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

by Mariya Lizunova, Jasper van Wezel

Submission summary

As Contributors: Jasper van Wezel
Arxiv Link: (pdf)
Date accepted: 2021-01-18
Date submitted: 2020-12-11 12:58
Submitted by: van Wezel, Jasper
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


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.

Published as SciPost Phys. Lect. Notes 23 (2021)

Author comments upon resubmission

Dear editor,

we are thankful to the referees for their positive reports and supportive comments.
Please find below a list of changes we made to the manuscript, addressing all of the minor issues raised by the referees.

Best regards,
Mariya Lizunova and Jasper van Wezel

List of changes

We would like to thank both referees for their positive comments and useful suggestions.
In response to their remarks, we made the following changes to the manuscript:

First referee:
1. We thank the referee for pointing out the typo, and changed "loose" into "lose", as suggested.
2. We thank the referee for proposing the change in section name, and adopt their suggestion.
3. We have added all recommended citations in relevant places in the text.

Second referee:
1. We appreciate the suggestion of the referee that hints and solutions to exercises may be useful. However, several exercises already have the answers outlined in the main text, and others are of a more practical nature, asking students to produce their own numerical codes. To stimulate actual effort from students in working through derivations and writing code, we prefer not to provide detailed answers to most exercises. We do however appreciate the usefulness of some further help along the way, and we therefore added hints to Exercises 3.1, 4.1, and 6.1.

2. Although we agree that discussions of physical applications can be useful and interesting, the choice not to include them in the present lecture notes was intentional. The aim of the lecture notes is to provide the mathematical and computational basis needed for understanding kink dynamics and its simulations in general, without the bias of having an application in mind. The vast scala of applications in which these dynamics feature can be easily be explored by interested students, starting from the citations that we provide in the Introduction.

Reports on this Submission

Report 2 by Amit Dutta on 2021-1-13 Invited Report


I have gone through the revised version. The authors have appropriately
taken care of my comments. I therefore, recommend its publication
in the present form.

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Anonymous Report 1 on 2021-1-5 Invited Report


I am happy with the small changes made.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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