Anomalous Behavior of Spin Systems with Dipolar Interactions

We study the properties of spin systems realized by cold polar molecules interacting via dipole-dipole interactions in two dimensions. Using a spin wave theory, that allows for the full treatment of the characteristic long-distance tail of the dipolar interaction, we find several anomalous features in the ground state correlations and the spin wave excitation spectrum, which are absent in their counterparts with short-range interaction. The most striking consequence is the existence of true long-range order at finite temperature for a two-dimensional phase with a broken $U(1)$ symmetry.

Figure 1

(a) Mean-field phase diagram for the $XXZ$ model with dipolar interactions, where $\tan \theta$ is the ratio between the $XY$ and the Ising spin couplings. (b) Ground state energy per particle: the dashed lines show the mean-field predictions, while the solid lines include the contributions from the spin waves. At the critical values ${\theta }_c$ and ${\tilde{\theta }}_c$, the ground state energy exhibits the jump $\Delta e_c\approx 0.14J$ and $\Delta {\tilde{e}}_c\approx 0.06J$, indicating the potential formation of an intermediate phase.

Figure 2

Spin wave excitations with $\Gamma =(0,0)$, $M=(0,\pi /2)$, and $K=(\pi /a,\pi /a)$ for different $\theta$ angles. (a) Spectrum of the I-F phase which also shows the behavior of the dipolar dispersion ${ϵ}_{\mathbf{q}}$ for $\theta =-3\pi /4$. (b)–(d) Spectrum for the $XY\mathrm{\text{−}}\mathrm{F}$, I-AF, and $XY\mathrm{\text{−}}\mathrm{AF}$ phases.

Figure 3

Time evolution for localized spin excitations described by the Gaussian wave packet ${\psi }_0(\mathbf{r})=e^{-|\mathbf{r}{|}^2/2{\sigma }^2}/\sqrt{\pi {\sigma }^2}$ with $\sigma \gg a$ in the continuum description. (a) For a linear dispersion $c|\mathbf{q}|$ in the I-F phase, the dynamics is described by cylindrical symmetric wave packets (see inset) traveling with velocity $c$, instead of the conventional quantum mechanical spreading for massive systems. (b) For an anomalous dispersion with $\alpha \sqrt{|\mathbf{q}|}$ in the $XY\mathrm{\text{−}}\mathrm{F}$ phase, the behavior at long times $t\gg \sqrt{\sigma }\alpha$ reduces to a scaling function $\xi (z)$ via $|\psi (x,\tau ){|}^2=\xi (x/\tau -1/2)/{\tau }^2$ (see inset) using rescaled time $\tau =t\alpha /\sqrt{\sigma }$ and space $x=|\mathbf{r}|/\sigma$ coordinates. It describes a cylindrical symmetric wave front with velocity $\alpha \sqrt{\sigma }$.