Spin Correlation and Discrete Symmetry in Spinor Bose-Einstein Condensates

We study spin correlations in Bose-Einstein condensates of spin 1 bosons with scatterings dominated by a total spin equal $2$ channel. We show that the low energy spin dynamics in the system can be mapped into an $o(n)$ nonlinear sigma model. $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ at the zero magnetic field limit and $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$ in the presence of weak magnetic fields. In an ordered phase, the ground state has a discrete ${Z}_{2}$ symmetry and is degenerate under the group $[U(1)\ifmmode\times\else\texttimes\fi{}{S}^{n\ensuremath{-}1}]{/Z}_{2}$. We explore consequences of the discrete symmetry and propose some measurements to probe it.