Chiral magnetism: a geometric perspective

Abstract

We discuss a geometric perspective on chiral ferromagnetism. Much like gravity becomes the effect of spacetime curvature in theory of relativity, the Dzyaloshinski-Moriya interaction arises in a Heisenberg model with nontrivial spin parallel transport. The Dzyaloshinskii-Moriya vectors serve as a background SO(3) gauge field. In 2 spatial dimensions, the model is partly solvable when an applied magnetic field matches the gauge curvature. At this special point, solutions to the Bogomolny equation are exact excited states of the model. We construct a variational ground state in the form of a skyrmion crystal and confirm its viability by Monte Carlo simulations. The geometric perspective offers insights into important problems in magnetism, e.g., conservation of spin current in the presence of chiral interactions.

Superconducting Triplet Rim Currents in a Spin-Textured Ferromagnetic Disk

“…In terms of f it then becomes J j γ ∝ | f SRT | 2 ( m × ∇ j m ) γ (here γ = x , y , z is the index in spin space and j = x , y , z is the index in coordinate space). A ferromagnetic vortex texture yields J j z ∝ | f SRT | 2 ∇ i θ v = | f SRT | 2 m j / r v which is in accordance with the general gauge-invariant expression for the spin current …”

“…A ferromagnetic vortex texture yields J j z ∝ | f SRT | 2 ∇ i θ v = | f SRT | 2 m j / r v which is in accordance with the general gauge-invariant expression for the spin current. 58 …”

“…which is in accordance with the general gauge-invariant expression for the spin current. 58 Mechanism for Generating LRT Rim Currents. Having established the equivalency between spin texture and SOC, we now provide a possible mechanism that relates the ESC to the emergence of LRT rim currents (see Supporting Information Figure S7 for a schematic representation).…”

Chiral magnetism: a geometric perspective

“…This is not a serious problem. In the thermodynamic limit, the bulk energy scales with the area of our two-dimensional sample, whereas the boundary energy scales with its perimeter and can therefore be neglected for large systems [30]. Therefore, if we could construct a Bogomolny state with a finite density of antiskyrmions, in the thermodynamic limit the energy density would be −4π per unit cell.…”

“…Ansatz (16) can be readily extended to other regular lattices, allowing a straightforward way to compute the elastic properties of the antiskyrmion crystal and to determine the spectrum of low-frequency spin waves [31]. These results and further details will be described in a forthcoming paper [30].…”