4
u 4  (M)
0.62
0.64
0.66
130 131 132 133 134 135 136 137 138
T (K)
0.5
0.55
0.6
0.65
u 4  (M HM )
0 0.2 0.4 0.6 0.8 q z
S(q z ) (a.u.)
(a)
(b)
0.66
0.64
0.62
FIG. 3: (color online) Binder cumulants at thickness n =
16, colors as in Fig. 2. (a): Binder cumulant for the order
parameter deﬁned in Eq. (3). (b): Binder cumulant extracted
from the integral of the structure factor (see Sec. II). Inse t:
structure factor for L = 64 between T = 131 K (upper curve)
and T = 140 K (lower), with 1 K temperature step.
III. RESULTS
The results obtained by MC simulations of the model
introduced in Sec. II will be presented starting from
n= 16, i.e. the highest investigated ﬁlm thickness which
still displays a bulk-like behaviour. In Fig. 2 the spe-
ciﬁc heat for samples withn = 16 and lateral dimension
L = 24 , 32, 48, 64 is shown. The location of the speciﬁc
heat maximum shows a quite deﬁnite evolution toward
the bulk transition temperature,T Ho
N ≃ 132 K10 (it is
worthwhile to note that for this XY model the mean ﬁeld
theory predicts a critical temperature T Ho
N,MF ≃ 198 K).
The intensity of the maximum of cv has been analyzed
by the MH technique for the same lateral dimensions (see
inset of Fig. 2): it clearly appears as it increases withL
in a smooth way.
The Binder cumulant for the average order parameter
deﬁned in Eq. (3) was obtained close to the cv peak and is
reported in Fig. 3a; its analysis leads to an estimate of the
critical temperature of the sample (given by the location
of the common crossing point of the diﬀerent curves re-
ported in the ﬁgure) ofTC (16) = 133 .2(5) This value can
be considered in a rather good agreement with the exper-
imental ordering temperature of HolmiumT Ho
N , the rel-
ative diﬀerence being about 1%. Even such a mismatch
betweenT Ho
N and TC (16) could be completely eliminated
by slightly adjusting the in-plane coupling constant J0,
but, as discussed in Sec. II, we shall preserve the value
reported in Refs. 13, and 12 in order to allow for a correct
comparison with the results reported in those papers.
The development of the helical arrangement of magne-
tization along the ﬁlm growth direction was investigated
by looking at the integral of the structure factorS(⃗ q)
along the z-direction, i.e. by taking ⃗ q= (0 , 0, qz), and
making again use of the cumulant analysis in order to
locate the helical transition temperature atTN (16) =
20 40 60 80 100 120 140
T (K)
χ κ (a.u.)
0
0.2
0.4
0.6
κ
0
0.5
1
1.5
2
c v  / k B
20 40 60 80 100 120 140
T (K)
0.1
0.2
0.3
0.4
0.5
0.6
u 4 (κ)
132 134 136
T (K)
∂ β κ  (a.u.)
(a)
(c)
(b)
(d)
FIG. 4: (color online) Thermodynamic quantities obtained f or
thickness n = 8 in the temperature range 0-150 K. Colors and
symbols as in Fig. 2. (a): speciﬁc heat; (b): chirality order
parameter. (c): susceptibility χκ . (d): Binder cumulant for
κ.
133.1(3) K (see Fig. 3b). The crossing points of the
Binder’s cumulants of the helical order parameter imme-
diately appear to be located, within the error bars, at the
same temperature of those for the average magnetization
previously discussed. In addition, it is worthwhile to ob-
serve that the peak evolution ofS(0, 0, qz), in particular
close to TN (16) (inset of Fig. 3b), displays the typical
behaviour expected for an helical structure. We can thus
conclude that forn = 16, as it is commonly observed
in bulk samples, the establishment of the in-plane order
coincides with onset of the perpendicular helical arrange-
ment atTN (16). However, due to helix distortion in the
surface regions, the maximum of S(0, 0, qz) stabilizes at
values of qz sensibly smaller (e.g. Qz(TN (16)) ≈ 16◦,
and Qz(T = 10 K) ≈ 28◦) with respect to the bulk one
(QHo
z= 30 .5◦).
The MC simulations outcomes for n = 16 we just pre-
sented appear quite diﬀerent with respect to those ob-
tained at the same thickness for the model with six cou-
pling constants along thez direction14,15. Indeed, for
the J1-J2 model here investigated, we observe that all