FIG. 8: (color online) Q$_{z}$ , position of the maximum of S ( ⃗ q ), vs. temperature for thickness n = 8. Inset: magnetic vector ( m x l , m y $_{l}$) profile for some temperatures for L = 64. Colors and symbols as in Fig. 2. FIG. 7: (color online) Δ ϕ$_{l}$ ( T ) vs. temperature for the surface planes, l = 1 (triangles), l = 2 (squares), l = 3 (diamonds), l = 4 (circles). Straight lines and full symbols: n = 8. Dashed lines and open symbols: n = 16. FIG. 9: Δ ϕ$_{l}$ for a BCT lattice and n = 12, when the six coupling constants set employed in Ref. 14,15 (see text) is used. The temperature range has been chosen around T$_{C}$ ( n ) (error bars lye within point size). 6 0 20 40 60 80 100 120 140 T  (K) 0 10 20 30 l,l+1 ( T ) (deg.) T$_{N}$(16) T$_{N}$(8) 0 20 40 60 80 100 T  (K) 0 5 10 15 20 25 30 Q z, max (deg.) 0 1 2 3 4 5 6 7 8 9 10 11 12 l 0 20 40 60 80 Δϕ$^{l}$ (deg) T =100K 130K 135K 140K 145K film leads to an effective 2d-like trend. Region ii ) looks however more intriguing, and requires a more accurate discussion, which can benefit from a careful comparison of the behaviour of a given quantity in regions i ) and ii ). gled out, with the high-temperature, paramagnetic phase separated from the low-temperature, long-range ordered one, by an intermediate-temperature block phase where outer ordered 4-layers blocks coexist with some inner dis- ordered ones. Moreover, it was observed that the phase transition of such inner layers turns out to have the sig- natures of a Kosterlitz-Thouless one. For this purpose, we look at the temperature depen- dence of the rotation angle of the magnetization between NN planes. In Fig. 7, Δ ϕ$_{l}$ ( T ) for n = 8 and n = 16 (continuous and dashed lines, respectively), is plotted for the outermost planes, l = 1 . . . 4. For both thicknesses, a monotonic trend is observed for all l , but at variance with what happens for the highest thickness, for n = 8 we see, starting from a temperature T ≲ T$_{N}$ (8), an abrupt drop of Δ ϕ$_{3}$ and Δ ϕ$_{4}$ , which rapidly reach an almost con- stant value, only slightly larger than Δ ϕ$_{1}$ . In the tem- perature range T$_{N}$ (8) ≲ T < T$_{C}$ (8) we thus substantially observe the same small magnetic phase shifts between all NN layers, testifying an energetically stable quasi -FM configuration giving no contribution to the helical order parameters. The latter point can be made clearer by looking at the the peak position Q$_{z,max}$ of the structure factor S (0 , 0 , q$_{z}$ ). In Fig. 8 the average of Q$_{z,max}$ vs T is reported, again for n = 8 and for different lateral dimen- sions L $^{26}$. As expected from the previous argument, we see that Q$_{z,max}$ = 0 for T$_{N}$ (8) < T < T$_{C}$ (8), while it be- gins to shift to higher values as soon as the temperature decreases below T$_{N}$ (8), making apparent a progressive fan stabilization with Q$_{z,max}$ = 0 and reaching a value of about 21 ◦ for T = 10 K. The absence of the block phase in the J$_{1}$ − J$_{2}$ model here investigated has to be attributed to the different range of interactions, rather than to the different lattice structure. We came to this conclusion by doing some simulations using the same set of interaction constants employed in Refs. 14,15, but using a BCT lattice: the results we obtained for Δ ϕ$_{l}$ with n = 12 are reported in Fig. 9. The latter is absolutely similar to Fig.7 of Ref. 15 and clearly displays the footmarks of the block phase (see down-triangle), with two external blocks of ordered layers ( l =1. . . 5 and 8. . . 12 ), where Δ ϕ$_{l}$ is roughly 10 $^{◦}$, sep- arated by a block of disordered layers, and with almost In a previous study, where the magnetic properties of Ho thin films were investigated by MC simulations of a Heisenberg model with easy-plane single-ion anisotropy and six out-of-plane coupling constants (as obtained by experimental neutron scattering measurements $^{16}$) on a HCP lattice $^{14,15}$, it was found that for thicknesses compa- rable with the helical pitch the phase diagram landscape is quite different from what we find here. Indeed, for n = 9 − 16, three different magnetic phases could be sin-