6
0 20 40 60 80 100 120 140
T (K)
0
10
20
30
 ∆ϕ l,l+1 ( T )   (deg.)
T N (16)T N (8)
FIG. 7: (color online) ∆ ϕl(T ) vs. temperature for the surface
planes, l = 1 (triangles), l = 2 (squares), l = 3 (diamonds),
l = 4 (circles). Straight lines and full symbols: n = 8. Dashed
lines and open symbols: n = 16.
ﬁlm leads to an eﬀective 2d-like trend. Region ii) looks
however more intriguing, and requires a more accurate
discussion, which can beneﬁt from a careful comparison
of the behaviour of a given quantity in regionsi) and ii).
For this purpose, we look at the temperature depen-
dence of the rotation angle of the magnetization between
NN planes. In Fig. 7, ∆ϕl(T ) for n = 8 and n = 16
(continuous and dashed lines, respectively), is plotted for
the outermost planes,l = 1 . . . 4. For both thicknesses, a
monotonic trend is observed for all l, but at variance with
what happens for the highest thickness, for n = 8 we see,
starting from a temperature T ≲ TN (8), an abrupt drop
of ∆ ϕ3 and ∆ ϕ4, which rapidly reach an almost con-
stant value, only slightly larger than ∆ ϕ1. In the tem-
perature range TN (8) ≲ T < T C (8) we thus substantially
observe the same small magnetic phase shifts between all
NN layers, testifying an energetically stablequasi-FM
conﬁguration giving no contribution to the helical order
parameters. The latter point can be made clearer by
looking at the the peak positionQz,max of the structure
factor S(0, 0, qz). In Fig. 8 the average of Qz,max vs T is
reported, again for n = 8 and for diﬀerent lateral dimen-
sions L26. As expected from the previous argument, we
see that Qz,max = 0 for TN (8) < T < T C (8), while it be-
gins to shift to higher values as soon as the temperature
decreases belowTN (8), making apparent a progressive
fan stabilization with Qz,max ̸= 0 and reaching a value
of about 21 ◦ for T = 10 K.
In a previous study, where the magnetic properties of
Ho thin ﬁlms were investigated by MC simulations of a
Heisenberg model with easy-plane single-ion anisotropy
and six out-of-plane coupling constants (as obtained by
experimental neutron scattering measurements16) on a
HCP lattice14,15, it was found that for thicknesses compa-
rable with the helical pitch the phase diagram landscape
is quite diﬀerent from what we ﬁnd here. Indeed, for
n= 9 − 16, three diﬀerent magnetic phases could be sin-
0 20 40 60 80 100
T (K)
0
5
10
15
20
25
30
Q z, max (deg.)
FIG. 8: (color online) Qz, position of the maximum of S(⃗ q),
vs. temperature for thickness n = 8. Inset: magnetic vector
(mx
l, my
l) proﬁle for some temperatures for L = 64. Colors
and symbols as in Fig. 2.
0 1 2 3 4 5 6 7 8 9 10 11 12
l
0
20
40
60
80
∆ϕ l (deg)
T =100K
     130K
     135K
     140K
     145K
FIG. 9: ∆ ϕl for a BCT lattice and n = 12, when the six
coupling constants set employed in Ref. 14,15 (see text) is
used. The temperature range has been chosen aroundTC (n)
(error bars lye within point size).
gled out, with the high-temperature, paramagnetic phase
separated from the low-temperature, long-range ordered
one, by an intermediate-temperature block phase where
outer ordered 4-layers blocks coexist with some inner dis-
ordered ones. Moreover, it was observed that the phase
transition of such inner layers turns out to have the sig-
natures of a Kosterlitz-Thouless one.
The absence of the block phase in the J1 − J2 model
here investigated has to be attributed to the diﬀerent
range of interactions, rather than to the diﬀerent lattice
structure. We came to this conclusion by doing some
simulations using the same set of interaction constants
employed in Refs. 14,15, but using a BCT lattice: the
results we obtained for ∆ϕl with n = 12 are reported in
Fig. 9. The latter is absolutely similar to Fig.7 of Ref. 15
and clearly displays the footmarks of the block phase (see
down-triangle), with two external blocks of ordered layers
(l =1. . . 5 and 8. . . 12 ), where ∆ ϕl is roughly 10 ◦, sep-
arated by a block of disordered layers, and with almost