Thermodynamic Casimir Force in Models of
4
He Films
Based on vortex-loop renormalization group techniques
(VLRGT), Williams [1] has predicted the finite-size scal-
ing functions (SF) describing the excess free energy (EFE)
and the (thermodynamic) Casimir force (CF) for the three-
dimensional XY model in a film geometry with thickness L
and under periodic boundary conditions (PBC). These pre-
dictions concern the region close to and below the critical
temperature T
b
c
of the bulk system. Within this approach
the Casimir amplitude (CA), characterizing the film at T
b
c
,
is given as
XY
0:155 while the "-expansion (EE)
result [2] is
XY
0:20. Rescaling the high temperature
results from the EE [2] with the ratio of the above ampli-
tudes, Williams has constructed a SF for the force in the
whole critical region by patching together the EE curve
with the low temperature VLRGT results, which renders a
rather smooth matching at T
b
c
[1]. In Fig. 1 we compare this
approach with recent Monte Carlo (MC) data for the same
system [3]. The data for the VLRGT approach shown in
Fig. 1 are obtained from a numerical solution of the equa-
tions given in Ref. [1] by the same method. An inspection
of Fig. 1 and of Fig. 2 in Ref. [1] shows that the MC data
are at variance both qualitatively and quantitatively with
VLRGT for T & T
b
c
where one expects for ON 2 mod-
els to see the pronounced effect of the Goldstone modes
(GM). This is born out by the MC results, but is completely
missing within the VLRGT approach. Furthermore, if,
using the Williams approach, one matches the MC and
the VLRGT data at T T
c
, a new (artificial) cusplike
singularity will be created in the SF of the CF. Because
of the contribution due to the GM, the CF (and the EFE) SF
tends to a negative constant. The VLRGT approach, in its
present form [1], does not account for this effect. Note that
there is no theoretical proof available that the decomposi-
tion of the CF into a sum of a noncritical ‘‘Goldstone part’’
and a ‘‘critical part’’ to be covered by VLRGT (Fig. 1)
gives a good approximation to the total force.
Experimental data (ED) for
4
He films [4] require a
theoretical description based on Dirichlet boundary con-
ditions (DBC). Above T
b
c
the EE results [2] available for
this case are in good agreement with the ED. Because of
inherent difficulties in that case there are, however, neither
analytical nor MC results available for the critical region
below T
b
c
. Very recently, for DBC and T well below T
b
c
the
asymptotic regime of large L has been analyzed in Ref. [5].
These calculations take into account the capillary wavelike
surface fluctuations that occur on one of the bounding
surfaces of the actual
4
He wetting film studied in
Ref. [4]. One finds a suppression of the SF for x ! 1
which is comparable with the ED.
Whereas there are significant discrepancies between the
MC data and the VLRGT approach for PBC, there is
agreement between the actual ED [4] described by DBC
and corresponding theoretical descriptions within their
range of applicability, i.e., for T T
b
c
[2] and T T
b
c
[5]. It remains as a challenge to confirm and to understand
the deep minimum (ca. 50 times deeper than the value at
T
b
c
) of the experimental SF for temperatures slightly below
T
b
c
. For this temperature range for DBC there are presently
no theoretical predictions available.
D. Dantchev,
1,2,3
M. Krech,
2,3
and S. Dietrich
2,3
1
Institute of Mechanics
Bulgarian Academy of Sciences
Academic G. Bontchev Street 4
1113 Sofia, Bulgaria
2
Max-Planck-Institut fu¨r Metallforschung
Heisenbergstrasse 1
D-70569 Stuttgart, Germany
3
Institut fu¨r Theoretische und Angewandte Physik
Universita¨t Stuttgart
D-70569 Stuttgart, Germany
Received 18 June 2004; published 14 December 2005
DOI: 10.1103/PhysRevLett.95.259701
PACS numbers: 74.78.w, 64.60.Fr, 75.40.s
[1] G.A. Williams, Phys. Rev. Lett. 92, 197003 (2004).
[2] M. Krech and S. Dietrich, Phys. Rev. A 46, 1886 (1992);
46, 1922 (1992).
[3] D. Dantchev and M. Krech, Phys. Rev. E 69, 046119
(2004).
[4] R. Garcia and M. H. W. Chan, Phys. Rev. Lett. 83, 1187
(1999).
[5] R. Zandi, J. Rudnick, and M. Kardar, Phys. Rev. Lett. 93,
155302 (2004).
[6] M. Krech, Phys. Rev. E 56, 1642 (1997).
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0
-10 -7.5 -5.0 -2.5 0 2.5 5.0 7.5 10
x = L /
MC,L = 12
MC,L = 16
MC,L = 20
MC,L = 24
vortex loop [1]
- expansion [2]
spherical model [4]
ξ
θ
(x) / 2
per
ε
FIG. 1 (color). Scaling function
per
x L==2 of the CF for
the d 3XY model in a slab obtained from MC [3] and EE [2]
for PBC. Here T T
b
c
is the bulk correlation length
with
t T T
b
c
=T
b
c
& 0
0
jtj
, ’ 0:67. In order
to be able to make contact with Ref. [1], as a measure of the
temperature below T
b
c
we define T T
b
c
jtj. The
vertical scale of the MC data has been adjusted such that there is
agreement with the estimate
per
0 2
XY
0:56 [3] which
is obtained by using EE results for the ratio
Ising
=
XY
of the CA
of the Ising and the XY model, and the best known numerical
value of
Ising
’0:153 [6]. Note that both the (properly
normalized) [3] EE and the MC results practically coincide
with the (exact) spherical model results for x * 2, which for
PBC renders the limit N !1 of the ON models.
PRL 95, 259701 (2005)
PHYSICAL REVIEW LETTERS
week ending
16 DECEMBER 2005
0031-9007= 05=95(25)=259701(1)$23.00 259701-1 © 2005 The American Physical Society