Holstein-Primakoff method

There are two ways to calculate dispersions one is the Holstein-Primakoff method, and the other is the equations of motion method. Usually, these two methods yield similar results. However, in the presence of spin-vortices, the latter method yields a new-type mode that is not obtained by the former method. In the following we employ the latter method [18,23]. 

In Fig. 15, magnetic excitation spectra calculated by the spin-vortex model are depicted. Results obtained by the Holstein-Primakoff method are also depicted. The dispersions exhibit hourglass-shapes. As the number of spin-vortices are increased, the peaks of the dispersion become blur. The neck energy increases with the increase of the number of holes in agreement with experiment. 

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