Pions versus Magnons: From QCD to Antiferromagnets and Quantum Hall Ferromagnets

The low-energy dynamics of pions and magnons -- the Goldstone bosons of the strong interactions and of magnetism -- are analogous in many ways. The electroweak interactions of pions result from gauging an SU(2)_L x U(1)_Y symmetry which then breaks to the U(1)_{em} gauge symmetry of electromagnetism. The electromagnetic interactions of magnons arise from gauging not only U(1)_{em} but also the SU(2)_s spin rotational symmetry, with the electromagnetic fields E and B appearing as non-Abelian vector potentials. Pions couple to electromagnetism through a Goldstone-Wilczek current that represents the baryon number of Skyrmions and gives rise to the decay \pi^0 to \gamma \gamma. Similarly, magnons may couple to an analogue of the Goldstone-Wilczek current for baby-Skyrmions which induces a magnon-two-photon vertex. The corresponding analogue of photon-axion conversion is photon-magnon conversion in an external magnetic field. The baryon number violating decay of Skyrmions can be catalyzed by a magnetic monopole via the Callan-Rubakov effect. Similarly, baby-Skyrmion decay can be catalyzed by a charged wire. For more than two flavors, the Wess-Zumino-Witten term enters the low-energy pion theory with a quantized prefactor N_c -- the number of quark colors. The magnon analogue of this prefactor is the anyon statistics angle \theta which need not be quantized.