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We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates(BECs) in the presence of a quadratic Zeeman effect, and derive theLee-Huang-Yang (LHY) corrections to the ground-state energy, sound velocity,and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECsthat can be realized experimentally. We also examine the stability of eachphase against quantum fluctuations and the quadratic Zeeman effect.Furthermore, we discuss a relationship between the number of symmetrygenerators that are spontaneously broken and that of Nambu-Goldstone (NG)modes. It is found that in the spin-2 nematic phase there are specialBogoliubov modes that have gapless linear dispersion relations but do notbelong to the NG modes.

Bogoliubov Theory and Lee-Huang-Yang Corrections in Spin-1 and Spin-2 Bose-Einstein Condensates in the Presence of the Quadratic Zeeman Effect