GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s53> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s104> GLYPH<s99>GLYPH<s114> FIG. 6: The dependence of the critical magnetic field on anisotropy. Here both characteristic fields, h cr , 0 and h$_{cr}$$_{,}$$_{11}$ are plotted as functions of β , and the stability region of the AFM state is doubly patterned region under both curves. D. Density of states. Let us consider positions and form of Van Hove singularities for the spectral density of magnons of both branches as shown in Fig. 7. In a weak field, frequency bands corresponding to the upper and lower branches of collective oscillations overlap. In this case it is convenient to introduce the partial density of states as shown in Fig. 7a for zero field. As for the aforementioned FM state of the array nonstandard behavior of D ( ω ) for both branches can merge caused by long-range character of interaction. As the previously discussed example, when the discontinuity of the derivative of density of states has the form D ( ω ) = C$_{1}$ + C$_{2}$ ( ω − ω$_{0}$ )Θ( ω − ω$_{0}$ ), this corresponds to the linear dependence of ω ( ⃗ q ) in the vicinity of the center of the Brillouin zone, ω ( ⃗ q ) − ω$_{0}$ ∝ | ⃗ q | . For the AFM case at various values of magnetic field this possibility can be realized, but the situation can be richer compared to the FM case. In particular, this peculiarity is manifested differently in the density of states for the upper and lower bands. It happens that the standard scheme of extrema (a minimum in the centre of the Brillouin zone, maxima on the edges of the Brillouin zone, saddle points in for symmetrical points like X(1,0)) take place only for the upper branch and only at large enough field. In this case in the upper band there are only three Van Hove singularities (see Fig. 4b,c). For the upper branch the point ⃗ q = 0 at any field always defines the global minimum and peculiarities like D ( ω ) ∝ ( ω − ω$_{0}$ )Θ( ω − ω$_{0}$ ) define the density of states near the lower edge of the upper band. For all fields the absolute maximum of the function ω$_{+}$ ( ⃗ k ) having a parabolic dependence is located at the corners of the Brillouin zone at points such as M (1,1). For this reason, 20