# Gauged Sigma Models and Magnetic Skyrmions

### Submission summary

 As Contributors: Bernd Schroers Preprint link: scipost_201905_00004v3 Date accepted: 2019-08-28 Date submitted: 2019-07-30 Submitted by: Schroers, Bernd Submitted to: SciPost Physics Discipline: Physics Subject area: Mathematical Physics Approach: Theoretical

### Abstract

We define a gauged non-linear sigma model for a 2-sphere valued field and a $SU(2)$ connection on an arbitrary Riemann surface whose energy functional reduces to that for critically coupled magnetic skyrmions in the plane, with arbitrary Dzyaloshinskii-Moriya interaction, for a suitably chosen gauge field. We use the interplay of unitary and holomorphic structures to derive a general solution of the first order Bogomol'nyi equation of the model for any given connection. We illustrate this formula with examples, and also point out applications to the study of impurities.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 7, 030 (2019)

In response to the comments by referees 1 and 3 I have made major changes to the paper, detailed and justified below. I hope that they address the referees' concerns. I thank referee 2 for their positive remarks, and for pointing out typos, which I corrected.

### List of changes

1. The introduction is substantially re-written and extended to bring out clearly that this paper goes far beyond the earlier paper [8] (in the new ordering of references), both in terms of applications (in deals with the most general DM interaction for magnetic skyrmions and includes impurities) and in terms of the underlying geometry (the underlying mathematical reason for solvability was not clear in [8] but is fully understood in this paper). The difference between this model and gauged non-linear sigma models studied elsewhere in the literature is also clarified.

2. I have made minor changes to Sections 2 (some more details on boundary terms in 2.4) and 3 (a new final section 3.3 to highlight the general formula for solutions). Despite the reservations the referees have expressed about the use of differential forms I have not replaced them. I find them by far the most efficient tools for the calculations in these sections, and rely on them to bring out the mathematical structures which underpin the model. This would be harder and less clear in traditional vector calculus notation. I would also make the case that this paper has a strong interdisciplinary element, connecting condensed matter physics with differential geometry and mathematical physics, and that some compromise in the notation is therefore inevitable. Finally, I have made only very minor use of differential forms in the new Section 4 which deals with applications.

3. I have introduced an entirely new Section 4 which spells out the application to magnetic skyrmions and impurities in much more detail then the previous version. This should bring out clearly the range of applications and also the extent to which the gauged sigma models provide a unifying picture for magnetic skyrmions and impurities.

### Submission & Refereeing History

Resubmission scipost_201905_00004v3 on 30 July 2019
Submission scipost_201905_00004v1 on 23 May 2019

## Reports on this Submission

### Report

The author addressed all the points raised by the referees. At least from my side there are no further concerns. I recommend the publication of the revised version in SciPost.

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

### Report

The author has adopted the minor changes I recommended in my first report

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

### Report

The revised version includes a substantial addition to the paper. This goes a long way towards alleviating my previous concerns that the paper did not contain enough novel material. I am happy to now recommend publication.

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