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A theoretical model of a Bose-Einstein condensate with angular spin-orbitcoupling has recently been proposed and it has been established that ahalf-skyrmion represents the ground state in a certain regime of spin-orbitcoupling and interaction. Here we investigate low-lying excitations of thisphase by using the Bogoliubov method and numerical simulations of thetime-dependent Gross-Pitaevskii equation. We find that a sudden shift of thetrap bottom results in a complex two-dimensional motion of the system's centerof mass that is markedly different from the response of a competing phase, andcomprises two dominant frequencies. Moreover, the breathing mode frequency ofthe half-skyrmion is set by both the spin-orbit coupling and the interactionstrength, while in the competing state it takes a universal value. Effects ofinteractions are especially pronounced at the transition between the twophases.

Excitation spectra of a Bose-Einstein condensate with an angular spin-orbit coupling