Field theory of collinear and noncollinear magnetic order

The lecture notes give the basics of the theory needed to study ferromagnets and antiferromagnets. The audience should be beginning doctoral students in this field, or it could also serve as a basis for part of a specialized graduate course. Since there is quite a significant number of students being trained in this area, it is very useful to have such lecture notes available.

The flow is smooth and reasonable steps are followed in going from one section (or subsection) to the next.
The general approach is to start with a discrete lattice and pass later to a continuum description. This is reasonable and also a helpful point of view for students.
Only a few examples of solutions of the equations are given, such as domain walls. This is sufficient given that these lecture notes are kept to a basic level.
The most advanced part is in Sec. 5, referring to a 3-sublattice antiferromagnet. The theory for this system is not easily accessible in other texts.

Remarks are following.
1. Eq. (48) gives m in terms of n, for antiferromagnets. This indicates that m=0 for all non-time dependent configurations, e.g., for domain walls. This contradicts various works. For example, in the recent paper H.T. Hirose, et al, Sci. Rep. 7:42440, 2016 (DOI: 10.1038/srep42440), significant net magnetization is observed for antiferromagnetic domain walls. References in this paper indicate that this phenomenon was clear already in previous experimental and theoretical works using models identical to the one explained in the present lecture notes.

2. p 16. The phrase "only use the paramagnetic energy in Eq. (66) in the Landau–Lifshitz equations and leaving out the gradient terms:" is not clear. It would need some reformulation and probably should become more explicit.

3. Eq. (50) and (81) are called in the text "general Landau-Lifshitz equation". The right hand side has the form of a Landau-Lifshitz equation but the left hand side does not. The explanation following Eqs. (81,82) gives a justification for the terminology. But, the behavior of Eq. (81) could be very different than the behavior (solutions, etc) given by the usual Landau-Lifshitz equation. If the author would like to call this a Landau-Lifshitz equation, there would need to be some more justification or explanation or clarification about the terminology.

4. After Eq. (13), a complex amplitude psi is used and the magnetization component mz is set to zero. It should be explained why it is consistent to set mz=0. Note also that, in the beginning of Sec. 3.2.1, it is assumed mz /= 0 for a similar calculation.

Here are some minor or technical remarks.
5. After Eq. (29), it is stated (correctly) that "they can be separated into 4 topological sectors". But, there is no previous explanation about topological sectors.
6. After Eq. (45), the expression "The ellipsis" is unclear.
7. Fields in Eq. (65) have been defined also in Eq. (62,63), although only for a 3-spin system. If possible, it would be better to avoid this repetition (or connect clearly the two definitions).

Typos:
"with broken inversion", "a similar permutations", "tou", "sublattce"

I recommend these lecture notes for publication in SciPost as they are well-written and they will be useful to an audience of graduate students.