Abstract
Little is known about finite-size effects of critical
systems below the bulk transition temperature $T_c$
for realistic boundary conditions, such as of Dirichlet
or Neumann type.
However, the exactly solvable mean spherical model can
play a significant role in elucidating finite-size
effects in the entire scaling regime from above to below
$T_c$.
We present the first investigation of finite-size effects
in this model with nonperiodic boundary conditions for
temperatures below $T_c$.
To avoid nonscaling effects present in $d=3$ dimensions
1, we continue the model to $2<d<3$, where scaling is
known to hold even for nonperiodic boundary conditions 2.
Explicit results are obtained in film geometry for the
universal scaling functions of the excess free energy and
the Casimir force for Dirichlet and Neumann boundary
conditions.
For appropriate bulk correlation lengths $\xi_+$ and $\xi_-$
above and below $T_c$, respectively, we find an unexpected
size dependence $\propto\xi_\pm^-1/\nuL^-2$ of the
Casimir force for large $L/\xi_\pm$.
The behavior above $T_c$ originates from the combined
action of the spherical constraint and the presence of a
surface free energy.
It differs from the predictions of standard finite-size
theories 3.
The behavior below $T_c$ arises from a nonzero energy of
the lowest mode.
The results for both above and below $T_c$ differ qualitatively
from the predictions of current finite-size theories for the
critical Casimir force above and below the lambda point of
$^4$He 4.
Possible applications of our results include the weakly
interacting Bose gas in the precritical regions above and
below $T_c$.
Support by DLR is gratefully acknowledged.
1 M.N.~Barber and M.E.~Fisher,
Annals of Physics (N.Y.) 77, 1 (1973).
2 X.S.~Chen and V.~Dohm,
Phys.\ Rev.\ E67, 056127 (2003).
3 V.~Privman, P.C.~Hohenberg, and A.~Aharony,
in Phase Transitions and Critical Phenomena,
C.~Domb and J.L.~Lebowitz, eds., Vol.\ 14
(Academic, NY, 1991);
M.~Krech,
The Casimir Effect in Critical Systems,
(World Scientific, Singapore, 1994).
4 D.~Dantchev, M.~Krech, and S.~Dietrich,
Phys.\ Rev.\ Lett.\ 95, 259701 (2005);
R.~Zandi, A.~Shackell, J.~Rudnick, M.~Kardar, and L.P.~Chayes,
cond-mat/0703262.
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