Field theories of micromagnetism in XY ferromagnet and antiferromagnet
Stewart Blusson Quantum Matter Institute, University of British Columbia
Abstract. — Micromagnetic field theories effectively capture the long-range static structures and dynamics of ordered spin systems at temperatures below their ordering temperatures. The field theory, if expressed in the correct form, further elucidates hidden features in the order. We discuss two such instances. First, we take a look at the 2+1D XY ferromagnet whose continuum field theory has been extensively studied in the context of the Kosterlitz-Thouless phase transition . We show how the field theory describes an interpolation of the quantum statistics of a magnetic vortex–from bosonic to fermionic–using a duality map to 2+1D electromagnetism.
Second, we examine the field theory of a generic 3-sublattice antiferromagnet in 2D, exemplified by the Heisenberg model on the triangular  and kagome  lattices. In a ground state, spins from the 3 sublattices are coplanar and at angles of 120° to one another such that S1+S2+S3=0. The six normal modes, either keep the spins in this plane (the a modes) or take them out of the plane (the b modes). The soft modes bx, by, and a0 respect the ground-state condition S1+S2+S3=0 and are the Goldstone modes of the spontaneously broken SO (3) symmetry. The hard modes ax, ay, and b0 generate a net magnetization and are hence energetically costly.
The a0 singlet has a simple scalar field theory. The field theory of the b doublet is reminiscent of the elasticity theory of a 2-dimensional isotropic solid with two distinct “speeds of sound”. Thus the 3 branches of low-frequency spin waves generally have 3 distinct velocities. The nearest-neighbor Heisenberg models on the triangular and kagome lattices are exceptional in that sense. The former exhibits an accidental degeneracy of the spin-wave velocities between the two b modes. The nearest-neighbor kagome model is similar to a two-dimensional solid with a vanishing shear modulus and thus a zero speed for the transverse part of the b doublet (the weather-vane mode) while the longitudinal part of the doublet is degenerate with the a0 mode. The 3 speeds can be readily calculated for any lattice model. The doublet a = (ax,ay) mediates the coupling between external perturbations – such as an applied magnetic field – and the antiferromagnetic order parameter. We apply this field theory to the hexagonal antiferromagnet Mn3Ge [5,6,7].
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