Energy-momentum tensor of a ferromagnet

TL;DR

This paper identifies gauge-dependence issues in the traditional energy-momentum tensor of ferromagnets and proposes a gauge-invariant tensor derived from the Wess-Zumino action, involving an extra spatial dimension.

Contribution

It introduces a gauge-invariant energy-momentum tensor for ferromagnets using the Wess-Zumino action, resolving unphysical artifacts present in previous formulations.

Findings

Traditional tensor is gauge-dependent and unphysical.

The new tensor is gauge-invariant and physically consistent.

Requires an extra spatial dimension for its formulation.

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 gauge-dependent. 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.