Field theory of a hexagonal antiferromagnet with 3 sublattices
Sayak Dasgupta, John’s Hopkins University, Department of Physics
We present a classical field theory of magnetization dynamics in a generic 3-sublattice antiferromagnet in 2 spatial dimensions 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, shown in Fig. 1, 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. They can be safely integrated out to generate kinetic energy for the soft modes.
The 3 Goldstone modes can be grouped into the trivial singlet a0 and the irreducible doublet b = (bx, by) of the point group D3. The a0 singlet obeys 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” (longitudinal and transverse). 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 hard doublet a = (ax,ay) plays an important role in mediating 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 [3,4].
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