Striped ferronematic ground states in a spin-orbit-coupled $S=1$ Bose gas

We theoretically establish the mean-field phase diagram of a homogeneous spin-1, spin-orbit-coupled Bose gas as a function of the spin-dependent interaction parameter, the Raman coupling strength, and the quadratic Zeeman shift. We find that the interplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of ferromagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling, increasing attractive spin-dependent interactions (as in ${}^{87}\mathrm{Rb}$ or ${}^7\mathrm{Li}$) induces a transition from a uniform to a stripe $XY$ ferromagnet (with no nematic order). For repulsive spin-dependent interactions (as in ${}^{23}\mathrm{Na}$), however, we find a transition from an $XY$ spin spiral phase ($\langle S_z\rangle =0$ and uniform total density) with uniaxial nematic order to a biaxial ferronematic, where the total density, spin vector, and nematic director oscillate in real space. We investigate the stability of these phases against the quadratic Zeeman effect, which generally tends to favor uniform phases with either ferromagnetic or nematic order but not both. We discuss the relevance of our results to ongoing experiments on spin-orbit-coupled, spinor Bose gases.

Figure 1

Schematic phase diagram for a spin-orbit-coupled spin-1 Bose gas as a function of spin-dependent interaction strength and quadratic Zeeman shift, showing translation symmetry-breaking phases. The underlying single-particle dispersion is shown above. For sufficiently large attractive spin-dependent interactions, an $XY$ spin-density-wave phase occurs, with oscillations in the total density (FM stripe). For repulsive spin-dependent interactions, an $XY$ spiral phase occurs, which has uniaxial nematic order (UN) but in which the total density remains uniform. For sufficiently large repulsive spin-dependent interactions, a biaxial nematic phase (BN) is present, where the total density, spin vector, and nematic director oscillate in real space. A sufficiently large positive or negative quadratic Zeeman effect favors homogeneous phases with either uniaxial nematic [${\mathrm{UN}}_{\perp }(\psi =\{0,1,0\})$ or ${\mathrm{UN}}_{\parallel }(\psi =\frac{1}{\sqrt{2}}\{1,0,1\})$] or ferromagnetic order [${\mathrm{FM}}_{\perp }(\psi =\{1,0,0\})$ or ${\mathrm{FM}}_{\parallel }(\psi =\frac{1}{2}\{1,\sqrt{2},1\})$] but not both.

Figure 2

Critical attractive spin-dependent interaction $|c_2|/c_0$ for the onset of the stripe ferromagnetic phase as a function of Rabi coupling $\Omega /E_R$ at $q=0$. We set $c_0{n}_0/E_R=0.4$. The dashed line shows $|c_2|/c_0$ for ${}^7\mathrm{Li}$; all values of $\Omega \lesssim 0.9E_R$ support the ferromagnetic phase with stripes in the total density. The inset shows the density in real space for $\Omega /E_R=0.8$ at $|c_2|/c_0$ corresponding to ${}^7\mathrm{Li}$.

Figure 3

(top) Spatial spin texture in the polar regime of a spin-orbit-coupled spin-1 Bose gas for strong spin-dependent interactions $c_2/c_0=1.1$ and $\Omega /E_R=0.8$, which corresponds to the three-minimum regime. We set $q=0$ here, although this phase is stable for moderate values of $q$ (see Fig. 1). The arrows indicate the projection of the spin in the $x\text{−}z$ plane in spin space at each point in real space. (bottom) The $x$ (green dotted line), $y$ (red dashed line), and $z$ (black solid) components of spin in real space. Oscillations in the $z$ direction (at wave vector $k=k_1)$ and the $x\text{−}y$ direction (at two wave vectors, $k=k_1,2k_1)$ are different from one another. The total density (not shown) also shows anharmonic oscillations at two dominant wave vectors, $k=k_1$ and $k=2k_1$. Note that due to the Raman coupling term, time reversal is explicitly broken, and the spatially averaged spin along the $x$ direction is finite.

Figure 4

Total spin (red dashed line) and biaxial order parameter (black solid line) shown as a function of space in the biaxial, spin-density-wave phase of a polar spin-orbit-coupled gas. The minimum in the biaxiality corresponds to a minimum in the total spin, and here a nearly uniaxial nematic appears, which illustrates the competing nature of the nematic and spin-order parameters.