7
-140 -139 -138 -137 -136 -135 -134 -133 -132 -131
e
0
0.5
1
 P e
90K
91K
92K
93K
94K
95K
-94 -92 -90 -88 -86 -84 -82 -80 -78
e
0
0.2
0.4
P e
129K
130K
131K
132K
133K
134K
T C (8) = 133.3(3)K
T N (8) = 92(2)K
(a)
(b)
FIG. 10: (colors online) Equilibrium probability distribu tion
of the energy for the thickness n = 8 for some temperatures
around TN (8), (a), and TC (8), (b), respectively.
opposite magnetization. We can thus conﬁdently assert
that, regardless of the underlying lattice structure, by
decreasing the number of the out-of-plane interactions,
for thicknesses close to the helical bulk pitch, the block
phase is replaced by a quasi-FM conﬁguration in the in-
termediate temperature range TN (n) < T < T C (n) .
As a ﬁnal issue we address the problem of the order
of the transitions observed at TN (n) and TC (n), respec-
tively. In particular, we focus our attention to the thick-
ness ranges where the chiral order parameter is relevant,
i.e. regionsi) and ii) as deﬁned at the beginning of
this Section. In Fig. 10 the equilibrium probability dis-
tribution of the energy for temperatures aroundTN (8)
(Fig. 10a) and TC (8) (Fig. 10b) is plotted: for both
temperatures, no double peak structure is observed, so
that we have no direct indication for a ﬁrst order tran-
sition even if, according to precedent studies of Loison
and Diep17,18, the presence of a ﬁrst-order transition at
TN (n), cannot be completely excluded, as it could reveal
itself only when the lateral dimension L are much larger
than the largest correlation length. The same conclusion
about the order of transition is reached for any other in-
vestigated ﬁlm thickness, as the energy probability distri-
bution shape does not qualitatively change. This ﬁndings
agree with the results we got in previous MC simulations
discussed in Ref. 15, so that we may conclude that the
order of the observed transitions is not aﬀected by the
range of interactions.
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