https://doi.org/10.1007/s100510051127
Relation between bulk order-parameter correlation function and finite-size scaling
1
Institut für Theoretische Physik, Technische Hochschule Aachen,
52056 Aachen, Germany
2
Institute of Particle Physics, Hua-Zhong Normal University,
Wuhan 430079, P.R. China
Received:
1
December
1999
Published online: 15 May 2000
We study the large-distance behavior of the bulk order-parameter correlation
function for
within the lattice version of the
theory including lattice effects. We also study the large-L
behavior of the susceptibility χ for
of the confined
lattice system of linear size L with periodic boundary conditions. We find
that the structure of the large-L behavior of χ of the confined system
is closely related to the structure of the large-distance behavior of
of the bulk system. Explicit results are derived in the spherical
(large-n) limit and in one-loop order for general dimensions d > 2. For
the lattice model with cubic symmetry we find that finite-size scaling must
be formulated in terms of the anisotropic bulk correlation length (exponential
correlation length) that governs the exponential decay of
for large
r rather than in terms of the ordinary isotropic bulk correlation length ξ
defined via the second moment of
. We show that it is the
exponential bulk correlation length
in the direction of the cubic axes
that determines the exponential finite-size scaling behavior of lattice systems
in a rectangular geometry. This result modifies a recent interpretation concerning
an apparent violation of finite-size scaling in terms of the second-moment
correlation length
. Exact results for the one-dimensional Ising
model illustrate our conclusions. Furthermore we show for general d> 2 that a
description of finite-size effects for finite n in the entire region
requires
different perturbative approaches
that are applicable either to the region
or
, respectively. In particular we show that the
exponential finite-size behavior for
above
is not
captured by the standard perturbation approach that separates the homogeneous
lowest mode from the inhomogeneous higher modes. Consequences for the theory of
finite-size effects above four dimensions are discussed. We show that the two-variable
finite-size scaling form predicts an exponential approach
to the bulk critical behavior above
whereas the reduction to a
single-variable scaling form implies a power-law approach
.
PACS: 05.70.Jk – Critical point phenomena / 64.60.-i – General studies of phase transitions
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000