content_image/331416.jpg content_image/331415.jpg content_image/331414.jpg content_image/331413.jpg content_image/331412.jpg
3 denote the projections on the set of orts, for example, ⃗ e$_{x}$ , ⃗ e$_{y}$ , ⃗ e$_{z}$ , for ⃗ m = m$_{0}$⃗ e$_{z}$ or ⃗ e$_{x}$ , − ⃗ e$_{y}$ , − ⃗ e$_{z}$ for ⃗ m = − m$_{0}$⃗ e$_{z}$ . 

## III. FERROMAGNETIC STATE OF ARRAY. 

For the FM state of the dot array ( ⃗ m = ⃗ e$_{z}$ for all dots) the same set of orts in Eq. (2), ⃗ e$_{1}$ = ⃗ e$_{x}$ , ⃗ e$_{2}$ = ⃗ e$_{y}$ and ⃗ e$_{3}$ = ⃗ e$_{z}$ should be used. In the quadratic approximation over the operators a † ⃗ l and a$_{⃗}$ l the Hamiltonian reads 

<img src='content_image/331416.jpg'>

where ⃗ δ = l$_{x}$⃗ e$_{x}$ + l$_{y}$⃗ e$_{y}$ is a dimensionless lattice vector, H$_{a}$ = κm$_{0}$ is the anisotropy field, M = m$_{0}$/a 3 is the characteristic value defining the dipolar interaction intensity and having the same dimension as usual the 3D magnetization. The collective modes are introduced via states a$_{k}$ and a † k of definite quasi-momentum ⃗ k 

<img src='content_image/331415.jpg'>

where N is the total number of dots in an array. The collective modes are defined by the quadratic Hamiltonian over a$_{k}$ , a † k which acquires the standard form 

<img src='content_image/331414.jpg'>

When the form of coefficients A$_{k}$ and B$_{k}$ are established, the collective excitation en- ergy ε ( ⃗ k ) = ℏ ω ( ⃗ k ) may be found by means of u − v Bogolyubov transformations (see, for example, $^{35}$) and universally reads 

<img src='content_image/331413.jpg'>

where γ = 2 µ$_{B}$/ ℏ is the gyromagnetic ratio. The concrete forms of A$_{k}$ and B$_{k}$ are defined by the distribution of the magnetic moments within the array. For the case of interest (parallel ordering of dot magnetization) one can find 

<img src='content_image/331412.jpg'>

