Emergence of a Spin Microemulsion in Spin-Orbit Coupled Bose-Einstein Condensates

We report the first numerical prediction of a “spin microemulsion”—a phase with undulating spin domains resembling classical bicontinuous oil-water-surfactant emulsions—in two-dimensional systems of spinor Bose-Einstein condensates with isotropic Rashba spin-orbit coupling. Using field-theoretic numerical simulations, we investigated the melting of a low-temperature stripe phase with supersolid character and find that the stripes lose their superfluidity at elevated temperature and undergo a Kosterlitz-Thouless-like transition into a spin microemulsion. Momentum distribution calculations highlight a thermally broadened occupation of the Rashba circle of low-energy states with macroscopic and isotropic occupation around the ring. We provide a finite-temperature phase diagram that positions the emulsion as an intermediate, structured isotropic phase with residual quantum character before transitioning at higher temperature into a structureless normal fluid.

Figure 1

Density profiles ${\rho }_{\uparrow }(\mathbf{r})$ of pseudospin bosons in the $|\uparrow ⟩$ state at $\tilde{\kappa }=0.4$ and interaction conditions $\tilde{g}=0.05$, and ${\eta }_g=1.1$. (a), (b) The stripe phase with Rashba SOC transitions into an isotropic spin microemulsion above ${\tilde{T}}_c$. Profiles are snapshots normalized by the spatially averaged density.

Figure 2

A spin microemulsion phase with $⟨N⟩\sim 4.8\times {10}^5$ particles at $\tilde{T}=17.9$, $\tilde{\kappa }=1.2$, $\tilde{g}=0.05$, and ${\eta }_g=1.1$. (a) Density profile ${\rho }_{\uparrow }(\mathbf{r})$ of pseudospin bosons in the $|\uparrow ⟩$ basis state, normalized by the spatially averaged density. A highlighted section is enlarged to show the local emulsionlike structure. (b) Thermal average momentum state occupation $N(\mathbf{k})$.

Figure 3

Pseudospin textures and topological defects visualized via $\mathbf{M}(\mathbf{r})$ at $\tilde{\kappa }=0.4$, $\tilde{g}=0.05$, and ${\eta }_g=1.1$ in (a) the stripe phase, $\tilde{T}=1.1$, and (b) the microemulsion phase, $\tilde{T}=10.25$. The color bar shows the magnitude of $M_z(\mathbf{r})$. Arrows represent the planar vector $[M_x(\mathbf{r}),M_y(\mathbf{r})]$. Boxed regions are enlarged to show planar pseudospin domain walls and vortices for the stripe and microemulsion phases, respectively. The plotted local magnetization vector is normalized by the local magnitude $|M(\mathbf{r})|=\sqrt{M_x^2(\mathbf{r})+M_y^2(\mathbf{r})+M_z^2(\mathbf{r})}$.

Figure 4

Finite-temperature phase diagram in the $\tilde{T}\text{−}\tilde{\kappa }$ plane of interacting bosons with isotropic SOC in two dimensions, constructed for ${\eta }_g=1.1$, $\tilde{g}=0.05$. Lines are a guide for the eye, and error bars are discussed in Supplemental Material [39] [see also Refs. [40, 41, 42] therein]. The blue solid line denotes the Kosterlitz-Thouless-like critical transition, while the red line denotes a first order phase transition into the $\widehat{z}$-ferromagnet (zFM) phase. The black dashed line depicts a crossover between the spin microemulsion and homogeneous normal fluid.

Figure 5

Finite-temperature phase transitions at $\tilde{\kappa }=0.5$, $\tilde{g}=0.05$, and ${\eta }_g=1.1$. (a)–(c) Momentum distribution $N(\mathbf{k})$ for the (a) stripe phase at $\tilde{T}=3.3$, (b) spin microemulsion phase at $\tilde{T}=13.2$, and (c) homogeneous normal fluid at $\tilde{T}=40.0$. (d) Superfluid stiffness tensor across the stripe to emulsion thermal phase transition at various system sizes. The ${\rho }_{\mathrm{SF}}^{xx}$ (${\rho }_{\mathrm{SF}}^{yy}$) fraction is shown with blue (red) markers. (e) Isothermal compressibility at different system sizes, scaled for data collapse. The universal peak near $\tilde{T}=19.4$ (gray shading) highlights the crossover from a structured microemulsion to a homogeneous normal fluid. All error bars are standard errors of the mean calculated during the Langevin time sample averaging process.