Abstract
This is the first of two chapters covering the mathematical and theoretical preliminaries one needs to tackle the skyrmion problems. In particular, the materials of this chapter comprise fundamental aspects of the modern theory of magnetism. Nowadays, there exists a mostly uniform treatment of these topics across the many condensed matter theory books available. Despite the risk of banality, I have tried to include a self-contained chapter on the Berry phase for spin problems so that readers need not look elsewhere to garner the requisite background.
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- 1.
We will not necessarily make a notational distinction between Lagrangian and its density; both are designated with L. More often than not, the spatial integral symbol will be missing in what I call the “Lagrangian”.
- 2.
It is an interesting coincidence that Witten’s 1983 paper basically dealt with the Berry’s phase term, or the topological term, in field theory. A year before Witten, the Thouless-Kohmoto-Nightingale-denNijs (TKNN) paper on the derivation of the topological number for the integer quantum Hall effect made essential use of the idea of geometric connection [4]. Berry’s own work, which came to stand for many (if not most) of geometric effects in condensed matter physics, was published in 1984 [2]. In retrospect, the early 80 s might go down as an era of the massive infusion of topological ideas in theoretical physics. Of course, Dirac, Aharonov, and Bohm had laid the ground a long time before.
- 3.
The word “coherent state” appears in many other contexts of quantum physics. Here, for clarity, we use the term “spin coherent state”.
- 4.
We revert to using \((\theta , \phi )\) to designate the angles of the unit vector.
- 5.
In the context of superfluid \(^3\)He this identity is known as the Mermin-Ho relation.
- 6.
Competing nomenclatures include Wess–Zumino–Witten action, and the Wess–Zumino–Novikov–Witten action. For convenience, we will simply call it the Wess–Zumino action.
- 7.
When not restricted to the surface of the sphere in three-dimensional space, the vector potential \(\mathbf{A} = - (\hat{r}_0 \times \hat{r} )/ ( r( 1- \hat{r}_0 \cdot \hat{r} )) \) yields a magnetic field of the form \(\varvec{\nabla }\times \mathbf{A} = \hat{r}/r^2\).
References
Landau, L.D., Lifshitz, E.M.: Theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowietunion 8, 153 (1935)
Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45 (1984)
Witten, E.: Global aspects of current algebra. Nuc. Phys. B 223, 422 (1983)
Thouless, D.J., Kohmoto, M., Nightingale, M.P., Den Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)
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Han, J.H. (2017). Geometric Phases. In: Skyrmions in Condensed Matter. Springer Tracts in Modern Physics, vol 278. Springer, Cham. https://doi.org/10.1007/978-3-319-69246-3_1
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DOI: https://doi.org/10.1007/978-3-319-69246-3_1
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