Quantum Lifshitz transitions generated by order from quantum disorder in strongly correlated Rashba spin-orbital coupled systems

We study the system of strongly interacting spinor bosons in a square lattice subject to the isotropic Rashba SOC $\alpha=\beta$. It supports collinear spin-bond correlated magnetic Y-x phase, a gapped in-commensurate (IC-) co-planar IC-XY-y phase, a non-coplanar commensurate (C-) $3 \times 3$ Skyrmion crystal phase (SkX). The state at the Abelian point $\alpha=\beta=\pi/2$ is just an AFM state in a rotated basis. Slightly away from the point, we identify a spurious $U(1)$ symmetry, develop a novel and non-perturbative method to calculate not only the gap, but also the excitation spectrum due to the order from quantum disorder (OFQD) mechanism. We construct a symmetry based effective action to investigate the quantum Lifshitz transition from the Y-x state to the IC-XY-y state and establish the connection between the phenomenological parameters in the effective action and those evaluated by the microscopic non-perturbative OFQD analysis in the large $S$ limits. Experimental implications on cold atoms and some 4d or 5d Kitaev materials are discussed.