&=& -\alpha \sum_j
|\Delta_j|^2 - \tilde{\kappa} h (|\Delta_{\uparrow\!\uparrow}|^2 -
|\Delta_{\downarrow\!\downarrow}|^2)\nonumber\\&+& [K_{jk;lm} (D_j \Delta_l) (D_k \Delta_m)^* + {\rm c.c.}]\nonumber\\\end{aligned}$$ with respect to one component of the order parameter $\Delta_{s}^*$. This gives an equation of the form $$\alpha \Delta_s = K_{kl;ss'}^* D_k D_l \Delta_s' -
\tilde{\kappa}H\Delta_s \label{EQ:genLinGL}$$ (note that we are ignoring the difference between $h$ and $H$ in this approximation). Since the gradient terms in Eq. cannot in general be reduced into a $D_x^2 + D_y^2$ form, the lowest Landau level wave functions are not sufficient for calculating the condensate wave function in general. However, the solution of this equation can still be expressed in terms of Landau level wave functions: $$\begin{aligned}
\phi_n ({\bf r}) &=& [2^n \pi^{1/2} (n!)]^{-1/2}\nonumber\\ &\times& \sum_m q_m e^{i k_m x'} e^{-(y'-k_m)^2/2} H_n (y'-k_m),\nonumber\\\end{aligned}$$ where $H_n$ is the Hermite polynomial of n$^{th}$ order, $x'$ and $y'$ are $x$, $y$ coordinates in the unit of the magnetic length $l = (\Phi_0/2\pi H)^{1/2}$. This is because $D_i$’s can be expressed as a linear combination of the raising and lowering operators of the Landau levels $\Pi_\pm$, since $\Pi_\pm = l(D_x \pm
iD_y)/\sqrt{2}$.

This wave function describes a vortex lattice when $|\phi_n ({\bf r})|$ is periodic in the lattice vectors ${\bf 