Topological Defects and the Superfluid Transition of the $s=1$ Spinor Condensate in Two Dimensions

The $s=1$ spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We analyze the topological defects of the polar condensate, correcting previous studies, and show that the polar condensate in two dimensions is unstable at any finite temperature; instead, there is a nematic or paired superfluid phase with algebraic order in $\exp (2i\theta )$, where $\theta$ is the superfluid phase, and no magnetic order. The Kosterlitz-Thouless transition out of this phase is driven by unbinding of half-vortices (the spin-disordered version of the combined spin and phase defects found by Zhou), and the anomalous universal $8T_c/\pi$ stiffness jump at the transition is confirmed in numerical simulations. The anomalous stiffness jump is a clear experimental signature of this phase and the corresponding phase transition.

Figure 1

(Top) Helicity modulus $\Gamma$ as a function of temperature for a set of fixed values of the parameters, $\alpha =0.5$, $a_0=4.0$, $T_c^{MF}=1.0$, $c_0=10.0$, $c_2=3.0$, $g_2=0$ [see Eq. (10)] for different system sizes. The two lines are $8T/\pi$ and $2T/\pi$. (Bottom) The goodness of fit $R^2$ to Eq. (11) for different sets of system sizes as a function of temperature. The asterisks and crosses correspond to sets with system sizes 6, 8, 10, 12, 14, 16, 18, 20 and 10, 12, 14, 16, 18, 20, 24, 32, respectively.

Figure 2

The phase diagram of the ground state of the $F=1$ spinor condensate with uniaxial anisotropy [Eq. (3)]. The order-parameter manifold is shown for each phase. Here $\parallel$ and $\perp$ denote in-plane and out-of-plane ordering, respectively.