k · G ( ⃗ k ), ω ( ⃗ k ) = γ √ H eff a + H$_{0}$ + M 2 Σ$_{(+)}$( ⃗ k ) · √ H eff a + H$_{0}$ + M 2 Σ$_{(}$$_{−}$$_{)}$( ⃗ k ) , (9) where we introduced the combinations of dipole sums Σ$_{(}$$_{±}$$_{)}$( ⃗ k ) Σ$_{(}$$_{±}$$_{)}$( ⃗ k ) = σ (0) − σ ( ⃗ k ) ± 3 ∣ σ$_{c}$ ( ⃗ k ) ∣ . (10) These sums are plotted in Fig. 1 as a function of ⃗ k for symmetrical directions of the reciprocal lattice. The field, H eff a = H$_{a}$ − (3 / 2) Mσ (0) is the z -projection of the effective anisotropy field, comprising an anisotropy field for the single particle and demagnetization field of the array as a whole, and σ (0) (3 m$_{0}$ /2 a $^{3}$) ∼ = 1 . 08 · (4 πM ). GLYPH<s45>GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s45> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s50>GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s40> GLYPH<s107> GLYPH<s41> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s40> GLYPH<s107> GLYPH<s41> GLYPH<s107>GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s107> GLYPH<s66> GLYPH<s107>GLYPH<s32> GLYPH<s124>GLYPH<s124>GLYPH<s32>GLYPH<s40>GLYPH<s49>GLYPH<s44>GLYPH<s48>GLYPH<s41> GLYPH<s107> GLYPH<s32>GLYPH<s124>GLYPH<s124>GLYPH<s32>GLYPH<s40>GLYPH<s49>GLYPH<s44>GLYPH<s49>GLYPH<s41> FIG. 1: The dependence of the combinations of the dipole sums Σ$_{(}$$_{±}$$_{)}$( ⃗ k ) on the quasimomentum for symmetric directions of the square lattice, full symbols depict Σ$_{(}$$_{−}$$_{)}$( ⃗ k ) = σ (0) − σ ( ⃗ k ) − 3 ∣ σ$_{c}$ ( ⃗ k ) ∣ , open circles represent Σ$_{(+)}$( ⃗ k ) = σ (0) − σ ( ⃗ k ) + 3 ∣ σ$_{c}$ ( ⃗ k ) ∣ . k$_{B}$ is the maximal value of the wave vector modulus for a given direction, corresponding to the border of the Brillouin zone, k$_{B}$ = π/a for ⃗ k || (0 , 1) || (1,0) and k$_{B}$ = π / a and k$_{B}$ = $_{π}$√$_{2}$$_{/a}$ for ⃗ k || (1 , 1). A. Dispersion relation and stability conditions. It is remarked that, the expression for the frequency can be rewritten in the universal form ω $^{2}$( ⃗ k ) = [ ω$_{0}$ + ω$_{int}$ Σ$_{(+)}$( ⃗ k ) ] [ ω$_{0}$ + ω$_{int}$ Σ$_{(}$$_{−}$$_{)}$( ⃗ k ) ] , (11) 8