x z y J J J 1  0 2  2 FIG. 1: (colors online) (a) : body-centered tetragonal (BCT) lattice with J$_{0}$ in-plane coupling constant, and out-of-plane J$_{1}$ , and J$_{2}$ competing interactions. be achieved with different number of interacting layers: notably, nearest and next-nearest layers competitive in- teractions are enough to get a helical structure with a whatever pitch wavevector. Such observation gives us a possible way to solve the conundrum previously emerged, as we have the possibility of varying the range of inter- actions without modifying the helical pitch, thus decou- pling the two relevant length scales along the film growth direction, and making accessible a range of n of the or- der of, or smaller than, the helical pitch, but still large enough that a substantial number of layers can behave as “bulk” layers. Therefore, while in the previous papers we have studied the properties of ultrathin magnetic films of Ho assuming a model with six interlayer exchange in- teractions, here we investigate by MC simulations the properties of the same system by making use of the sim- plest model Hamiltonian able to describe the onset of a helical magnetic order in Holmium, i.e. we consider only two inter-layer coupling constants, as previously done in Ref. 11. ⃗ S$_{i}$ are classical planar unit vectors representing the di- rection of the total angular momentum of the magnetic ions, whose magnitude √ j ( j + 1) ( j = 8 for Holmium ions) is already encompassed within the definition of the interaction constants J$_{0}$$_{,}$$_{1}$$_{,}$$_{2}$ . As sketched in Fig. 1, the magnetic ions are located on the sites of a body-centered tetragonal (BCT) lattice; the first sum appearing in the Hamiltonian describes the in-plane ( xy ) nearest neigh- bor (NN) interaction, which is taken ferromagnetic (FM), with exchange strength J$_{0}$ > 0; the second sum rep- resents the coupling, of exchange strength J$_{1}$ , between spins belonging to nearest neighbor (NN) planes along the z -direction (which we will assume to coincide with the film growth direction); finally, the third sum takes into account the interaction, of exchange strength J$_{2}$ , be- tween spins lying on next-nearest neighbor (NNN) planes along z . In order to have frustration, giving rise to non- collinear order along z in the bulk, NN interaction J$_{1}$ can be taken both ferro- or antiferromagnetic, but NNN coupling J$_{2}$ has necessarily to be antiferromagnetic, and the condition | J$_{2}$ | > | J$_{1}$ | / 4 must be fulfilled. Such simpli- fied Hamiltonian was already employed to simulate he- lical ordering in bulk systems by Diep 1,17 and Loison $^{18}$. In the bulk limit, the state of minimal energy of a sys- tem described by Eq.(1) corresponds to a helical arrange- ment of spins. The ground state energy per spin is equal to e$_{g}$ ( Q$_{z}$ ) = [ − 4 J$_{0}$ − 2 J$_{1}$ (4 cos ( Q$_{z}$c $^{′}$) + δ cos (2 Q$_{z}$c $^{′}$))] where c ′ is the distance between NN layers, δ = J$_{2}$ $_{J$_{1}$}$, and Q$_{z}$c ′ = arccos ( − 1 δ ) is the angle between spins ly- ing on adjacent planes along the z -direction. The ob- served helical arrangement in bulk holmium corresponds to Q$_{z}$c ′ ≃ 30 . 5 ◦ $^{10}$: such value can be obtained from the formula above with the set of coupling constants J$_{0}$ =67.2 K, J$_{1}$ =20.9 K, and J$_{2}$ = − 24.2 K, that we have employed in our simulations. The given values for the ex- change constants are the same already used by Weschke et al. in Ref. 13 to interpret experimental data on Holmium films on the basis of a J$_{1}$ − J$_{2}$ model, after a proper scaling by the numbers of NN and NNN on neighboring layers of a BCT lattice. The paper is organized as follows: In Sec. II the model Hamiltonian will be defined, and the MC techniques, and all the thermodynamic quantities relevant for this study, will be introduced. In Sec. III the results obtained for different thicknesses will be presented, both in the matter of the critical properties of the model and of the magnetic ordered structures observed. Finally, in Sec. IV we shall discuss such results, drawing also some conclusions. In the following we will denote with n the film thick- ness, i.e. the number of spin layers along the z direction, and with L × L the number of spins in each layer (i.e., L is the lattice size along both the x and y directions). In our simulations thickness values from 1 to 24 were con- sidered, while the range of lateral size L was from 8 to 64. Periodic boundary conditions were applied along x and y , while free boundaries were obviously taken along the film growth direction z . Thermal equilibrium was attained by the usual Metropolis algorithm $^{19}$, supplemented by the over- relaxed technique 20 in order to speed-up the sampling of the spin configuration space: a typical “Monte Carlo step” was composed by four Metropolis and four-five over-relaxed moves per particle. Such judicious mix of moves is able both to get faster the thermal equilibrium and to minimize the correlation “time” between succes- sive samples, i.e. the undesired effects due to lack of in- The model Hamiltonian we use in our simulations is the minimal one able to describe helimagnetic structures: II. MODEL HAMILTONIAN AND MONTE CARLO OBSERVABLES H = −   J$_{0}$ ∑ 〈 ij 〉 ⃗ S$_{i}$ · ⃗ S$_{j}$ + J$_{1}$ ∑ 〈 ik 〉 ⃗ S$_{i}$ · ⃗ S$_{k}$ + J$_{2}$ ∑ 〈 il 〉 ⃗ S$_{i}$ · ⃗ $_{S$_{l}$}$  . (1)