Controlling knot topology in magnetic hopfions via spin-orbit torque

Knots formed by the intertwining of strings have attracted broad interest across various scientific disciplines owing to their rich topology. This concept has recently gained increasing importance in condensed matter physics, as exemplified by a magnetic hopfion labeled by a topological invariant called the Hopf number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>H</a:mi> </a:math> . Here, we show that spin-orbit torque (SOT) enables dynamic manipulation of the Hopf number of magnetic hopfions. We investigate the SOT-driven evolution of hopfions, revealing the splitting of a high- <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>H</b:mi> </b:math> hopfion into multiple lower- <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>H</c:mi> </c:math> ones, a process that can be quantified by an effective tension picture. Comparative analysis across different <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>H</d:mi> </d:math> uncovers a hierarchy of instabilities that dictates these dynamical topological transitions. These findings not only indicate potential applications of hopfions in SOT-driven multilevel memory devices, but also provide a paradigm for the dynamical control of knot topology.