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We numerically study the Berezinskii-Kosterlitz-Thouless (BKT) transition ofa spin-1 spinor Bose gas under the quadratic Zeeman effect. A calculation ofthe mass and spin superfluid densities shows that (i) the BKT transition occursonly when vortices are classified by the integer group $\mathbb{Z}$, and$\mathbb{Z}_2$ vortices do not contribute to the BKT transition, (ii) the twoBKT transition temperatures for mass and spin superfluid densities aredifferent for a positive quadratic Zeeman effect and equal for a negativequadratic Zeeman effect, and (iii) the universal relation of the superfluiddensities at the BKT transition temperature is changed when multiple kinds ofvortices contribute to the transition. We have further found that (iv)spin-singlet pairs in non-magnetic states show thequasi-off-diagonal-long-range order at the different temperature lower than theBKT transition temperature, giving the new universal relation of the superfluiddensity.

Berezinskii-Kosterlitz-Thouless transition of spin-1 spinor Bose gases in the presence of the quadratic Zeeman effect