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Do the magnetic hopfions have tails?

Konstantin L. Metlov, Maksim M. Gordei

Abstract

Magnetic hopfions in chiral magnets are topological solitons, localized in three dimensions. But is their localization strong? To address this question we derive an asymptotic expansion for the isolated hopfion's spatial profile. It becomes starting point for a simple analytical model, which is asymptotically correct both near the hopfion center and far away from it. Region of equilibrium hopfions on the phase diagram of a helimagnet is computed and material requirements for supporting movable isolated magnetic hopfions on uniform background are discussed.

Do the magnetic hopfions have tails?

Abstract

Magnetic hopfions in chiral magnets are topological solitons, localized in three dimensions. But is their localization strong? To address this question we derive an asymptotic expansion for the isolated hopfion's spatial profile. It becomes starting point for a simple analytical model, which is asymptotically correct both near the hopfion center and far away from it. Region of equilibrium hopfions on the phase diagram of a helimagnet is computed and material requirements for supporting movable isolated magnetic hopfions on uniform background are discussed.
Paper Structure (6 sections, 18 equations, 2 figures)

This paper contains 6 sections, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hopfion energy
  3. Hopfion tails
  4. Isolated hopfion stability
  5. Summary
  6. Energy functions

Figures (2)

  • Figure 1: (left) Cross section of the equilibrium hopfion \ref{['eq:ansatz']} by the $X=0$ plane through its central $OZ$ axis; (top right) the energy of a uniform magnetization background at an angle $\theta$ to the anisotropy axis, inset shows the saturation-to-saturation hysteresis loop, where dot marks the field, experienced by the hopfion on the left; (bottom right) equilibrium hopfion profile functions from the Eq. \ref{['eq:trialE']} (solid) and numerically computed (dotted). The dimensionless magnetic field $h$ and the anisotropy quality factor $q$ are defined in the text.
  • Figure 2: Phase diagram of a classical helimagnet (with conical, helical, uniform and skyrmion phases) shown in the background; in the shaded region the isolated hopfions on uniform background are energetically favorable; dots show the parameters for which the hopfion profiles were computed numerically and shown in Fig. \ref{['fig:hopfion']} lower right; the inset shows the dependence of the equilibrium hopfion radius along a slice of its stability region.