The AF order parameter \(N_{z}\) is related to the coherent state order parameter \(\beta\) by \(\left\langle N_{z}\right\rangle=2 \Omega\left|b_{2}\right|\left(f-\beta^{2}\right)^{1 / 2} \beta\), where \(b_{2}\) is a coupling strength and \(f=2 \Omega\) is the fractional occupation. Prove that for the graphene SU(4) coherent state the maximum of \(N_{z}\) occurs for the \(\beta\) value that minimizes total energy. Show that \(\left\langle N_{z}\right\rangle_{\max }=\) \(\Omega\left|b_{2}\right| f\), so that the maximum value of the AF order parameter occurs for half filling of the \(n=0\) Landau level. The \(\mathrm{SO}(8)\) model is particle-hole symmetric, so \(f\) counts electrons up to half filling and holes (absence of electrons) after half filling.