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## 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. 

<img src='content_image/362748.jpg'>

To describe the magnetic oscillations for two sublattices, it is sufficient to use two Bose operators as follows, 

<img src='content_image/362749.jpg'>

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. 

