Effects of Rashba-Dresselhaus competition in 2D two-component spinor BEC
"Dancing bands": as the spin-orbit coupling mixing angle changes, the two bands dance periodically.

The total Hamiltonian of my model,
\begin{align*} H &= \frac{k_{x}^{2}+k_{y}^{2}}{2m}+V(r)\\ &+\kappa_{R}(\sigma_{x}k_{y}-\sigma_{y}k_{x}) + \kappa_{D}(\sigma_{x}k_{x}-\sigma_{y}k_{y}) \\ & + h\sigma_{z}+k_{x}d \\ &:= H_{\text{harmonic}}+H_{\text{soc}}+H_{\text{ex}} \end{align*}

Neglecting the harmonic potential, we have the homogeneous model,
\begin{eqnarray} H(\mathbf{k}) & = & \frac{\mathbf{k}^{2}}{2m}+H_{\text{soc}}+h\sigma_{z}+k_{x}d\nonumber \\ & = & \begin{pmatrix}\left(\frac{\mathbf{k}^{2}}{2m}+k_{x}d\right)+h & \kappa\left(k_{x}e^{i\alpha}+ik_{y}e^{-i\alpha}\right)\\ \kappa\left(k_{x}e^{-i\alpha}-ik_{y}e^{i\alpha}\right) & \left(\frac{\mathbf{k}^{2}}{2m}+k_{x}d\right)-h \end{pmatrix}\label{eq: H(k)} \end{eqnarray}
from which I obtained two branches
\[ E_{\mathbf{k}}^{\pm}=\left(\frac{\mathbf{k}^{2}}{2m}+k_{x}d\right)\pm\kappa\sqrt{\mathbf{k}^{2}+2k_{x}k_{y}\sin(2\alpha)+(h/\kappa)^{2}} \]
The lower branch. An O(2) phase exists between two distinct Z_2 phases.

There is a linear touching point at zero momentum point but a finite Zeeman field h will lift the degeneracy.

Next I demonstrated that the minima are always located along diagonal directions.

Rewrite the momenta in two directions in terms of a single complex quantity $q=k_{x}+ik_{y}:=ke^{i\beta}$.

I found,
\[ E_{q}^{\pm} = \left(\frac{k^{2}}{2m}+k_{x}d\right)\pm\kappa k\sqrt{1+\sin(2\beta)\sin(2\alpha)+(h/k\kappa)^{2}} \]
and finally I arrived at the condition