## SciPost Submission Page

# Gauged Sigma Models and Magnetic Skyrmions

### by Bernd J Schroers

### Submission summary

As Contributors: | Bernd Schroers |

Preprint link: | scipost_201905_00004v1 |

Date submitted: | 2019-05-23 |

Submitted by: | Schroers, Bernd |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | Mathematical Physics |

### 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 for a particular choice of the 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 point out applications to the study of impurities.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2019-6-11 Invited Report

### Strengths

The paper constructs a novel energy functional in which a gauge constraint ensures that all allowed configurations extremize the energy.

### Weaknesses

It's not clear what the applications of this energy functional are, beyond those pointed out in the previous paper.

The paper is written in a differential geometric language that does little to improve the rigor of the work but makes the paper difficult to read for the condensed matter audience.

### Report

The paper studies a class of solitons in field theories in two spatial dimensions. Among these solitons are magnetic Skyrmions, which are of some interest in condensed matter systems.

The paper writes down an energy functional for an SU(2) gauge field coupled to an adjoint scalar. The gauge field appears in the energy only algebraically and so its equation of motion gives rise to a constraint on the allowed field configurations. The energy functional has the peculiar property that all solutions of this constraint also obey the equations of motion of the scalar field. This means that all solutions to the constraint are automatically extrema of the action. The configurations can be labelled by a topological charge and this determines the energy.

This is an interesting observation. But it's not clear to me what can be done with it. The set of all field configurations that obey the constraint appear to be too large to be of interest. Instead, the author suggests that one should not view the gauge field as dynamical, but instead fix it to some particular configuration. With this interpretation, the constraint need not be imposed but can instead be interpreted as a Bogomoln'yi equation of a related field theory. This is in keeping with a previous paper by the author, listed as [1] in the bibliography, in which it was pointed out that, for a particular choice of this gauge field, the model reduces to a model for magnetic Skymions, tuned to a BPS limit.

This suggests that the model could be viewed as unifying system, in which different choices of gauge field give different theories. But no other interesting model is introduced in this way. There is a suggestion in the conclusions that this may be useful to study BPS impurities, but this is not pursued. A general solution to the constraint equation is given, but the main application seems to be a recapitulation of the results of [1].

In summary, there is an interesting observation in this paper. However, the implications of this observation remain unclear, and suggestions to find a novel application are not followed through. For this reason, I do not think that this paper contains enough material to warrant publication in SciPost.

I thank the referee for the careful report. However, I think it does not properly take into account that this is an interdisciplinary paper which does two things:

1. The paper provides a (new, as far as I know) geometrical framework which explains the results of paper [1] in terms of complex geometry. This allows for a vast generalisation of the results of [1], but, more importantly from the point of view of mathematical physics, provides a geometrical understanding of them: critically coupled magnetic skyrmions are holomorphic sections with respect to a connection defined by the DMI term! This is largely a mathematical advance, but experience shows that this kind of insight is also likely to be useful in applications.

2. The paper briefly illustrates the power of the new framework with applications to magnetic skyrmions and defects.

If feel that the referee judges the merit of the paper almost entirely on its usefulness, i.e. point 2. I do not think is is a fair assessment, but would be able to do my part for balancing the two aspects as follows. While I think that a full discussion of applications would best be done in a separate paper which emphasises the physics and downplayed the mathematics, I would be happy to include further comments on other applications (beyond the one of paper [1]) which are currently only sketched in equations 3.47 ff. In particular, this would include comments on in-plane magnets and bi-merons.