at the low experimental temperatures, we follow the common practice of considering only LLL states. Neglecting the frozen kinetic-energy degree of freedom, the Hamiltonian in second quantization is written as $${\cal H}={\cal H}_I+{\cal H}_{dis}+{\cal H}_Z\,,$$ where ${\cal H}_I$ is the interaction part of the Hamiltonian $$\begin{aligned}
{\cal H}_I&=&\frac{1}{2}\int d{{\bf r}}\int d{{\bf r}}' \sum_{\sigma\, \sigma'}
v_I({{\bf r}}-{{\bf r}}') \hat\psi^\dagger_\sigma({{\bf r}})
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma'}({{\bf r}}') \hat\psi_\sigma({{\bf r}})\,\,,\end{aligned}$$ ${\cal H}_{dis}$ is the external disorder part of the Hamiltonian $$\begin{aligned}
{\cal H}_{dis}&=&\int d{{\bf r}}\sum_{\sigma}v_{E}({{\bf r}})
\hat\psi^\dagger_\sigma({{\bf r}})\hat\psi_\sigma({{\bf r}})\,,\end{aligned}$$ and ${\cal H}_Z$ is the Zeeman term $${\cal H}_Z=-\frac{1}{2}g\mu_B \int d{{\bf r}}\sum_{\sigma \sigma'} 
\hat\psi^\dagger_{\sigma'}({{\bf r}}') \hat\psi_{\sigma}({{\bf r}}')
\vec{\tau}_{\sigma' \sigma}\cdot\vec{B}(\vec r)\,,$$ with $\sigma=\uparrow,\downarrow$, $v_I$ and $v_E$ being the Coulomb interaction and disorder potentials respectively, and $\tau_i$ being the Pauli matrices. Here we also define the Zeeman coupling strength as $\tilde g=g\mu_B B/(e^2/\epsilon l)$ for later reference. We chose the the Landau gauge elliptic theta functions as our basis $$\phi_m(x,y)=\frac{1}{\sqrt{L_y l \sqrt{\pi}}}
\sum_{s=-\infty}^\infty e^{i\frac{1}{l^2}x_{m,s}y}
e^{-\frac{1}{2 l^2}(x-x_{m,s})^2}\,\,,$$ where $x_{m,s}=\frac{2\pi m l^2}{L_y}+s L_x$, $m,m'=1,\dots,N_\phi$, $N_\phi=A/(2\pi l^2)$, and $l$ is the magnetic length which we set equal to $1$ for simplicity. These wavefunctions satisfy the semi-periodic boundary conditions $\phi_m(x,y)=\phi_m(x,y+L_y)$ and $\phi_m(x+L_x)=\exp(+iL_x y/l^2)\phi_m(x,y)$.

We consider the HF single particle states to be 