[Paper Review] Solvable Models of Magnetic Skyrmions

Bernd J. Schroers|arXiv (Cornell University)|Oct 30, 2019

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper presents solvable models of magnetic skyrmions by mapping the magnetic skyrmion problem to a gauged nonlinear sigma model with a fixed SU(2) background gauge field, where exact solutions are constructed from holomorphic functions on the Riemann sphere. The key result is that for each Dzyaloshinskii-Moriya (DM) interaction term, a specific combination of anisotropy and Zeeman potential yields explicit solutions, revealing that skyrmion charge and energy depend on the asymptotic behavior of the holomorphic data, with anti-skyrmions and skyrmions having equal energy in rank-one DM models.

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.

Motivation & Objective

• To construct analytically solvable models of magnetic skyrmions that capture key topological and energetic features of generic skyrmion configurations.
• To clarify the role of the Dzyaloshinskii-Moriya (DM) interaction in determining the relative energy of skyrmions and anti-skyrmions.
• To establish a connection between magnetic skyrmions and integrable gauged sigma models through a precise mapping of the DM term and potential to an SU(2) gauge field.
• To demonstrate that exact solutions can be generated from arbitrary holomorphic maps, enabling explicit construction of multi-skyrmion and anti-skyrmion configurations.

Proposed method

• The magnetic skyrmion energy functional is reformulated as the energy of a gauged nonlinear sigma model with a fixed SU(2) background gauge field, where the gauge field is derived from the spiralization tensor D.
• The DM interaction term is expressed as a covariant derivative term involving the gauge field, and the potential is chosen as $ V_A(\boldsymbol{n}) = \frac{1}{2}|\boldsymbol{A}_1 \times \boldsymbol{n}|^2 + \frac{1}{2}|\boldsymbol{A}_2 \times \boldsymbol{n}|^2 - (\boldsymbol{n}, \boldsymbol{A}_1 \times \boldsymbol{A}_2) $, which ensures the model is solvable.
• Solutions are constructed via a holomorphic function $ h(z) $, with the magnetization field $ \boldsymbol{n} $ determined implicitly through a transformation involving the inverse coordinate $ v = 1/w $, leading to the equation $ v = -\frac{i}{2}\kappa\bar{z} + h $.
• The topological charge $ Q $ is computed from the asymptotic behavior of $ h(z) $, with $ Q = M $ if $ L > 1 $, $ Q = N $ if $ L = 1 $, and $ Q = N-1 $ if $ L < 1 $, where $ L = \lim_{|z|\to\infty} |2h/(\kappa\bar{z})| $ and $ h = p(z)/q(z) $ with degrees $ M $ and $ N $.
• The energy of the solutions is shown to be $ 4\pi Q $, and the Bogomol’nyi bound is saturated, indicating the solutions are BPS states.
• The model is generalized to both rank-one and generic DM interactions, with the rank-one case corresponding to a flat gauge field and thus mapping to the standard Belavin-Polyakov model.

Experimental results

Research questions

• RQ1 How can the magnetic skyrmion problem be reformulated as a solvable gauged nonlinear sigma model with exact solutions?
• RQ2 What specific choice of potential ensures integrability for a given Dzyaloshinskii-Moriya interaction term?
• RQ3 How does the topological charge of a skyrmion configuration depend on the holomorphic data used in the solution construction?
• RQ4 What determines the relative energy of skyrmions and anti-skyrmions in models with different DM interaction structures?
• RQ5 Can the solvable models reveal general qualitative features of multi-skyrmion and anti-skyrmion configurations seen in numerical simulations?

Key findings

• For each DM interaction term, a specific potential $ V_A(\boldsymbol{n}) $ ensures the model is solvable and the energy functional matches that of a gauged sigma model with a fixed SU(2) gauge field.
• Solutions are explicitly constructed from arbitrary holomorphic functions $ h(z) $, with the magnetization field determined via a transformation involving the inverse coordinate $ v = 1/w $, leading to $ v = -\frac{i}{2}\kappa\bar{z} + h $.
• The topological charge $ Q $ of a solution is determined by the asymptotic behavior of $ h(z) $: $ Q = M $ if $ L > 1 $, $ Q = N $ if $ L = 1 $, and $ Q = N-1 $ if $ L < 1 $, where $ M $ and $ N $ are the degrees of the numerator and denominator of $ h(z) = p(z)/q(z) $.
• The energy of all solutions is $ 4\pi Q $, and the Bogomol’nyi bound is saturated, confirming that the solutions are BPS states.
• In rank-one DM models (where $ \boldsymbol{A}_1 \times \boldsymbol{A}_2 = 0 $), skyrmions and anti-skyrmions of opposite charge have equal energy $ 4\pi |Q| $, in contrast to generic models where the energy depends on the sign of $ Q $.
• The models reproduce complex configurations such as multi-anti-skyrmions, bags, and line defects, and explain features like $ Q+1 $ maxima in the energy density for charge $ Q > 0 $ configurations.