What is the dimension of the domain wall? Related Post:

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*Domain walls intersection


*How is domain wall formation related to spontaneous symmetry breaking?


*Wikipedia Domain wall (magnetism)
One encountered the concept of domain wall. It was interpreted as something that broke the symmetry, such as the case in the reference 1.
Consider the case where a symmetry $g$ who's representation was of dimension $D$, it seemed that in the case such as 1. the typical dimension of such domain wall should be less than $D$, but it was not specified, where in reference 2 the domain wall was of $D-1$ (if consider a clear cut) a boundary region of the space and exactly what  "a domain wall" sounded. (A joke: sounded was good.) However, a picture in the reference 3(Schematic representation of domain wall unpinning) represented the break of the symmetry by a ?topological operator? which was of dimension 1. Thus, there seemed to be more representation of how a domain was represented and in different dimensions.
If one was to consider more general cases not restricted to the reference 1, the domain wall does not have to be local with respect to the symmetry operator, i.e. if viewing the local translation generator $g$ in the direction of the circle theory as a symmetry (For simplicity $(S^1)$ every dimension), one could expand the the parameter $\theta$ of the Wilson line not only dependent on $X^\mu$ but also perhaps some theory or external free parameters $\theta(X^\mu,\gamma)$, and this would be a $D+1$ dimension domain wall for the $g$ since it was encountered by performing the integral over the circle and with a free parameter.
What is the dimension of the domain wall?
 A: I  will not give a canonical answer here since it seems of what you are asking is more related to a physical interpretation.
To answer this is actually linked with the question why a single dipole magnet has no such transition domain at the equator? The answer to this is that there is no abrupt change in the curl of the field at the equator of the magnet which magnetic flux lines are more or less parallel to its magnetic moment axis N-S.
However, in the case  of interacting chunks of matter (magnetic dipole domains) consisting a ferromagnetic bulk material, these microscopic dipole magnets have random orientation (i.e. assuming the material is not magnetized) inside the bulk materiel of their magnetic moment vectors  and their dipole magnetic fields are "touching" each other thus compressed against each other by the molecular forces keeping the condensed matter of its bulk together.
Ultimately, the thickness of the domain walls between the magnetic domains (chunks of ferromagnetic matter) in bulk assuming a uniform ferromagnetic material, should be a function of its magnetic coercivity value and its molecular structure (bonding forces) and molecular density.
Expect, the higher the material's  magnetic coercivity value, molecular bonds strength and molecular density are, the higher the compression between its magnetic domains to be and therefore the thinner the formed domain walls.
Nevertheless, the magnetic flux lines density is higher inside these domain wall the higher the compression (i.e thinner domain walls) in that region is.
Here for example is the real time flux imprint of the magnetic domains forming between dipole magnets in a magnetic array emulating the magnetic domains in a bulk ferromagnetic material as shown by a ferrolens. The black circles are the poles of the individual magnets and the white formed lines among them are the Bloch domain walls that consist of high density flux regions where the lines are running almost in parallel to each other and straight without significant curl. The gray spot area  at the center shown are magnetic cancellation areas (i.e. dead zones B=0 caused by like poles fields facing each other around a center area in space) as shown by the ferrolens:

We get similar imprint from a gel based magnetic field viewing film but without discrete magnetic flux lines display to be possible under magnification:

For a more canonical answer about the quantitative calculation of the domain wall width, I recommend this paper here and also this book reference:
"A. Hubert, R. Schäfer, Magnetic domains: the analysis of magnetic microstructures, Springer 1998".
