A. Brillouin zone, quasimomentum and magnon operators. Let us consider the simplest chessboard antiferromagnetic state, with the magnetic mo- ments ⃗ m$_{⃗}$ l = ( − 1) l$_{x}$ + l$_{y}$ m$_{0}$⃗ e$_{z}$ in the ground state. This system can be treated as the embedding of two quadratic sublattices with the lattice spacing $_{a}$√$_{2, as shown in Fig. 4. Let us denote}$ the sites of the first sublattice which have ⃗ m$_{⃗}$ δ = ⃗ e$_{z}$ as a ⃗ δ , ⃗ δ = √$_{2(}$$_{δ}$$_{1}$$_{⃗}$ ε$_{1}$ + δ$_{2}$⃗ ε$_{2}$ ), where δ$_{1}$ , δ$_{2}$ are integers and ⃗ ε$_{1}$ = ( ⃗ e$_{x}$ − ⃗ e$_{y}$ ) $_{/}$√$_{2,}$ ⃗ ε$_{2}$ = ( ⃗ e$_{x}$ + ⃗ e$_{y}$ ) $_{/}$√$_{2 are basis vectors for sublattices.}$ Using orts ⃗ e$_{x}$, ⃗ e$_{y}$ , the vector ⃗ δ can be expressed as ⃗ δ = δ$_{x}$⃗ e$_{x}$ + δ$_{y}$⃗ e$_{y}$ , where δ$_{x}$ = δ$_{1}$ + δ$_{2}$ , δ$_{y}$ = δ$_{2}$ − δ$_{1}$ . The sites for second sublattice which have ⃗ m$_{⃗}$ µ = − m$_{0}$⃗ e$_{z}$ can be expressed as a⃗ µ , ⃗ µ = √$_{2[(}$$_{µ}$$_{1}$$_{+ 1}$$_{/}$$_{2)}$$_{⃗}$ ε$_{1}$ + ( µ$_{2}$ + 1 / 2) ⃗ ε$_{2}$ ], or ⃗ µ = µ$_{x}$⃗ e$_{x}$ + µ$_{y}$⃗ e$_{y}$ , µ$_{x}$ = µ$_{1}$ + µ$_{2}$ + 1, µ$_{y}$ = µ$_{2}$ − µ$_{1}$ , where µ$_{1}$ and µ$_{2}$ are integers. FIG. 4: ) The scheme of the sublattices for the particle array with the chessboard magnetic order; the solid and dashed lines connect particles belonging to the first and second sublattices, correspondingly; dotted lines connect particles from different sublattices. b) the shape of the first Brillouin zone, the solid line restricts the Brillouin zone of the sublattice, the dashed line restricts the Brillouin zone for the whole lattice. The correspondence between ⃗ q and ˜ ⃗ q , introduced below, is also presented. To describe the magnetic oscillations for two sublattices, it is sufficient to use two Bose operators as follows, GLYPH<s113> GLYPH<s113> GLYPH<s47> GLYPH<s97> GLYPH<s47> GLYPH<s97> GLYPH<s98> GLYPH<s41> GLYPH<s97> GLYPH<s41> GLYPH<s32> GLYPH<s32> m$_{z,}$$_{⃗}$ δ = m$_{0}$ − 2 µ$_{B}$a † ⃗ $_{δ}$a$_{⃗}$ δ , m$_{x,}$$_{⃗}$ δ = ( a$_{⃗}$ δ + a † ⃗ δ ) √ m$_{0}$µ$_{B}$, m$_{y,}$$_{⃗}$ δ = i ( a † ⃗ δ − a$_{⃗}$ δ ) √ m$_{0}$µ$_{B}$ ; m$_{z,⃗}$ µ = 2 µ$_{B}$b † ⃗ $_{µ}$b$_{⃗}$ µ − m$_{0}$, (15) m$_{x,⃗}$ µ = b$_{⃗}$ µ + b † ⃗ µ √ m$_{0}$µ$_{B}$, m$_{y,⃗}$ µ = − i ( b † ⃗ µ − b$_{⃗}$ µ ) √ m$_{0}$µ$_{B}$ . Then the Hamiltonian takes the form, containing the terms describing the interaction of oscillations both within first and second sublattices, as well as between different sublattices. 14