## Giving Faraday's Law a Spin

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All of the fundamental forces of nature, Strong, Electromagnetic, Weak, with the possible exception of Gravity, correspond to local gauge theories. Everyday non-fundamental forces, such as the elastic forces involved when a ball bounces off the ground, are not obviously related to these four forces although clearly they derive from the electromagnetic forces between electrons. Such molecular forces can be determined within Born-Oppenheimer (1927) adiabatic approximation that separates the fast fermion charge degrees of freedom from the slow motion of the atoms. In terms of Berry phase (1984) physics, derived is an expression for a Berry connection, i.e., an effective four vector potential A_mu from which the molecular forces can be determined by differentiation (see e.g. Resta, 2000).

Conducting magnets add an extra dimension. Now an electrical change current leads to generalized forces, in particular torques, acting upon the slow magnetic degrees of freedom, but in addition there are spin forces acting upon the conduction electrons. This leads to a spin-motive force (SMF), that is a non-conservative force acting upon the conduction electron that reflects the requirements of the conservation of both energy and linear momentum. This non-fundamental motive force, although it does cause electrical currents to flow in an external circuit, is not an electro-motive force (EMF) since it has a different sign when acting upon the majority and minority electrons. Following the Feynman's (1948, see Dyson 1990) prescription there exist spin electric E_s and magnetic B_s fields which obey a generalized Faraday's law. Historically spin dependent forces in ferromagnets first appeared in Korenman et al. (1977) followed a decade later by the oft cited Volovik (1987) however their reduction of the gauge fields A_mu from SU(2) to U(1) throws away the SMF to leave only the conservative Stern-Gerlach force. Andy Stern (1992) erroneously christened as an SMF the same force. The same truncation to U(1) plagues most works in this cottage industry of ``emergent electromagnetism", e.g., for the Skyrmion lattice.

Barnes and Maekawa (2007) where the first to show the existence of both the conservative and non-conservative, i.e., SMF, forces and predicted an SMF of 100 microV/T for a moving magnetic domain wall driven by a magnetic field. This was soon verified by G. Beach (Yang et al., 2009) and a proof of principle ``spin-battery" was demonstrated by Hai et al (2009). The experimental implications of the SMF in spintronics, e.g., for MARM, and its real world potential for energy storage and recovery will be discussed as time permits.

Max Born and J. Robert Oppenheimer "Zur Quantentheorie der Molekeln" Annalen der Physik 389 457-484 (1927).

Berry, M. V. "Quantal Phase Factors Accompanying Adiabatic Changes". Proceedings of the Royal Society A 392 45-57 (1984).

Resta, Raffaele "Manifestations of Berry's phase in molecules and in condensed matter". J. Phys.: Condens. Matter 12 R107 (2000).

F. J. Dyson, "Feynman's Proof of the Maxwell Equations" Am. J. Phys 58, 209-211 (1990)

V. Korenman, J. L. Murray, and R. E. Prange, "Local-band theory of itinerant ferromagnetism. I. Fermi-liquid theory" Phys. Rev. B16 4032 (1977)

G. E. Volovik, "Linear momentum in ferromagnets", J. Phys. C 20, L83 (1987)

A. Stern, "Berry's Phase, Motive Forces, and Mesoscopic Conductivity", Phys. Rev. Lett., 68, 1022 (1992)

S. E. Barnes and S. Maekawa, "Generalization of Faraday's Law to Include Nonconservative Spin Forces", Phys. Rev. Lett. 98, 246601 (2007).

S. A. Yang, G. S. D. Beach, C. Knutson et al., "Universal electromotive force induced by domain wall motion", Phys. Rev. Lett. 102, 067201 (2009)

Hai, P. N., Ohya, S., Tanaka, M., Barnes, S. E. & Maekawa, S. "Electromotive force and huge magnetoresistance in magnetic tunnel junctions." Nature 458, 489-492 (2009)