Stoner Ferromagnetism in a Thermal Pseudospin-$1/2$ Bose Gas

We compute the finite-temperature phase diagram of a pseudospin-$1/2$ Bose gas with contact interactions, using two complementary methods: the random-phase approximation and self-consistent Hartree-Fock theory. We show that the spin-dependent interactions, which break the (pseudo-) spin-rotational symmetry of the Hamiltonian, generally lead to the appearance of a magnetically ordered phase at temperatures above the superfluid transition. In three dimensions, we predict a normal easy-axis (easy-plane) ferromagnet for sufficiently strong repulsive (attractive) interspecies interactions, respectively. The normal easy-axis ferromagnet is the bosonic analog of Stoner ferromagnetism known in electronic systems. For the case of interspecies attraction, we also discuss the possibility of a bosonic analog of the Cooper-paired phase. This state is shown to significantly lose in energy to the transverse ferromagnet in three dimensions, but is more energetically competitive in lower dimensions. Extending our calculations to a spin-orbit-coupled Bose gas with equal Rashba and Dresselhaus-type couplings (as recently realized in experiment), we investigate the possibility of stripe ordering in the normal phase. Within our approximations, however, we do not find an instability towards stripe formation, suggesting that the stripe order melts below the condensation temperature, which is consistent with the experimental observations of Ji et al. [Ji et al., Nat. Phys. 10, 314 (2014)].

Figure 1

3D finite-temperature phase diagram of a pseudospin-$1/2$ Bose gas. Intraspecies interactions are repulsive ($g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=g=4\pi {\hslash }^{2}a/m>0$; we set $a{n}^{1/3}=0.1$), while the interspecies interaction $g_{\uparrow \downarrow }$ varies. $T_c$ is the ideal-gas Bose-Einstein condensation temperature (only $T>T_c$ is shown), $a_{\uparrow \downarrow }$ is the interspecies scattering length, and $n$ is the total density. For $|g_{\uparrow \downarrow }|$ greater than a critical value, the system develops ferromagnetic order in the $z$ direction ($z\mathrm{FM}$) and in the $x\text{−}y$ plane (TFM) above the superfluid transition. Collapse occurs for sufficiently large negative $g_{\uparrow \downarrow }$ (see Fig. 2). Dashed line shows the transition between the unpolarized normal state (UN) and the paired state. TFM is always favored over pairing in 3D.

Figure 2

Thermodynamic instability in $g_{\uparrow \downarrow }<0$ region in 3D. UN is thermodynamically stable for $g>0$, while the TFM is stable in the region on the right side of the instability lines: dotted line ($a{n}^{1/3}=0.4$), dashed line ($a{n}^{1/3}=0.5$), and dashed-dotted line ($a{n}^{1/3}=0.6$).

Figure 3

Phase diagram in quasi-2D as a function of temperature and interactions for $\eta =0.1$. We set $a/a_z=0.02$ and define $T_0={\hslash }^{2}n/2m{k}_B$. Blue line shows transition between UN and $z\mathrm{FM}$ phase, red line shows transition between UN and TFM phase, dashed-dotted black line corresponds to BKT transition temperature calculated for $a_{\uparrow \downarrow }=0$. The dashed black line represents the transition between UN and the paired phase.