3
dependence of diﬀerent samples during the measurement
stage. For each temperature we have usually performed
three independent simulations, each one containing at
least 2×105 measurements, taken after discarding up to
5×104 Monte Carlo steps in order to assure thermal equi-
libration.
In the proximity of the critical region the multiple his-
togram (MH) technique was also employed 21, as it allows
us to estimate the physical observables of interest over a
whole temperature range in a substantially continuous
way by interpolating results obtained from sets of simu-
lations performed at some diﬀerent temperatures.
For all the quantities of interest, the average value and
the error estimate were obtained by the bootstrap re-
sampling method22 given that, as pointed out in Ref. 23,
for a large enough number of measurements, this method
turns out to be more accurate than the usual blocking
technique. In our implementation, we pick out randomly
a sizable number of measurements (typically, between 1
and 1×103 for the single simulation, and between 1 and
5×104 for the MH technique), and iterate the re-sampling
at least one hundred times.
The thermodynamic observables we have investigated
include the FM order parameter for each plane l:
ml =
√
(mx
l)2 + (my
l)2 , (2)
which is related to the SO(2) symmetry breaking. At the
same time, it turns out to be signiﬁcant also the average
order parameter of the ﬁlm, deﬁned as
M = 1
n
n∑
l=1
ml . (3)
Turning to the helical order, which is the relevant
quantity for the Z2 × SO(2) symmetry, we can explore
it along two diﬀerent directions. The ﬁrst one is by the
introduction of the chirality order parameter1,2
κ = 1
4(n − 1)L2 sin Qz
∑
⟨ij⟩
[
Sx
i Sy
j − Sy
i Sx
j
]
, (4)
where the sum refers to spins belonging to NN layers
iand j, respectively, while Qz is the bulk helical pitch
vector along the z direction. The second possibility is
that of looking at the integral of the structure factor:
MHM = 1
K
∫ π
0
dqzS(⃗ q) (5)
where S(⃗ q), with ⃗ q= (0 , 0, qz), is the structure factor 24
(i.e. the Fourier transform of the spin correlation func-
tion) along the z-direction of the ﬁlm, while the normal-
ization factorK is the structure factor integral at T = 0.
Although the use of the last observable can be seen as a
suitable and elegant way to overcome the intrinsic diﬃ-
culties met in deﬁning a correct helical order parameter,
free of any undue external bias (as the wave-vectorQz
0 20 40 60 80 100 120 140
T  (K)
0
0.5
1
1.5
2
2.5
c v  / k B
L = 24
L = 32
L  = 48
L  = 64
20 30 40 50 60 70
2.1
2.2
2.3
2.4
2.5
2.6
c v, max
L
FIG. 2: (color online) Speciﬁc heat cv per spin vs. temper-
ature for thickness n = 16 (for lateral dimension, see the
legend inside the ﬁgure). Inset: Maximum of cv vs. L ob-
tained through MH technique. The continuum red line is a
power law ﬁt.
entering the deﬁnition ofκ in Eq. (4)), we remind that
such quantity has generally to be managed with particu-
lar care, as discussed in details in Refs.14,15, where it was
shown that the presence of block structures prevents us
to unambiguously relate the evolution ofS(⃗ q) with the
onset of helical order. However, for the speciﬁc case of
the model under investigation such integrated quantity
can still be considered a fairly signiﬁcant order parame-
ter, as no block structures emerge from the simulations
(see below).
In order to get a clear picture of the critical region and
to give an accurate estimate of the critical temperature,
we look also at the following quantities
cv = nL2β2 (
⟨e2⟩ − ⟨e⟩2)
, (6)
χo = nL2β
(
⟨o2⟩ − ⟨o⟩2)
, (7)
∂β o = nL2 (⟨oe⟩ − ⟨o⟩⟨e⟩) , (8)
u4(o) = 1 − ⟨o4⟩
3⟨o2⟩2 , (9)
where β = 1 /kBT , and o is one of the relevant observ-
ables, i.e. ml, M, κ, M HM . In this paper, we shall mainly
locate the critical temperature by looking at the intersec-
tion of the graphs of the Binder cumulant25, Eq. (9), as a
function of T obtained at diﬀerent L. For clarity reasons,
we introduce also the following symbols: by TN (n) we