions, resonant scattering of kinks, and kink-impurity interactions.

Current status:
Has been resubmitted



Reports on this Submission

Anonymous Report 2 on 2020-11-10 Invited Report

Strengths

Detailed and transparent

Weaknesses

Lack of Physical examples

Report

This is an interesting and useful review article which is very carefully drafted.

This review deals solitons arising in $\phi^4$ which is not integrable and hence
interactions between the kinks lead to interesting behavior in comparison to the
(integrable) sine-Gordon model. Major focus of the review is on the dynamics and
interactions of kinks in the $\phi^4$ theory.
The basic theory of solitons and and kink collisions are nicely discussed
in chapters 2 and 3. Collective coordinate approximation, gluing static
solutions and kink-impurity interactions are discussed in subsequent
sections. Overall, the review is self-sufficient and mathematical
steps and arguments are quite transparent. The discussion in the
main text is supplemented by three appendices. Indeed, authors have
discussed a wide range of using numerical calculations and effective
models.

I have just two suggestions:

1. It is very heartening to that authors have provides some relevant
exercise in every section? What about providing some hints and answers
at the end

2.Secondly, I find this article a bit dry. This is not a criticism
but, I do believe, review would look far more enriched if examples of physical
situations are added wherever possible. For example, as
given in Chaikin Lubensky chapter 9. The article would then attract
a wider audience.

With these minor comments, I would accept this review for publication.

Requested changes

Already in report

  • validity: high
  • significance: -
  • originality: good
  • clarity: good
  • formatting: perfect
  • grammar: acceptable

Anonymous Report 1 on 2020-11-8 Invited Report

Strengths

The authors have done a nice job of producing a readable, even
pedagogical, summary of $\phi^4$ kinks. They have managed to do so
without neglecting mathematical aspects, including some that were new
to me (the elliptic sine solution). The standard of english is excellent.

Report

The authors have done a nice job of producing a readable, even
pedagogical, summary of $\phi^4$ kinks. They have managed to do so
without neglecting mathematical aspects, including some that were new
to me (the elliptic sine solution). The standard of english is excellent.

Requested changes

I would change the title of Section 3 to "Kink-antikink collisions".
On page 12, replace "loose" by "lose".

Although it is difficult to cite all literature, I do feel that
the tutorial would be enhanced by mentioning the following:

@article{kevrekidis2019dynamical,
title={A Dynamical Perspective on the $\varphi^4$ model},
author={Kevrekidis, Panayotis G and Cuevas-Maraver, Jes{\'u}s},
journal={Past, present and future. Nonlinear Systems and Complexity},
volume={26},
year={2019},
publisher={Springer}
}

@article{rice1983physical,
title={Physical dynamics of solitons},
author={Rice, MJ},
journal={Physical Review B},
volume={28},
number={6},
pages={3587},
year={1983},
publisher={APS}
}

@article{bishop1980solitons,
title={Solitons in condensed matter: a paradigm},
author={Bishop, AR and Krumhansl, JA and Trullinger, SE},
journal={Physica D: Nonlinear Phenomena},
volume={1},
number={1},
pages={1--44},
year={1980},
publisher={Elsevier}
}

@article{goodman2005kink,
title={Kink-Antikink Collisions in the $\phi^4$ Equation: The n-Bounce Resonance and the Separatrix Map},
author={Goodman, Roy H and Haberman, Richard},
journal={SIAM Journal on Applied Dynamical Systems},
volume={4},
number={4},
pages={1195--1228},
year={2005},
publisher={SIAM}
}

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Login to report


Comments

Anonymous on 2020-11-10

This is an interesting and useful review article which is very carefully drafted.

This review deals solitons arising in $\phi^4$ which is not integrable and hence interactions between the kinks lead to interesting behaviour in comparison to the (integrable) sine-Gordon model. Major focus of the review is on the dynamics and interactions of kinks in the $\phi^4$ theory. The basic theory of solitons and and kink collisions are nicely discussed in chapters 2 and 3. Collective coordinate approximation, gluing static solutions and kink-impurity interactions are discussed in subsequent sections. Overall, the review is self-sufficient and mathematical steps and arguments are quite transparent. The discussion in the main text is supplemented by three appendices. Indeed, authors have discussed a wide range of using numerical calculations and effective models.

I have just two suggestions:

  1. It is very heartening to that authors have provides some relevant exercise in every section? What about providing some hints and answers at the end

2.Secondly, I find this article a bit dry. This is not a criticism but, I do believe, review would look far more enriched if examples of physical situations are added wherever possible. For example, as given in Chaikin Lubensky chapter 9. The article would then attract a wider audience.

With these minor comments, I would accept this review for publication.