with $\Psi_{\alpha}$ being a variational parameter and neglecting the terms of the third and fourth order, one can obtain the Hamiltonian truncated up to quadratic terms in the creation and annihilation operators: $$\label{eq:3.5}
H(\Psi)\simeq H^{(0)}(\Psi)+H^{(2)}(\Psi),$$ where $H^{(0)}(\Psi)$ is the $c$-number part of the truncated Hamiltonian given by $$\label{eq:3.6}
{1\over{\cal V}}H^{(0)}(\Psi)={U(0)\over 2}(\Psi^{*}\Psi)^{2}+{J(0)\over 2}(\Psi^{*}\lambda^{a}\Psi)^{2}-h(\Psi^{*}\lambda^{2}\Psi).$$ The quadratic part reads $$\label{eq:3.6'}
H^{(2)}(\Psi)=H_{0}^{(2)}(\Psi)+V^{(2)}_{\rm p}(\Psi)+V^{(2)}_{\lambda}(\Psi),$$ where $H_{0}^{(2)}(\Psi)$ does not include the interparticle interactions, $$\label{eq:3.7}
H_{0}^{(2)}(\Psi)=\sum_{{\bf p}\neq 0}\varepsilon_{\bf p}(a^{\dagger}_{\bf p}a_{\bf p})-h\sum_{{\bf p}\neq 0}(a^{\dagger}_{\bf p}\lambda^{2}a_{\bf p}).$$ Two other terms describing the interaction have the form $$\label{eq:3.8}
V_{\rm p}^{(2)}(\Psi)=U(0)\sum_{{\bf p}\neq 0}(\Psi^{*}\Psi)(a^{\dagger}_{\bf p}a_{\bf p})+{1\over 2}\sum_{{\bf p}\neq 0}U({\bf p})\left[(\Psi a^{\dagger}_{\bf p})(\Psi^{*}a_{\bf p})+(\Psi a^{\dagger}_{\bf p})(\Psi a^{\dagger}_{-{\bf p}})+{\rm h.c.}\right]$$ and $$\begin{gathered}
V_{\lambda}^{(2)}(\Psi)=J(0)\sum_{{\bf p}\neq 0}(\Psi^{*}\lambda^{a}\Psi)(a^{\dagger}_{{\bf p}}\lambda^{a}a_{\bf p}) \nonumber\\
+{1\over 2}\sum_{{\bf p}\neq 0}J({\bf p})\left[(a^{\dagger}_{\bf p}\lambda^{a}\Psi)(\Psi^{*}\lambda^{a}a_{\bf p})+(a^{\dagger}_{\bf p}\lambda^{a}\Psi)(a^{\dagger}_{-{\bf p}}\lambda^{a}\Psi)+{\rm h.c.}\right], \label{eq:3.9}\end{gathered}$$ where we use the following notations $(\Psi^{*}\Psi)\equiv\Psi_{\alpha}^{*}\Psi_{\alpha}$, $(a^{\dagger}_{\bf p}a_{\bf p})\equiv a^{\dagger}_{{\bf p}\alpha}a_{{\bf p}\alpha}$, $(\Psi^{*}\lambda^{a}\Psi)\equiv \Psi_{\alpha}^{*}\lambda^{a}_{\alpha\beta}\Psi_{\beta}$, and so on, assuming matrix multiplication. Note that the replacement of creation and annihilation operators by $c$-numbers implies the gauge symmetry breaking and leads to non-conservation of the total number of atoms. Therefore, the problem should be considered in the grand canonical ensemble, where the chemical potential $\mu$, being a Lagrange multiplier, reflects the conservation of the total number of 