text: z 
  
  ~Si × ~Sj 
  h 
  . 
  
  The complex phase diagram unveiled by such MC sim- 
  ulations awaken however a further intriguing question: 
  to what extent the observed behavior may be considered 
  a simple consequence of the competition between helical 
  order and surface eﬀects? I.e., is it just a matter of hav- 
  ing such a competition or does the range of interactions 
  also play a relevant role? Indeed, when the range of the 
  interactions is large enough we have a greater number of 
  planes which can be thought of as ”surface planes”, i.e. 
  for which the number of interacting neighbors are sig- 
  niﬁcantly reduced with respect to the bulk layers; there- 
  fore, we expect that the larger the interaction range, the 
  stronger should be the surface eﬀects. But, at the same 
  time, the same modulation of the magnetic order can 
  
  i 
  
  For these rare-earth elements, the development of new 
  and sophisticated experimental methods11 has allowed to 
  obtain ultra-thin ﬁlms where the non-collinear modula- 
  tion is comparable with the ﬁlm thickness. Under such 
  conditions the lack of translational invariance due to the 
  presence of surfaces results decisive in order to observe 
  a drastic change of the magnetic structures12. Recent 
  experimental data on ultra-thin Holmium ﬁlms13 have 
  been lately interpreted and discussed14,15 on the basis 
  of detailed classical Monte Carlo (MC) simulations of a 
  spin Hamiltonian, which is believed to give a realistic 
  modeling of bulk Holmium. Such Hamiltonian, proposed 
  by Bohr et al.16, allows for competitive middle-range in- 