Toroidal Order in Magnetic Metamaterials

Chiral Magnetic Phases in Moire Bilayers of Magnetic Dipoles

“…[ 20,32 ] The classical toroidal moment is defined as $$$$\begin{equation} \mathbf {t}=\sum _i r_i\times m_i \end{equation}$$$$where r i and m i are the position and magnetic moment of the i‐th dipole respect to the center of the magnetic unit cell. [ 21,35 ] Since r i and m i are in the plane of the lattice, t can point along $$\pm \hat{z}$$. It is a pure real‐space quantity that is related to the helicity $${\mathbf {t}} \propto sin(\gamma)\hat{z}$$.…”

“…where r i and m i are the position and magnetic moment of the i-th dipole respect to the center of the magnetic unit cell. [21,35] Since r i and m i are in the plane of the lattice, t can point along ±ẑ. It is a pure real-space quantity that is related to the helicity t ∝ sin(𝛾)ẑ.…”

“…Recent experiments have shown that permalloy nanoislands interacting via dipolar coupling manifest the ferrotoroidic phase. In this experiment, local control of the ferrotoroidicity has been achieved by scanning a magnetic tip, thus widening the possibilities of control of chiral phases [ 35 ] in artificial dipolar lattices.…”