477 research outputs found
Pronounced minimum of the thermodynamic Casimir forces of O() symmetric film systems: Analytic theory
Thermodynamic Casimir forces of film systems in the O universality
classes with Dirichlet boundary conditions are studied below bulk criticality.
Substantial progress is achieved in resolving the long-standing problem of
describing analytically the pronounced minimum of the scaling function observed
experimentally in He films by R. Garcia and M.H.W. Chan, Phys. Rev.
Lett. and in Monte Carlo simulations for the
three-dimensional Ising model () by O. Vasilyev et al., EPL . Our finite-size renormalization-group approach yields
excellent agreement with the depth and the position of the minimum for
and semiquantitative agreement with the minimum for . Our theory also
predicts a pronounced minimum for the Heisenberg universality class.Comment: 1 figur
Non-universal critical Casimir force in confined He near the superfluid transition
We present the results of a one-loop calculation of the effect of a van der
Waals type interaction potential on the critical
Casimir force and specific heat of confined He near the superfluid
transition. We consider a He film of thickness . In the region (correlation length) we find that the van der Waals interaction
causes a leading non-universal non-scaling contribution of to the critical temperature dependence of the Casimir force
above that dominates the universal scaling contribution predicted by earlier theories. For the specific heat we find subleading
non-scaling contributions of and .Comment: 2 pages, submitted to LT23 Proceedings on June 14, 2002, accepted for
publication in Physica B on September 12, 200
Violation of Finite-Size Scaling in Three Dimensions
We reexamine the range of validity of finite-size scaling in the
lattice model and the field theory below four dimensions. We show that
general renormalization-group arguments based on the renormalizability of the
theory do not rule out the possibility of a violation of finite-size
scaling due to a finite lattice constant and a finite cutoff. For a confined
geometry of linear size with periodic boundary conditions we analyze the
approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this
analysis ordinary renormalized perturbation theory is sufficient. On the basis
of one-loop results and of exact results in the spherical limit we find that
finite-size scaling is violated for both the lattice model and the
field theory in the region . The non-scaling effects in the
field theory and in the lattice model differ significantly from each other.Comment: LaTex, 51 page
Relation between bulk order-parameter correlation function and finite-size scaling
We study the large- behavior of the bulk order-parameter correlation
function for within the lattice theory. We also
study the large- behavior of the susceptibility of the confined
lattice system of size with periodic boundary conditions. The large-
behavior of is closely related to the large- behavior of .
Explicit results are derived for . Finite-size scaling must be formulated
in terms of the anisotropic exponential correlation length that governs
the decay of for large rather than in terms of the isotropic
correlation length defined via the second moment of . This
result modifies a recent interpretation concerning an apparent violation of
finite-size scaling in terms of . Exact results for the
Ising model illustrate our conclusions. Furthermore, we show that the
exponential finite-size behavior for is not captured by the
standard perturbation approach that separates the lowest mode from the higher
modes. Consequences for the theory of finite-size effects for are
discussed. The two-variable finite-size scaling form predicts an approach
to the bulk critical behavior whereas a single-variable
scaling form implies a power-law approach .Comment: LaTex, 59 pages, accepted for publication in Eur. Phys. J.
Finite-Size Effects in the Field Theory Above the Upper Critical Dimension
We demonstrate that the standard O(n) symmetric field theory does
not correctly describe the leading finite-size effects near the critical point
of spin systems on a -dimensional lattice with . We show that these
finite-size effects require a description in terms of a lattice Hamiltonian.
For and explicit results are given for the susceptibility
and for the Binder cumulant. They imply that recent analyses of Monte-Carlo
results for the five-dimensional Ising model are not conclusive.Comment: 4 pages, latex, 1 figur
Nonmonotonic External Field Dependence of the Magnetization in a Finite Ising Model: Theory and MC Simulation
Using field theory and Monte Carlo (MC) simulation we investigate
the finite-size effects of the magnetization for the three-dimensional
Ising model in a finite cubic geometry with periodic boundary conditions. The
field theory with infinite cutoff gives a scaling form of the equation of state
where
is the reduced temperature, is the external field and
is the size of system. Below and at the theory predicts a
nonmonotonic dependence of with respect to at fixed and a crossover
from nonmonotonic to monotonic behaviour when is further increased. These
results are confirmed by MC simulation. The scaling function obtained
from the field theory is in good quantitative agreement with the finite-size MC
data. Good agreement is also found for the bulk value at .Comment: LaTex, 12 page
Lattice theory of finite-size effects above the upper critical dimension
We present a perturbative calculation of finite-size effects near of
the lattice model in a -dimensional cubic geometry of size with
periodic boundary conditions for . The structural differences between
the lattice theory and the field theory found previously in
the spherical limit are shown to exist also for a finite number of components
of the order parameter. The two-variable finite-size scaling functions of the
field theory are nonuniversal whereas those of the lattice theory are
independent of the nonuniversal model parameters.One-loop results for
finite-size scaling functions are derived. Their structure disagrees with the
single-variable scaling form of the lowest-mode approximation for any finite
where is the bulk correlation length. At , the large-
behavior becomes lowest-mode like for the lattice model but not for the
field-theoretic model. Characteristic temperatures close to of the
lattice model, such as of the maximum of the susceptibility
, are found to scale asymptotically as ,
in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising
model. We also predict asymptotically. On a
quantitative level, the asymptotic amplitudes of this large - behavior close
to have not been observed in previous MC simulations at because
of nonnegligible finite-size terms caused by the
inhomogeneous modes. These terms identify the possible origin of a significant
discrepancy between the lowest-mode approximation and previous MC data. MC data
of larger systems would be desirable for testing the magnitude of the
and terms predicted by our theory.Comment: Accepted in Int. J. Mod. Phys.
Minimal renormalization without \epsilon-expansion: Three-loop amplitude functions of the O(n) symmetric \phi^4 model in three dimensions below T_c
We present an analytic three-loop calculation for thermodynamic quantities of
the O(n) symmetric \phi^4 theory below T_c within the minimal subtraction
scheme at fixed dimension d=3. Goldstone singularities arising at an
intermediate stage in the calculation of O(n) symmetric quantities cancel among
themselves leaving a finite result in the limit of zero external field. From
the free energy we calculate the three-loop terms of the amplitude functions
f_phi, F+ and F- of the order parameter and the specific heat above and below
T_c, respectively, without using the \epsilon=4-d expansion. A Borel
resummation for the case n=2 yields resummed amplitude functions f_phi and F-
that are slightly larger than the one-loop results. Accurate knowledge of these
functions is needed for testing the renormalization-group prediction of
critical-point universality along the \lambda-line of superfluid He(4).
Combining the three-loop result for F- with a recent five-loop calculation of
the additive renormalization constant of the specific heat yields excellent
agreement between the calculated and measured universal amplitude ratio A+/A-
of the specific heat of He(4). In addition we use our result for f_phi to
calculate the universal combination R_C of the amplitudes of the order
parameter, the susceptibility and the specific heat for n=2 and n=3. Our
Borel-resummed three-loop result for R_C is significantly more accurate than
the previous result obtained from the \epsilon-expansion up to O(\epsilon^2).Comment: 29 pages LaTeX including 3 PostScript figures, to appear in Nucl.
Phys. B [FS] (1998
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