text: ~Si are classical planar unit vectors representing the di- 
  rection of the total angular momentum of the magnetic 
  ions, whose magnitude 
  j(j + 1) (j = 8 for Holmium 
  ions) is already encompassed within the deﬁnition of the 
  interaction constants J0,1,2. As sketched in Fig. 1, the 
  magnetic ions are located on the sites of a body-centered 
  tetragonal (BCT) lattice; the ﬁrst sum appearing in the 
  Hamiltonian describes the in-plane (xy) nearest neigh- 
  bor (NN) interaction, which is taken ferromagnetic (FM), 
  with exchange strength J0 > 0; the second sum rep- 
  resents the coupling, of exchange strength J1, between 
  spins belonging to nearest neighbor (NN) planes along 
  the z-direction (which we will assume to coincide with 
  the ﬁlm growth direction); ﬁnally, the third sum takes 
  into account the interaction, of exchange strength J2, be- 
  tween spins lying on next-nearest neighbor (NNN) planes 
  along z. In order to have frustration, giving rise to non- 
  collinear order along z in the bulk, NN interaction J1 
  can be taken both ferro- or antiferromagnetic, but NNN 
  coupling J2 has necessarily to be antiferromagnetic, and 
  the condition |J2| > |J1|/4 must be fulﬁlled. Such simpli- 
  ﬁed Hamiltonian was already employed to simulate he- 
  lical ordering in bulk systems by Diep1,17 and Loison18. 
  In the bulk limit, the state of minimal energy of a sys- 
  tem described by Eq.(1) corresponds to a helical arrange- 
  ment of spins. The ground state energy per spin is equal 
  to eg(Qz) = [−4J0 − 2J1 (4 cos (Qzc′) + δ cos (2Qzc′))] 
  is the distance between NN layers, δ = J2 
  where c′ 
  J1 , 
  and Qzc′ = arccos 
  is the angle between spins ly- 
  ing on adjacent planes along the z-direction. The ob- 
  served helical arrangement in bulk holmium corresponds 
  to Qzc′ ≃ 30.5◦10: 
  such value can be obtained from 
  the formula above with the set of coupling constants 
  J0=67.2 K, J1=20.9 K, and J2 = −24.2 K, that we have 
  employed in our simulations. The given values for the ex- 
  change constants are the same already used by Weschke 
  et al. 
  in Ref. 13 to interpret experimental data on 
  Holmium ﬁlms on the basis of a J1 − J2 model, after 
  a proper scaling by the numbers of NN and NNN on 
  neighboring layers of a BCT lattice. 
  
  p 