-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) 7 phase is replaced by a quasi -FM configuration in the in- termediate temperature range T$_{N}$ ( n ) < T < T$_{C}$ ( n ) . As a final issue we address the problem of the order of the transitions observed at T$_{N}$ ( n ) and T$_{C}$ ( n ), respec- tively. In particular, we focus our attention to the thick- ness ranges where the chiral order parameter is relevant, i.e. regions i ) and ii ) as defined at the beginning of this Section. In Fig. 10 the equilibrium probability dis- tribution of the energy for temperatures around T$_{N}$ (8) (Fig. 10a) and T$_{C}$ (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 Diep $^{17,18}$, the presence of a first-order transition at T$_{N}$ ( 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. 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 FIG. 10: (colors online) Equilibrium probability distribution of the energy for the thickness n = 8 for some temperatures around T$_{N}$ (8), (a) , and T$_{C}$ (8), (b) , respectively. 134420 (2009). 1 Frustrated spin Systems , edited by H. T. Diep (World Sci- entific, 2004). 2 H. Kawamura, J. Phys.: Cond. Matt. 10 , 4707 (1998). 3 T. Kimura et al. , Nature (London) 426 , 55 (2003). 4 F. Cinti et al. , Phys. Rev. Lett. 100 , 057203 (2008). 5 J.H. Park, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. Lett. 101 , 167202 (2008), and references therein. 6 S. W. Cheong and M. Mostovoy, Nature Materials (Lon- don) 6 , 13 (2007). 7 Minhyea Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, Phys. Rev. Lett. 102 , 186601 (2009) 8 P. Pedrazzini et al. , Phys. Rev. Lett. 98 , 047204 (2007). 9 H. Kawamura and M. S. Li, Phys. Rev. Lett. 87 , 187204 (2001). 10 P. J. Jensen, and A. R. Mackintosh, Rere Earth Mag- netism (Structure and Excitations) , Clarendon Press, Ox- ford (1991). 11 S. Konings, C. Schuessler-Langeheine, H. Ott, E. Weschke, E. Schierle, J. B. Goedkoop, arXiv 0707.2765v2 12 P.J. Jensen, and K.H. Bennemann, Surface Science Re- ports 61 , 129 (2006). 13 E. Weschke, et al. , Phys. Rev. Lett. 93 , 157204 (2004). 14 F. Cinti, A. Cuccoli, and A. Rettori, Phys. Rev. B 78 , 020402(R) (2008). 15 F. Cinti, A. Cuccoli, and A. Rettori, Phys. Rev. B 79 , 26 Such observable has been obtained from instantaneous evaluation of the structure factor during the stochastic process, and subsequently statistically analyzed as all the other macroscopic quantities. 25 K. Binder, Z. Phys. B 43 , 119 (1981). K. Binder, Phys. Rev. Lett. 47 , 693 (1981). 16 J. Bohr D. Gibbs, J. D. Axe, D. E. Moncton, K. L. D’Amico, C. F. Majkrzak, J. Kwo, M. Hong, C. L. Chien, and J. Jensen, Physica B 159 , 93 (1989). 17 H. T. Diep, Phys. Rev. B 39 , 397 (1989). 18 D. Loison, Physica A 275 , 207 (2000). 19 N. Metropolis, et al. , J. Chem. Phys. 21 , 1087 (1953). 20 F. R. Brown and T. J. Woch, Phys. Rev. Lett. 58 , 2394 (1987). 21 D. P. Landau, and K. Binder, A Guide to Monte Carlo Simulation in Statistical Physics , Cambridge University Press, Cambridge (2000). 22 M. E.J. Newman, and G. T. Barkema, Monte Carlo Meth- ods in Statistical Physics , Clarendon Press, Oxford (1999). 23 B. Efron, The Annals of Statistics 7 , 1 (1979). 24 P. M. Chaikin, T. C. Lubensky Principles of condensed matter physics , Cambridge University Press, New York (1995).