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 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 × 10 5 measurements, taken after discarding up to 5 × 10 4 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. FIG. 2: (color online) Specific heat c$_{v}$ per spin vs. temper- ature for thickness n = 16 (for lateral dimension, see the legend inside the figure). Inset: Maximum of c$_{v}$ vs. L ob- tained through MH technique. The continuum red line is a power law fit. For all the quantities of interest, the average value and the error estimate were obtained by the bootstrap re- sampling method 22 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 × 10 3 for the single simulation, and between 1 and 5 × 10 4 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 : entering the definition of κ 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 of S ( ⃗ 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). 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 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 Turning to the helical order, which is the relevant quantity for the Z$_{2}$ × SO (2) symmetry, we can explore it along two different directions. The first one is by the introduction of the chirality order parameter 1,2 where the sum refers to spins belonging to NN layers i and j , respectively, while Q$_{z}$ is the bulk helical pitch vector along the z direction. The second possibility is that of looking at the integral of the structure factor: where β = 1 /k$_{B}$T , and o is one of the relevant observ- ables, i.e. m$_{l}$, 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 cumulant $^{25}$, Eq. (9), as a function of T obtained at different L . For clarity reasons, we introduce also the following symbols: by T$_{N}$ ( n ) we will denote the helical/fan phase transition temperature for thickness n , T$_{C}$ ( n ) will instead indicate the order- ing temperature of the sample as deduced by looking at the behaviour of the average order parameter (3), while T l $_{C}$( n ) will be the l -th plane transition temperature re- lated to the order parameter defined in Eq. (2). where S ( ⃗ q ), with ⃗ q = (0 , 0 , q$_{z}$ ), 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 factor K 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-vector Q$_{z}$ c$_{v}$ = nL $^{2}$β2 ( 〈 e $^{2}$〉 − 〈 e 〉 $^{2}$) , (6) χ$_{o}$ = nL $^{2}$β 〈 o $^{2}$〉 − 〈 o 〉 , (7) ∂$_{β}$o = nL $^{2}$( 〈 oe 〉 − 〈 o 〉〈 e 〉 ) , (8) u$_{4}$ ( o ) = 1 − 〈 o $^{4}$〉 3 〈 o $^{2}$〉2 , (9) κ = 1 4( n − 1) L $^{2}$sin Q$_{z}$ ∑ 〈 ij 〉 [ S x i S y j − S y i S x j ] , (4) M$_{HM}$ = 1 K ∫ π 0 dq$_{z}$S ( ⃗ q ) (5) M = 1 n n ∑ l =1 m$_{l}$ . (3) m$_{l}$ = √ ( m x l ) 2 + ( m y l ) 2 , (2)