Spin-transfer torque, stress-energy tensor, and Magnus force*

In micromagnetics, equations of motion for the magnetic moments m(r,t) are given by the Landau Lifshitz Gilbert equation, which can be equivalently derived from a Lagrangian. The Lagrangian for ferro and antiferromagnets contains terms with a spatial derivative of the order parameter in the presence of a spin carrying electric current, (u●▽)n. Here u is the electron drift velocity proportional to the current density j. In terms of collective coordinates (q) this term appears as a vector potential coupling to u, A(q)●u. It is free of time derivatives and thus appears to be a form of potential energy. But it is explicitly gauge dependent and should not directly represent a physical quantity like a potential energy density. Treating it like one yields incorrect generalized forces and a gauge dependent stress energy tensor. Motivated by this anomaly we outline the procedure to correctly define the energy density and energy flow caused by the electric current interacting with the soliton and provide a conjectural form of a gauge independent stress energy tensor. We illustrate with examples of a 1D ferromagnetic wire and an antiferromagnetic vortex.