Solvable Models of Magnetic Skyrmions
543
4 Conclusion
We have shown that solvable models of magnetic skyrmions exist for any DM
interaction term. Even though they require fine-tuning of the potential, their exact
solutions shed light on qualitative properties of solutions in more general models.
These include general features of multi-(anti)-skyrmion configurations such as the
appearance of Q + 1 maxima in the energy density of certain charge Q > 0
configurations (as shown in the Q = 5 solution in Fig. 1), or the deformation of
a skyrmion to an anti-skyrmion via a line defect, as shown in the top row of Fig. 1
and discussed in some detail in [9].
The solvable models also shed light on the crucial influence of the DM
interaction on the relative energy of skyrmions compared to anti-skyrmions. Our
short discussion illustrates the more general findings of [8]. In the axisymmetric
models of Sect. 3.2, Q = −1 skyrmions have energy −4π , whereas Q = 1 antiskyrmions have the opposite energy 4π (this can be reversed by a different choice
of solvable model, see [9]). In rank one models, by contrast, skyrmions and antiskyrmions have the same energy. It was shown in [8] that models with generic
spiralization tensors should interpolate between these two extremes, and it would
be interesting to explore this in the solvable models with generic DM terms.
To end, we note that the language of gauged non-linear sigma models provides
a rare and rather beautiful link between pure mathematics and real physics by
connecting the geometry of holomorphic maps and vortex equations as discussed
in [14] with magnetic skyrmions. In fact, simply allowing the gauge field to depend
non-trivially on space may provide further applications, for example, to the study of
impurities as discussed in [18] and [10].
Acknowledgments I thank Bruno Barton-Singer for sharing his Python code for generating the
plots in Fig. 1, Calum Ross for pointing out an error in an earlier version of this paper, and the
referee for constructive comments.
References
1. A.N. Bogdanov, D.A. Yablonskii, Thermodynamically stable ‘vortices’ in magnetically
ordered crystals. The mixed state of magnets. Zh. Eksp. Teor. Fiz. 95, 178 (1989)
2. N. Nagaosa, Y. Tokura, Topological properties and dynamics of magnetic skyrmions. Nat.
Nanotechnol. 8, 899 (2013). https://doi.org/10.1038/NNANO.2013.243
3. I. Dzyaloshinskii, A thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics.
J. Phys. Chem. Solids 4, 241 (1958). https://doi.org/10.1016/0022-3697(58)90076-3
4. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120,
91 (1960). https://doi.org/10.1103/PhysRev.120.91
5. A. Fert, V. Cros, J. Sampaio, Skyrmions on the track. Nat. Nanotechnol. 8, 152 (2013).
https://doi.org/10.1038/nnano.2013.29
6. C. Melcher, Chiral skyrmions in the plane. Proc. R. Soc. A 470, 20140394 (2014). https://doi.
org/10.1098/rspa.2014.0394
543
4 Conclusion
We have shown that solvable models of magnetic skyrmions exist for any DM
interaction term. Even though they require fine-tuning of the potential, their exact
solutions shed light on qualitative properties of solutions in more general models.
These include general features of multi-(anti)-skyrmion configurations such as the
appearance of Q + 1 maxima in the energy density of certain charge Q > 0
configurations (as shown in the Q = 5 solution in Fig. 1), or the deformation of
a skyrmion to an anti-skyrmion via a line defect, as shown in the top row of Fig. 1
and discussed in some detail in [9].
The solvable models also shed light on the crucial influence of the DM
interaction on the relative energy of skyrmions compared to anti-skyrmions. Our
short discussion illustrates the more general findings of [8]. In the axisymmetric
models of Sect. 3.2, Q = −1 skyrmions have energy −4π , whereas Q = 1 antiskyrmions have the opposite energy 4π (this can be reversed by a different choice
of solvable model, see [9]). In rank one models, by contrast, skyrmions and antiskyrmions have the same energy. It was shown in [8] that models with generic
spiralization tensors should interpolate between these two extremes, and it would
be interesting to explore this in the solvable models with generic DM terms.
To end, we note that the language of gauged non-linear sigma models provides
a rare and rather beautiful link between pure mathematics and real physics by
connecting the geometry of holomorphic maps and vortex equations as discussed
in [14] with magnetic skyrmions. In fact, simply allowing the gauge field to depend
non-trivially on space may provide further applications, for example, to the study of
impurities as discussed in [18] and [10].
Acknowledgments I thank Bruno Barton-Singer for sharing his Python code for generating the
plots in Fig. 1, Calum Ross for pointing out an error in an earlier version of this paper, and the
referee for constructive comments.
References
1. A.N. Bogdanov, D.A. Yablonskii, Thermodynamically stable ‘vortices’ in magnetically
ordered crystals. The mixed state of magnets. Zh. Eksp. Teor. Fiz. 95, 178 (1989)
2. N. Nagaosa, Y. Tokura, Topological properties and dynamics of magnetic skyrmions. Nat.
Nanotechnol. 8, 899 (2013). https://doi.org/10.1038/NNANO.2013.243
3. I. Dzyaloshinskii, A thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics.
J. Phys. Chem. Solids 4, 241 (1958). https://doi.org/10.1016/0022-3697(58)90076-3
4. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120,
91 (1960). https://doi.org/10.1103/PhysRev.120.91
5. A. Fert, V. Cros, J. Sampaio, Skyrmions on the track. Nat. Nanotechnol. 8, 152 (2013).
https://doi.org/10.1038/nnano.2013.29
6. C. Melcher, Chiral skyrmions in the plane. Proc. R. Soc. A 470, 20140394 (2014). https://doi.
org/10.1098/rspa.2014.0394