of the microscopic constituents of matter. The many-body Hamiltonian of two-particle interaction includes the so-called spin-spin interaction given by $$\label{eq:2.3}
V={1\over 2\mathcal{V}}\sum_{{\bf p}_{1},\ldots{\bf p}_{4}}I({\bf p}_{1}-{\bf p}_{3})a^{\dagger}_{{\bf p}_{1}\alpha}a^{\dagger}_{{\bf p}_{2}\beta}S^{i}_{\alpha\gamma}S^{i}_{\beta\delta}a_{{\bf p}_{3}\gamma}a_{{\bf p}_{4}\delta}\,\delta_{{\bf p}_{1}+{\bf p}_{2},\,{\bf p}_{3}+{\bf p}_{4}}, \quad S^{i}_{\alpha\beta}={1\over 2 }\sigma^{i}_{\alpha\beta},$$ where $a_{{\bf p}\alpha}$, $a^{\dagger}_{{\bf p}\alpha}$ satisfy the permutation relations given by Eqs. (\[eq:2.2\]) and $I({\bf p}_{1}-{\bf p}_{3})$ denotes the exchange interaction. Here and below, the summation over the repeated indices related to internal symmetry is assumed. The above Hamiltonian commutes with the spin operator of a many-body system, $$S^{i}=\sum_{\bf p}a^{\dagger}_{{\bf p}\alpha}S^{i}_{\alpha\beta}a_{{\bf p}\beta}$$ and, consequently, the later represents the conserved quantity or the integral of motion. It is related to magnetization vector $M^{i}$ by $M^{i}=2\mu_{0}S^{i}$, where $\mu_{0}=e\hbar/2mc$ is the Bohr magneton. Note that three components of a particle spin (microscopic characteristic) generate the same number of macroscopic parameters necessary to describe a many-body system of spin-1/2 particles. The Hamiltonian given by Eq. (\[eq:2.3\]) can be applied to describe a gas of spin-1/2 atoms or interacting electron gas embedded in a solid state system. Note that in the lattice models, the creation and annihilation operators carry lattice site index instead of momentum ${\bf p}$.

Now we address the description of a many-body system of 