  Motivated by a recent experiment in an antiferromagnetic spin-1 Bose-Einstein
condensate of ${}^{23} \textrm{Na}$ atoms, we study the energetical stability
of a singly quantum vortex injected into the center of a quasi-two-dimensional
gas with zero total spin against dissocation into a pair of half-quantum
vortices. We find that the critical dissociation point of this
confinement-deconfinement type phase transition can be expressed in terms of
the ratio of density-density ($c_0$) and spin-spin ($c_2$) coupling constants.
The transition of bound to unbound vortices, in particular, sensitively depends
on (1) the ratio of system size ($R$) to density healing length ($\xi_d$), and
(2) the trap potential. Specifically, the critical ratio $(c_2 /
c_0)_{\textrm{cr}}$ increases when $R / \xi_d$ decreases, and is relatively
larger in a harmonic trap than in a box trap. Dissociation is energetically
generally favored for $c_2 / c_0 < (c_2 / c_0)_{\textrm{cr}}$, which as a
corollary implies that vortex dissociation is observed as well for negative
$c_2 < 0$, e.g., in a rubidium spin-1 BEC, whereas in a sodium spin-1 BEC
($c_2>0$) it is energetically blocked above the critical ratio $(c_2 /
c_0)_{\textrm{cr}}$. Tuning the coupling ratio $c_2/c_0$ by using microwave
control techniques, the dependence of the deconfinement phase transition on
coupling parameters, density, and system size we predict, can be verified in
experiments with ultracold spinor gases.
