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. 5: Transition temperatures TN (n) and TC (n) vs. ﬁlm
thickness n.
the same is true for the crossing point of the Binder cu-
mulant of the average magnetizationM (not reported in
ﬁgure), which is located at TC (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 thicknessesn below 16, so that TC (n)
results quite n-independent (see also Fig. 5) .
Although the layer subtraction does not seem to mod-
ify TC (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 susceptibilityχκ (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 ofTN (8) by looking at the
Binder cumulant u4(κ), reported in Fig. 4d. By making
use of the MH technique, we locate the crossing point at
TN (8) = 92(2) K. Finally, it is worthwhile to observe as
the speciﬁc heat does not show any anomaly at TN (8),
being the entropy substantially removed at TC (8).
The scenario just outlined for n = 8 results to be cor-
rect in the thickness range 6 ≤ n ≲ 15, where a clear
separation between TN (n) and TC (n) can be easily ﬁg-
ured out. In such temperature window, the strong sur-
face eﬀects produce aquasi-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 ofTN (n) and TC (n) vs. n for all
the simulated thicknesses. The separation between the
two critical temperatures is maximum forn = 6, where
TN (6) = 38(4), that is TN (6) ∼ 1
3 TC (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 belowTC (n). As
a representative quantity we ﬁnally look at the rotation
0 1 2 3 4 5 6
0
5
10
15
20
 ∆ϕ 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
FIG. 6: Rotation angle ∆ ϕl 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.
angle of the magnetization between nearest planes:
∆ ϕl = ϕl+1 − ϕl = arccos
[
Mx
l Mx
l+1 + My
l My
l+1
]
(10)
where ( Mx
l , M y
l ) is the magnetic vector proﬁle for each
plane l. ∆ ϕl 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. forn = 5, ∆ ϕl keeps an
almost temperature independent very small value; what’s
more, ∆ϕl 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 atTC (n), i.e. at the temperature where the order
parameters deﬁned in Eqs. (2) and (3) show a critical
behaviour.
IV. DISCUSSION AND CONCLUSION
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-