Quantum field theory of nematic transitions in spin orbit coupled spin-1 polar bosons in one dimension

We will discuss our recent theoretical study of an ultra-cold gas of spin-1 polar bosons in one spatial dimension, which are subject to a quadratic Zeeman field and a Raman induced spin-orbit coupling. We analytically solve the model in its low-energy sector characterizing the relevant phases and the quantum phase transitions between them. Depending on the sign of the effective quadratic Zeeman field $\epsilon$, two superfluid phases with distinct nematic order appear. In addition, we uncover a spin-liquid superfluid state at strong coupling. We employ a combination of renormalization group calculations, duality transformations, and fermionization to access the nature of phase transitions. At $\epsilon = 0$, a line of spin-charge separated pairs of Luttinger liquids divides the two nematic phases and the transition to the spin disordered state at strong coupling is of the Berezinskii-Kosterlitz-Thouless type. In contrast, at $\epsilon \neq 0$, the quantum critical theory separating nematic and strong coupling spin disordered phases maps to a quantum critical Ising model that is coupled to the charge Luttinger liquid. We will discuss the experimental signatures of our findings that are relevant to ongoing experiments in ultra-cold atomic gases of $^{23}$Na.