Elementary excitations in quadrupolar systems

We consider $\text{S ?}\text{3}\text{2}$ isotropic quadrupolar ordered systems and derive elementary excitations at low temperature. A Holstein-Primakoff type transformation and a linear approximation are used. For $\text{S =}\text{3}\text{2}$, the spectrum is made of four degenerate acoustic branches. For S ? 2, only two degenerate branches satisfy the Goldstone theorem: they describe Δm = ± 1 excitations similar to librons in molecular crystals. The two degenerate branches describing Δm = ± 2 excitations have a gap at k = 0 although the hamiltonian is isotropic. For a special $\text{S =}\text{3}\text{2}$ cubic hamiltonian, a Goldstone mode is found in the spectrum and related to a continuous degeneracy of the ground state. A comparison between $\text{S =}\text{1}\text{2}$ dipolar and $\text{S =}\text{3}\text{2}$ quadrupolar systems is presented.