Solvable Models of Magnetic Skyrmions
Bernd Schroers
Abstract We give a succinct summary of the recently discovered solvable models
of magnetic skyrmions in two dimensions, and of their general solutions. The
models contain the standard Heisenberg term, the most general translation invariant
Dzyaloshinskii–Moriya (DM) interaction term and, for each DM term, a particular
combination of anisotropy and Zeeman potentials. We argue that simple mathematical features of the explicit solutions help understand general qualitative properties
of magnetic skyrmion configurations in more generic models.
Keywords Magnetic skyrmions · Gauged sigma models
1 Introduction
Magnetic skyrmions are topological solitons in two-dimensional field theories for
the magnetisation field n of a magnetic material [1, 2]. In the continuum version, the
energy functional consists of the Dirichlet energy (quadratic in derivatives), a potential which includes anisotropy terms and a Zeeman contribution (no derivatives),
and the crucial Dzyaloshinskii–Moriya (DM) interaction term (linear in derivatives)
[3, 4]. Such an energy functional has stationary configurations which are stable
under Derrick scaling provided the DM term is negative for those configurations.
Magnetic skyrmions have been the subject of intensive experimental and numerical studies in recent years because they combine interesting physics with potential
technological applications in magnetic information storage and manipulation [5].
More recently, rigorous analytical studies have established conditions for the
existence of solutions as well as energy bounds in different topological sectors [6, 7],
and have clarified the interesting way in which the relative energy of skyrmions and
anti-skyrmions depends on the DM term [8].
B. Schroers ()
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt
University, Edinburgh, UK
e-mail: b.j.schroers@hw.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_50
535
Bernd Schroers
Abstract We give a succinct summary of the recently discovered solvable models
of magnetic skyrmions in two dimensions, and of their general solutions. The
models contain the standard Heisenberg term, the most general translation invariant
Dzyaloshinskii–Moriya (DM) interaction term and, for each DM term, a particular
combination of anisotropy and Zeeman potentials. We argue that simple mathematical features of the explicit solutions help understand general qualitative properties
of magnetic skyrmion configurations in more generic models.
Keywords Magnetic skyrmions · Gauged sigma models
1 Introduction
Magnetic skyrmions are topological solitons in two-dimensional field theories for
the magnetisation field n of a magnetic material [1, 2]. In the continuum version, the
energy functional consists of the Dirichlet energy (quadratic in derivatives), a potential which includes anisotropy terms and a Zeeman contribution (no derivatives),
and the crucial Dzyaloshinskii–Moriya (DM) interaction term (linear in derivatives)
[3, 4]. Such an energy functional has stationary configurations which are stable
under Derrick scaling provided the DM term is negative for those configurations.
Magnetic skyrmions have been the subject of intensive experimental and numerical studies in recent years because they combine interesting physics with potential
technological applications in magnetic information storage and manipulation [5].
More recently, rigorous analytical studies have established conditions for the
existence of solutions as well as energy bounds in different topological sectors [6, 7],
and have clarified the interesting way in which the relative energy of skyrmions and
anti-skyrmions depends on the DM term [8].
B. Schroers ()
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt
University, Edinburgh, UK
e-mail: b.j.schroers@hw.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_50
535