Theory of elastic interaction between axially symmetric 3D skyrmions in confined chiral nematic liquid crystals and in skyrmion bags

ABSTRACT

We study axially symmetrical 3D skyrmionic particles (torons, Hopfions) in a thin homeotropic cell filled with a cholesteric liquid crystal. We show that a small 3D skyrmion asymptotically can be presented as a particle with six multipole moments. In this case, the odd moments – dipole, octupole and the fifth-order moments – are helicoidal moments, generated by chirality. And even moments – quadrupole, hexadecapole and sixth order moments – have the usual form, as colloidal particles in nematics. Thus, we find the exact analytical solution for the far field director configuration around the 3D skyrmion and the exact analytical expression for the elastic interaction potential between 3D skyrmions in a homeotropic cell with thickness $L$. We show that it has an exponential decay length equal ${\lambda }_{SS}=L/\pi$. The comparison with the experiment allows us to recover approximately one set of multipole coefficients for the small 3D skyrmion. We also describe the expansion of the skyrmion bag analytically depending on the number $N$ of anti-skyrmions in it and obtain an agreement with the experiment. Along the way, we determined the director field and the elastic interaction potential betweenchiral axially symmetric colloidal particles in the unlimited nematic and in the homeotropic cell, taking into account higher order elastic terms.

GRAPHICAL ABSTRACT

KEYWORDS:

• Skyrmion
• interaction of skyrmions
• chiral nematic
• chiral magnet
• hopfion
• topological soliton

Acknowledgments

One of the authors (SBC) used the help of Google Translator and Grammarly to write this article in English.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.