Phase Diagram of Two-Dimensional Polar Condensates in a Magnetic Field

Spin-1 condensates in the polar (antiferromagnetic) phase in two dimensions are shown to undergo a transition of the Ising type, in addition to the expected Kosterlitz-Thouless (KT) transition of half-vortices, due to the quadratic Zeeman effect. We establish the phase diagram in terms of temperature and the strength of the Zeeman effect using Monte Carlo simulations. When the Zeeman effect is sufficiently strong, the Ising and KT transitions meet. For very strong Zeeman field the remaining transition is of the familiar integer KT type.

Figure 1

The phase diagram for a two-dimensional polar condensate with quadratic Zeeman effect. The dashed red line and the solid blue line mark the KT transition and Ising transition, respectively. In the normal phase ($N$) half-vortices of equal or opposite charge are joined by domain walls (blue lines). In the superfluid phase ($S$) vortices of opposite charge are bound (red ellipse) and there are no domain walls. In the pair superfluid phase (PS), $q/t\lesssim 4$, vortices are bound but domain walls remain.

Figure 2

A half-vortex pair connected by a domain wall (shaded region) in the $\mathbf{n}$ vector [light gray (red)] arising from easy-axis anisotropy (the easy axis is horizontal). The dark gray (blue) arrows denote the configuration of the phase $\theta$.

Figure 3

The specific heat $C$, for $L=16$, showing the merging of the peaks associated with the two transitions as $q$ increases. Inset: $C$ for different system sizes at $q=1$.

Figure 4

Intersection of the Binder cumulants $B[\varphi ]$ for different $L$ at $q=1$. Inset: Enlarged region of intersection.

Figure 5

Left: Intersection of the Binder cumulants $B[\varphi ]$ for different $L$ at $q=1$. Right: The same data but plotted against $(T-T_c)L$. Note that the data for $L=8$ are not within the range of validity for finite size scaling.

Figure 6

Helicity for $q=1$. The upper and lower dashed lines are $8T/\pi$ and $2T/\pi$, respectively. The jump is clearly commensurate with $\Delta {\rm Y}=8T/\pi$.