\left(|\Delta _{\uparrow\uparrow}({{\bf q}})|^2
+|\Delta _{\downarrow\downarrow}({{\bf q}})|^2+|\Delta_{\uparrow
\downarrow}({{\bf q}})|^2+| \Delta_{\downarrow\uparrow}({{\bf q}})|^2\right)\\
&&-\frac{\gamma}{2} \tilde g \hat B\cdot \vec{P}_{tot}\,,\end{aligned}$$ where $ \vec{P}_{tot}$ is the total global spin polarization. The local spin magnetization density, which we calculate as well, is given by $$\langle S_i({{\bf r}})\rangle=
\frac{\hbar}{2}
\sum_{m',m,\sigma,\sigma'} \rho_{\sigma \sigma'}(x_{m'}|x_m)\tau^i_{\sigma \sigma'}
\phi_m^*({{\bf r}})\phi_{m'}({{\bf r}})\,\,.$$ We define the local spin polarization as $\langle P_i({{\bf r}})\rangle=2\langle S_i({{\bf r}})\rangle/(\hbar 
\langle \rho({{\bf r}})\rangle)$. Note that in this case $\langle |\vec P({{\bf r}})| \rangle$ does not have to be equal to $1$ since the system is compressible except in the limit of very large $\gamma$ where $\langle |\vec{P}({{\bf r}})|\rangle\rightarrow 1$ for all ${{\bf r}}$.

Our criteria for convergence is that $$\delta \Delta\equiv\frac{1}{N_\phi^2} \sum_{{{\bf q}}\in {\bf BZ}}
\sum_{\sigma \sigma'} 
|\Delta_{\sigma \sigma'}^{i}({{\bf q}})-\Delta_{\sigma \sigma'}^{i-1}({{\bf q}})|^2< 1\times
10^{-6}\,,$$ where $i$ stands for the $i$th iteration. We have performed calculations for several disorder realizations at different values of $\nu$ for system sizes of $N_\phi=16-32$. The finite size effects come mainly from the effective exchange potential $U_F({{\bf q}})$ and have been studied in detail previously.[@Allan_finite] The main effect is on the interaction part of the energy per particle and it is well understood and easily corrected. We believe that the physics of the phase transitions 