Elliptical stability of hopfions in bulk helimagnets

Magnetic hopfions are three-dimensional topological solitons with nonzero Hopf index <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="script">H</a:mi></a:math> in the vector field of material's local magnetization. Here elliptical stability of hopfions with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi mathvariant="script">H</c:mi><c:mo>=</c:mo><c:mn>1</c:mn></c:mrow></c:math> in a classical helimagnet is studied on the basis of a variational model. It is shown that, depending on their internal structure (vortex and antivortex tubes ordering), the hopfions can be either stable in a bulk magnet or unstable with respect to elongation along their central axis. It is found that the energy of stable hopfions is always below the energy of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>2</e:mn><e:mi>π</e:mi></e:mrow></e:math>-skyrmion lattice in the same material, suggesting the possibility to use <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mn>2</f:mn><f:mi>π</f:mi></f:mrow></f:math>-skyrmions as a precursor for hopfion nucleation. Stability diagram for hopfions on the magnetic anisotropy-field phase diagram is computed numerically. Explicit analytical expressions for some of its critical lines are derived.