Energy-momentum tensor of a ferromagnet

Abstract

The energy-momentum tensor of a ferromagnet derived according to the standard prescription of Noether's theorem has a major flaw: the term originating from the spin Berry phase is gaugedependent. As a consequence, some physical quantities computed from the tensor show unphysical behavior. For example, the presence of a spin-polarized current does not affect the energy of the domain wall in the commonly accepted gauge, which implies-incorrectly-the absence of the adiabatic spin torque. In other gauges, the spin torque shows unphysical glitches occurring when the plane of magnetization crosses the Dirac string associated with a magnetic monopole in spin space. We derive a gauge-invariant energy-momentum tensor that is free from these artifacts but requires the addition of an extra spatial dimension, with the ferromagnet living on its boundary. It can be obtained most directly from the Wess-Zumino action for spins, which relies on the same extra dimension.

Chiral magnetism: a geometric perspective

“…Although the continuity equation ( 16) follows directly from Noether's theorem, its standard field-theoretic derivation is complicated by a gauge dependence of the Berry-phase term in the action of a ferromagnet [19,20]. Instead, we can be obtain it from the Landau-Lifshitz equation of motion for the magnetization field,…”

“…The advantage of the spin-current conservation in its gauged form (20) is that the dominant source of anisotropy-the DM interaction-is included in the definition of spin current. The spin current is automatically conserved to the first order in the relativistic spin-orbit coupling.…”

Conserved momenta of ferromagnetic solitons through the prism of differential geometry

“…By extension, any two homotopic 1 paths with the same endpoints will yield the same conserved momentum (8). However, two non-homotopic paths may give different answers for the conserved momentum [8].…”

“…We mentioned in the introduction that a conserved momentum P a (8) coincides with the canonical momentum p a (7) in cases when the vector potential A i is symmetric under translations of coordinate q a with the generator ∂ a . This argument applies more broadly to any Cartan symmetry generated by a vector field…”

“…Through the mechanism of adiabatic spin transfer, conduction electrons passing through an area of inhomogeneous magnetization exchange angular momentum with the ferromagnet, thereby applying what's known as the adiabatic spin-transfer torque to the magnetization [18,19]. As one of us discussed elsewhere [8], the adiabatic spin-transfer torque can be viewed as an instance of the gyroscopic force if we expand the dynamical system to include the electric charge Q of the injected current I = Q. The current creates a gyroscopic force F αQ Q for the α coordinate, with the gyroscopic coefficient F αQ = Pħ h/e, where e is the electron charge and P is the degree of spin polarization in the current.…”