a linear combination of the up and down spin states of these basis functions $$|\alpha\rangle=\sum_{m,\sigma}\langle m \sigma|\alpha\rangle \,\,
|m \sigma\rangle\,\,.$$ Before writing down the Hamiltonian matrix in the HF approximation, we introduce several notation simplifying definitions, closely following previous HF studies. [@olderhfdis] The expectation value of the particle density (in momentum space) is given by $$\begin{aligned}
\langle \rho({{\bf q}})\rangle&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha)\langle \alpha
|e^{-i{{\bf q}}\cdot{{\bf r}}}|\alpha\rangle\\
&\equiv&N_\phi e^{-\frac{1}{4}q^2}
\sum_{\sigma,\sigma'} \delta_{\sigma,\sigma'} \Delta_{\sigma'\,\sigma}({{\bf q}})
e^{-i\frac{q_x q_y}{2}}\,\,,\end{aligned}$$ where we define $$\Delta_{\sigma'\,\sigma}({{\bf q}})\equiv\frac{1}{N_\phi}
\sum_{m,m'}\delta_{(x_{m'},x_m+q_y)}e^{-iq_x x_m}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m),$$ and $$\begin{aligned}
\rho_{\sigma'\,\sigma}(x_{m'}|x_m)
&=&\sum_\alpha n_F(\epsilon_F-\epsilon_\alpha) \langle
m'\sigma'|\alpha\rangle\langle\alpha|m\sigma\rangle\,.\end{aligned}$$ Here $\delta_{(x_{m'},x_m+q_y)}$ is a periodic Kronig delta function, [*i.e.*]{} it is nonzero for $x_{m'}=x_m+q_y+s\,L_x$ for any integer s. With these definitions we can write the Hamiltonian matrix in the HF approximation in a compact form which is simple to diagonalize numerically $$\begin{aligned}
&&\langle m \sigma|{\cal H}|m' \sigma'\rangle_{HF} =
\sum_{{{\bf q}}\epsilon {\rm BZ}}
\left\{(\gamma\Delta_0({{\bf q}}){\rm U_H}({{\bf q}})+{\rm U_{\rm dis}}({{\bf q}})) {\rm I}+\right.
\nonumber\\
&&\left.\frac{\gamma}{2}(\Delta_0({{\bf q}}){\rm I}+\vec{\Delta}({{\bf q}})\cdot\vec{\tau})
{\rm U_F}({{\bf q}})\right\}
\delta_{(x_m,x_{m'}+q_y)}e^{+iq_x x_{m'}}\nonumber\\
&&-\frac{1}{2}\tilde g \hat B\cdot \vec{\tau}\,,\end{aligned}$$ where $\Delta_\alpha({{\bf q}})={\rm Tr}\{\Delta_{\sigma \sigma'}({{\bf q}})
\tau^{\alpha}\}$, and $\hat B$ specifies the orientation of the external magnetic field. The various effective potentials which appear here are defined as $${\rm U_{dis}}({{\bf q}})=\frac{1}{A}\sum_{{\bf G}}e^{-\frac{1}{4}|{{\bf q}}+{{\bf G}}|^2}v_E({{\bf q}}+{{\bf G}}) e^{
