Low-energy dynamics of spinor condensates

We present a derivation of the low-energy Lagrangian governing the dynamics of the spin degrees of freedom in a spinor Bose condensate for any phase in which the average magnetization vanishes. This includes all phases found within mean-field treatments except for the ferromagnet for which the low-energy dynamics has been discussed previously. The Lagrangian takes the form of a sigma model for the rotation matrix describing the local orientation of the spin state of the gas.

Figure 1

(Color online) Majorana representation of a $s=3$ spin state with arrows representing the principal spinors.

Figure 2

(Color online) Top: disphenoid corresponding to the Majorana representation of the state with $\chi =\pi /4$, $\eta =\pi /2$ in Eq. (16). Bottom left: rectangular state with $\chi =0$, $\eta =\pi /2$. Bottom right: tetrahedron with $\chi =\eta =\pi /2$.

Figure 3

(Color online) Tetrahedron with threefold axis marked, see Eq. (22).

Figure 4

(Color online) Distortion of the tetrahedral state shown in Fig. 2 due to boosts in the $z$ direction