51 research outputs found
Crossover from Goldstone to critical fluctuations: Casimir forces in confined O symmetric systems
We study the crossover between thermodynamic Casimir forces arising from
long-range fluctuations due to Goldstone modes and those arising from critical
fluctuations. Both types of forces exist in the low-temperature phase of O
symmetric systems for in a -dimensional
slab geometry with a finite aspect ratio . Our
finite-size renormalization-group treatment for periodic boundary conditions
describes the entire crossover from the Goldstone regime with a nonvanishing
constant tail of the finite-size scaling function far below up to the
region far above including the critical regime with a minimum of the
scaling function slightly below . Our analytic result for
agrees well with Monte Carlo data for the three-dimensional XY model. A
quantitative prediction is given for the crossover of systems in the Heisenberg
universality class.Comment: 2 figure
Critical Casimir force in slab geometry with finite aspect ratio: analytic calculation above and below
We present a field-theoretic study of the critical Casimir force of the Ising
universality class in a -dimensional slab
geometry with a finite aspect ratio above, at, and below
. The result of a perturbation approach at fixed dimension is
presented that describes the dependence on the aspect ratio in the range . Our analytic result for the Casimir force scaling function for
agrees well with recent Monte Carlo data for the three-dimensional
Ising model in slab geometry with periodic boundary conditions above, at, and
below .Comment: 4 figure
Finite-size effects on the thermal conductivity of ^4He near T_\lambda
We present results of a renormalization-group calculation of the thermal
conductivity of confined He in a geometry above and
at within model F with Dirichlet boundary conditions for the order
parameter. We assume a heat flow parallel to the boundaries which implies
Neumann boundary conditions for the entropy density. No adjustable parameters
other than those known from bulk theory and static finite-size theory are used.
Our theoretical results are compared with experimental data by Kahn and Ahlers.Comment: 2 pages, 2 figure
Multiparameter universality and intrinsic diversity of critical phenomena in weakly anisotropic systems
Recently a unified hypothesis of multiparameter universality for the critical
behavior of bulk and confined anisotropic systems has been formulated [V. Dohm,
Phys. Rev. E {\bf 97}, 062128 (2018)]. We prove the validity of this hypothesis
on the basis of the principle of two-scale-factor universality for isotropic
systems. We introduce an angular-dependent correlation vector and a generalized
shear transformation that transforms weakly anisotropic systems to isotropic
systems. As examples we consider the -symmetric , Gaussian,
and -vector model. We determine the structure of the bulk order-parameter
correlation function, of the singular bulk part of the critical free energy,
and of critical bulk amplitude relations of anisotropic systems. It is shown
that weakly anisotropic systems exhibit a high degree of intrinsic diversity
due to independent parameters. Exact results are derived for the
Ising universality class and for the spherical and Gaussian universality
classes. For the Ising universality class we identify the universal
scaling function of the isotropic bulk correlation function from the
nonuniversal result of the functional renormalization group. A proof is
presented for the validity of multiparameter universality of the exact critical
Casimir amplitude in a rectangular geometry of weakly anisotropic systems with
periodic boundary conditions in the Ising universality class. This confirms the
validity of recent predictions of self-similar structures of finite-size
effects at derived from conformal field theory. This also substantiates
the previous notion of an effective shear transformation for anisotropic
two-dimensional Ising models. Our theory paves the way for a quantitative
theory of nonuniversal critical Casimir forces in anisotropic superconductors
Exact Critical Casimir Amplitude of Anisotropic Systems from Conformal Field Theory and Self-Similarity of Finite-Size Scaling Functions in Dimensions
The exact critical Casimir amplitude is derived for anisotropic systems
within the Ising universality class by combining conformal field theory
(CFT) with anisotropic theory. Explicit results are presented for
the general anisotropic scalar model and for the fully anisotropic
triangular-lattice Ising model in finite rectangular and infinite strip
geometries with periodic boundary conditions (PBC). These results demonstrate
the validity of multiparameter universality for confined anisotropic systems
and the nonuniversality of the critical Casimir amplitude. We find an
unexpected complex form of self-similarity of the anisotropy effects near the
instability where weak anisotropy breaks down. This can be traced back to the
property of modular invariance of isotropic CFT for . More generally, for
we predict the existence of self-similar structures of the finite-size
scaling functions of -symmetric systems with planar anisotropies and PBC
both in the critical region for as well as in the
Goldstone-dominated low-temperature region for
Minimal renormalization without epsilon-expansion: Amplitude functions in three dimensions below T_c
Massive field theory at fixed dimension d<4 is combined with the minimal
subtraction scheme to calculate the amplitude functions of thermodynamic
quantities for the O(n) symmetric phi^4 model below T_c in two-loop order.
Goldstone singularities arising at an intermediate stage in the calculation of
O(n) symmetric quantities are shown to cancel among themselves leaving a finite
result in the limit of zero external field. From the free energy we calculate
the amplitude functions in zero field for the order parameter, specific heat
and helicity modulus (superfluid density) in three dimensions. We also
calculate the q^2 part of the inverse of the wavenumber-dependent transverse
susceptibility chi_T(q) which provides an independent check of our result for
the helicity modulus. The two-loop contributions to the superfluid density and
specific heat below T_c turn out to be comparable in magnitude to the one-loop
contributions, indicating the necessity of higher-order calculations and
Pade-Borel type resummations.Comment: 41 pages, LaTeX, 8 PostScript figures, submitted to NPB [FS
Critical free energy and Casimir forces in rectangular geometries
We study the critical behavior of the free energy and the thermodynamic
Casimir force in a block geometry in
dimensions with aspect ratio above, at, and below on
the basis of the O symmetric lattice model with periodic boundary
conditions (b.c.). We consider a simple-cubic lattice with isotropic
short-range interactions. Exact results are derived in the large - limit
describing the geometric crossover from film () over cubic to
cylindrical () geometries. For , three perturbation
approaches are presented that cover both the central finite-size regime near
for and the region outside the central
finite-size regime well above and below for arbitrary . At bulk
of isotropic systems with periodic b.c., we predict the critical Casimir
force in the vertical direction to be negative (attractive) for a slab
(), and zero for a cube
. We also present extrapolations to the cylinder limit
() and to the film limit () for and . Our
analytic results for finite-size scaling functions in the minimal
renormalization scheme at fixed dimension agree well with Monte Carlo
data for the three-dimensional Ising model by Hasenbusch for and by
Vasilyev et al. for above, at, and below .Comment: 23 pages, 14 figure
Diversity of critical behavior within a universality class
We study spatial anisotropy effects on the bulk and finite-size critical
behavior of the O symmetric anisotropic lattice model with
periodic boundary conditions in a -dimensional hypercubic geometry above, at
and below . The absence of two-scale factor universality is discussed for
the bulk order-parameter correlation function, the bulk scattering intensity,
and for several universal bulk amplitude relations. For the confined system,
renormalization-group theory within the minimal subtraction scheme at fixed
dimension for is employed. For the case of cubic symmetry and for
our perturbation approach yields excellent agreement with the Monte Carlo
(MC) data for the finite-size amplitude of the free energy of the
three-dimensional Ising model at by Mon [Phys. Rev. Lett. {\bf 54}, 2671
(1985)]. Below a minimum of the scaling function of the excess free
energy is found. We predict a measurable dependence of this minimum on the
anisotropy parameters. The relative anisotropy effect on the free energy is
predicted to be significantly larger than that on the Binder cumulant. Our
theory agrees quantitatively with the non-monotonic dependence of the Binder
cumulant on the ferromagnetic next-nearest neighbor (NNN) coupling of the
two-dimensional Ising model found by MC simulations of Selke and Shchur [J.
Phys. {\bf A 38}, L739 (2005)]. Our theory also predicts a non-monotonic
dependence for small values of the {\it antiferromagnetic} NNN coupling and the
existence of a Lifschitz point at a larger value of this coupling. The
nonuniversal anisotropy effects in the finite-size scaling regime are predicted
to satisfy a kind of restricted universality. The tails of the large-
behavior at violate both finite-size scaling and universality
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