Actullaly, there is no universal theory capable to describe accurately the structure and energetics of all skyrmions. The main barriers are the complexity of non-local stray field interactions, the reduced size of such structure and the boundaries conditions. Howover, the most adopted approach is based on maicrmagnetics which assumes :

1) a fixed magnetization modulus m = M/Ms where |m|=1

2) a slow variations at the atomic scale permitting to adopt a continuous model m = m(r, t)

The problem then consists in finding the spatial orientation of the magnetization, which is given by the magnetization direction vector  m = M/Ms by minimising the functional energy.

No clue.

Maybe I should clarify my question a bit. There is no problem when we are considering magnetization vector M in continuous body, even if such a body is inherently anisotropic. M is then well defined at its every point.

By definition, M is the spatial density of magnetic moments. This is why it becomes suspected at nanoscale. My feeling is that asking about M vector at the point between two neighboring spins is like asking about gas pressure somewhere between two close particles of ideal gas. Consequently, the vector M should never appear in micromagnetic calculations, at least in those dealing directly with individual spins. But we are often doing this kind of calculations under the name of 'finite elements method ' and we tend to use as small elements as our computing machinery is able to carry (just to get most 'precise results'). Where is the safe border between the two approaches? Which one of them, and when, should be trusted?

Dear Marek Wojciech Gutowski

>By definition, M is the spatial density of magnetic moments.

Correct. Physical unit for M is the Ampere per meter, which can be thought of as Ampere meter squared per meter cubed, making the meaning more obvious.

>This is why it becomes suspicious at nanoscale.

I don't agree. The nanoscale is large enough for the continuum approximation of the exchange energy to be valid. It definitely loses meaning at scales smaller than nano.

>My feeling is that asking about M vector at the point between two neighboring spins is like asking about gas pressure somewhere between two close particles of ideal gas.

A valid point, however this only occurs at smaller scales.

>Consequently, the vector M should never appear in micromagnetic calculations, at least in those dealing directly with individual spins.

Micromagnetics, understood as the continuum theory developed by WF Brown Jr. does not deal with individual actual spins. Instead, the magnetization vector field in micromagnetics represents at every point the integral of spin magnetic moments over a "suitable small volume". Since the spin magnetic moment directions do not change drastically from one lattice site to the other due to ferromagnetic coupling, the integral over any of the "suitable small volumes" is a constant (in magnitude) called the saturation magnetization, with units ampere per meter. This explains the constant magnitude of the magnetization vector at any spatial point in micromagnetics.

>But we are often doing this kind of calculations under the name of 'finite elements method ' and we tend to use as small elements as our computing machinery is able to carry (just to get most 'precise results'). Where is the safe border between the two approaches?

As a continuum theory it is perfectly valid to use a discretized method like FEM for the micromagnetic equations. However, as far as I know, most researchers do not try to use elements as small as possible. There is an intrinsic length, called the exchange length (dependent on a material's magnetic parameters) at which we can expect "important" changes in the direction of the magnetization vector. It is unnecessary to have elements (in a FEM mesh) smaller than this length and, as you mention, at some point ever finer discretizations will make the model meaningless as our "suitable small volumes" become too small to contain an appropriate number of lattice sites.

I hope this clarifies what M means and its use in micromagnetics. As for skyrmions, I'm no expert but I don't see why the magnetic field should be singular? The magnetization vector, as I see it, should be well defined everywhere

Dear Miguel,

Thank you for the very clear description of micromagnetism as it was proposed by W.F. Brown Jr. Especially important is well justified using exchange length as a lowest sensible size of finite element being used in this kind of calculations. This size is not known precisely but certainly the volume under study is roughly equal to elementary cell (perhaps slightly bigger), at least in crystalline materials. The vector M makes sense. Unfortunately, in ~1960, when this procedure was created, skyrmions (and antiskyrmions as well!) were not known and the procedure was very satisfactory. But today we have a problem, at least I have. Even if there is no singularity of magnetic field around the skyrmion then the field is certainly not approximately constant as assumed in micromagnetic calculations. Now individual spins making skyrmion 'feel' the magnetic induction created by other spins (and orient themselves in space accordingly, at least in classical, non-quantum picture). I don't see where is the place for vector M.

Magnetization is simply the alignment of rotating dipoles along their mutual rotation axes. It can happen on any scale. If the dipoles are simple, the electrostatic attraction between positive and negative results in a double helix around the mutual rotation axis. This would explain magnetic attraction in terms of Coulomb's law if space is dielectric.

Article The Double Helix and the Electron-Positron Aether

My problem does not exist during micromagnetic calculations. But what about the atomistic point of view? What is the M vector at position of a single (not necessarily magnetic) atom/ion? Is it the sum of dipolar magnetic fields generated by its magnetic neighbors or we should consider somehow exchange interactions too?

Normally 'atomistic' (some people do not like this term as it's also commonly used to refer to molecular dynamics models in Chemistry) spin dynamics simulations are an implementation of the classical Heisenberg ferromagnet which undergoes evolution by the discrete Landau-Lifshitz-Gilbert equation. Here you assume that electrons are localised around each atom, and each atom is a point dipole made up of the total of the electronic contributions. It's well known, however that this is not a very good approximation. From what I remember the book 'Simple Models of Magnetism' by Ralph Skomski [1] discusses this quite nicely. There's also a review article by Ulrik Nowak which is quite nice [2] in the practical details for writing an atomistic spin dynamics simulation.

The biggest issue with doing numerical studies of real materials like this is that parameterising models for real materials is very difficult, both for parameterising the models and in terms of computational time.

It's worth noting that many studies come from a theoretical angle rather than from a drive to study a particular material, because they are motivated by looking at universality of models, and studying for e.g. the critical behaviour of such systems.

[1] Article Simple Models of Magnetism

[2]Chapter Classical Spin Models