5 0 1 2 3 4 5 6 0 5 10 15 20 l (deg.) T =10K T =20K T =30K T =40K T= 50K 0 1 2 3 4 5  l 0 1 2 3 4 5 (a) n =  6 (b) n =  5 0 2 4 6 8 10 12 14 16 18 20  n 0 20 40 60 80 100 120 140 T N  (n) , T C  (n) (K) T$_{N}$ (n) T$_{C}$ (n) T$_{N}$ bulk FIG. 6: Rotation angle Δ ϕ$_{l}$ between magnetic moments on NN layers ( l + 1 , l ) at some low temperatures, for thickness n = 5 and n = 6, and lateral dimension L = 64. FIG. 5: Transition temperatures T$_{N}$ ( n ) and T$_{C}$ ( n ) vs. film thickness n . Δ ϕ$_{l}$ = ϕ$_{l}$$_{+1}$ − ϕ$_{l}$ = arccos [ M x l M x l +1 + M y l M y l +1 ] (10) IV. DISCUSSION AND CONCLUSION angle of the magnetization between nearest planes: the same is true for the crossing point of the Binder cu- mulant of the average magnetization M (not reported in figure), which is located at T$_{C}$ (8) = 133 . 3(3) K. These data give a first rough indication that also for n = 8 all the planes of the sample are still ordering almost at the same temperature; such property has been observed for all the investigated thicknesses n below 16, so that T$_{C}$ ( n ) results quite n -independent (see also Fig. 5) . where ( M x l , M y l ) is the magnetic vector profile for each plane l . Δ ϕ$_{l}$ is displayed in Fig. 6a and Fig. 6b, for n = 6 and n = 5, respectively. In Fig. 6a, a quite clear fan stabilization is observed when the temperature de- creases, while in Fig. 6b, i.e. for n = 5, Δ ϕ$_{l}$ keeps an almost temperature independent very small value; what’s more, Δ ϕ$_{l}$ seems to loose any temperature dependence as T = 0 is approached. We attribute the absence of fan arrangement for n ≤ 5 as simply due to the lack of “bulk planes” inside the film, so that we are left with only a 2d trend at T$_{C}$ ( n ), i.e. at the temperature where the order parameters defined in Eqs. (2) and (3) show a critical behaviour. Although the layer subtraction does not seem to mod- ify T$_{C}$ ( n ), the onset of helical arrangement is observed to shift at lower temperatures as n decreases. The chirality κ defined in Eq. (4) is reported in Fig 4b for n = 8. As the temperature decreases, around T ∼ 80 K we can identify a finite-size behaviour of κ which, at variance with the previous one, can be easily recognized as typical of an effective phase transition. Such conclusion is confirmed by the analysis of the chiral susceptibility χ$_{κ}$ (Fig. 4c), which for the largest L has a maximum at T = 85 K. As- suming that the order parameter (4) is the relevant one to single out the onset of the fan arrangement, we can get a more accurate estimate of T$_{N}$ (8) by looking at the Binder cumulant u$_{4}$ ( κ ), reported in Fig. 4d. By making use of the MH technique, we locate the crossing point at T$_{N}$ (8) = 92(2) K. Finally, it is worthwhile to observe as the specific heat does not show any anomaly at T$_{N}$ (8), being the entropy substantially removed at T$_{C}$ (8). A possible framework to analyze the results presented in the previous Section is suggested by Fig. 5, where we can easily distinguish three significant regions: i ) high thickness, n ⩾ 16, where the films substantially display a bulk behaviour, with the single planes ordering tempera- ture coinciding with the helical phase transition one; ii ) intermediate thickness, 6 ≤ n ≲ 15, where the tempera- ture corresponding to the onset of in-plane order, T$_{C}$ ( n ), is still ≃ T Ho N , but where the helical/fan arrangement sta- bilizes only below a finite temperature T$_{N}$ ( n ) < T$_{C}$ ( n ); iii ) low thickness,1 ≤ n ≤ 5, where T$_{C}$ ( n ) ≲ T Ho N but no fan phase is present at any temperature. The scenario just outlined for n = 8 results to be cor- rect in the thickness range 6 ≤ n ≲ 15, where a clear separation between T$_{N}$ ( n ) and T$_{C}$ ( n ) can be easily fig- ured out. In such temperature window, the strong sur- face effects produce a quasi -FM set-up of the magnetic film structure along the z -direction. While leaving to the next Section a more detailed discussion of this regime, we report in Fig. 5 a plot of T$_{N}$ ( n ) and T$_{C}$ ( n ) vs. n for all the simulated thicknesses. The separation between the two critical temperatures is maximum for n = 6, where T$_{N}$ (6) = 38(4), that is T$_{N}$ (6) ∼ 1 $_{3}$T$_{C}$ (6). For films with less than six layers no fan order is observed, i.e. for n = 5 and below the chirality does not display any typical fea- ture of fan ordering at any temperature below T$_{C}$ ( n ). As a representative quantity we finally look at the rotation The observed behaviour in region iii ) can be reason- ably attributed to the decreasing relevance of the con- tribution to the total energy of the system coming from the competitive interactions among NNN planes as the film thickness decreases; moreover, the thinness of the