35 research outputs found
Transfer matrix and Monte Carlo tests of critical exponents in Ising model
The corrections to finite-size scaling in the critical two-point correlation
function G(r) of 2D Ising model on a square lattice have been studied
numerically by means of exact transfer-matrix algorithms. The systems have been
considered, including up to 800 spins. The calculation of G(r) at a distance r
equal to the half of the system size L shows the existence of an amplitude
correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very
small magnitude also has been detected, as it can be expected from our recently
developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of
the squared magnetization of 3D Ising model have been performed by Wolff's
algorithm in the range of the reduced temperatures t =< 0.000086 and system
sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase
above the currently accepted numerical values. The critical coupling
K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96
=< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling
of the derivatives of the Binder cumulant, tends to decrease slightly below the
RG value 1.587 for the largest system sizes. The finite-size scaling of
accurately simulated maximal values of the specific heat C_v in 3D Ising model
confirms a logarithmic rather than power-like critical singularity of C_v.Comment: 34 pages, 8 figures. New references ([9],[18],[29],[37]) are added
and the text is changed to reflect partly the status of ar
Critical behavior of n-vector model with quenched randomness
We consider the Ginzburg-Landau phase transition model with O(n) symmetry
(i.e., the n-vector model) which includes a quenched randomness, i.e., a random
temperature disorder. We have proven rigorously that within the diagrammatic
perturbation theory the quenched randomness does not change the critical
exponents at n tending to 0, which is in contrast to the conventional point of
view based on the perturbative renormalization group theory.Comment: 6 pages, no figure
Perturbative renormalization of the Ginzburg-Landau model revisited
The perturbative renormalization of the Ginzburg-Landau model is reconsidered
based on the Feynman diagram technique. We derive renormalization group (RG)
flow equations, exactly calculating all vertices appearing in the perturbative
renormalization of the phi^4 model up to the epsilon^3 order of the
epsilon-expansion. In this case, the phi^2, phi^4, phi^6, and phi^8 vertices
appear. All these vertices are relevant. We have tested the expected basic
properties of the RG flow, such as the semigroup property. Under repeated RG
transformation R_s, appropriately represented RG flow on the critical surface
converges to certain s-independent fixed point. The Fourier-transformed
two-point correlation function G(k) has been considered. Although the
epsilon-expansion of X(k)=1/G(k) is well defined on the critical surface, we
have revealed an inconsistency of the perturbative method with the exact
rescaling of X(k), represented as an expansion in powers of k at k --> 0. We
have discussed also some aspects of the perturbative renormalization of the
two-point correlation function slightly above the critical point. Apart from
the epsilon-expansion, we have tested and briefly discussed also a modified
approach, where the phi^4 coupling constant u is the expansion parameter at a
fixed spatial dimensionality d.Comment: 37 pages, no figure
Critical two-point correlation functions and "equation of motion" in the phi^4 model
Critical two-point correlation functions in the continuous and lattice phi^4
models with scalar order parameter phi are considered. We show by different
non-perturbative methods that the critical correlation functions <phi^n(0)
phi^m(x)> are proportional to at |x| --> infinity for any
positive odd integers n and m. We investigate how our results and some other
results for well-defined models can be related to the conformal field theory
(CFT), considered by Rychkov and Tan, and reveal some problems here. We find
this CFT to be rather formal, as it is based on an ill-defined model. Moreover,
we find it very unlikely that the used there "equation of motion" really holds
from the point of view of statistical physics.Comment: 15 pages, 3 figure
Monte Carlo test of critical exponents and amplitudes in 3D Ising and phi^4 lattice models
We have tested the leading correction-to-scaling exponent omega in
O(n)-symmetric models on a three-dimensional lattice by analysing the recent
Monte Carlo (MC) data. We have found that the effective critical exponent,
estimated at finite sizes of the system L and L/2, decreases remarkably within
the range of the simulated L values. This shows the incorrectness of some
claims that omega has a very accurate value 0.845(10) at n=1. A selfconsistent
infinite volume extrapolation yields row estimates omega=0.547, omega=0.573,
and omega=0.625 at n=1, 2, and 3, respectively, in approximate agreement with
the corresponding exact values 1/2, 5/9, and 3/5 predicted by our recently
developed GFD (grouping of Feynman diagrams) theory. We have fitted the MC data
for the susceptibility of 3D Ising model at criticality showing that the
effective critical exponent eta tends to increase well above the usually
accepted values around 0.036. We have fitted the data within [L;8L], including
several terms in the asymptotic expansion with fixed exponents, to obtain the
effective amplitudes depending on L. This method clearly demonstrates that the
critical exponents of GFD theory are correct (the amplitudes converge to
certain asymptotic values at L tending to infinity), whereas those of the
perturbative renormalization group (RG) theory are incorrect (the amplitudes
diverge). A modification of the standard Ising model by introducing suitable
"improved" action (Hamiltonian) does not solve the problem in favour of the
perturbative RG theory.Comment: As compared to the first version from June 2001, three figures and
additional discussion has been added, including an estimation of the critical
exponent omega for O(n)-symmetric models with n=2 and 3. Now there are 18
pages and 9 figure
Longitudinal and transverse Greens functions in phi^4 model below and near the critical point
We have extended our method of grouping of Feynman diagrams (GFD theory) to
study the transverse (G_t) and longitudinal (G_l) Greens functions in phi^4
model below the critical point (T<T_c) in presence of an infinitesimal external
field. Our method allows a qualitative analysis not cutting the perturbation
series. We have shown that the critical behavior of the Greens (correlation)
functions is consistent with a general scaling hypothesis, where the same
critical exponents, found within the GFD theory, are valid both at T<T_c and
T>T_c. The long-wave limit k->0 has been studied at T<T_c, showing that the
transverse and the longitudinal correlation functions diverge as 1/k in the
power of lambda_t and lambda_l, respectively, where d/2< lambda_t < 2 and
lambda_l = 2 lambda_t - d holds at the spatial dimensionality 2<d<4. It is the
physical solution of our equations, which coincides with the asymptotic
solution at T -> T_c as well as with a non-perturbative renormalization group
(RG) analysis provided in our paper. It is confirmed also by Monte Carlo
simulation. The exponents, as well as the ratio bM^2/a^2 (where M is
magnetization, a and b are the amplitudes of G_t and G_l at k->0) are
universal. The results of the perturbative RG method are reproduced by formally
setting lambda_t=2. Nevertheless, we disprove the conventional statement that
lambda_t=2 is the exact result.Comment: The paper has been completed by a wider discussion of literature
(Sec.9.1), as well as by Monte Carlo simulations (Sec.10). Now are 32 pages
and 2 figure
Nonexistence of the non-Gaussian fixed point predicted by the RG field theory in 4-epsilon dimensions
The Ginzburg-Landau phase transition model is considered in d=4-epsilon
dimensions within the renormalization group (RG) approach. The problem of
existence of the non-Gaussian fixed point is discussed. An equation is derived
from the first principles of the RG theory (under the assumption that the fixed
point exists) for calculation of the correction-to-scaling term in the
asymptotic expansion of the two-point correlation (Green's) function. It is
demonstrated clearly that, within the framework of the standard methods (well
justified in the vicinity of the fixed point) used in the perturbative RG
theory, this equation leads to an unremovable contradiction with the known RG
results. Thus, in its very basics, the RG field theory in 4-epsilon dimensions
is contradictory. To avoid the contradiction, we conclude that such a
non-Gaussian fixed point, as predicted by the RG field theory, does not exist.
Our consideration does not exclude existence of a different fixed point.Comment: 5 pages, no figure
Monte Carlo test of critical exponents in 3D Heisenberg and Ising models
We have tested the theoretical values of critical exponents, predicted for
the three--dimensional Heisenberg model, based on the published Monte Carlo
(MC) simulation data for the susceptibility. Two different sets of the critical
exponents have been considered - one provided by the usual (perturbative)
renormalization group (RG) theory, and another predicted by grouping of Feynman
diagrams in phi^4 model (our theory). The test consists of two steps. First we
determine the critical coupling by fitting the MC data to the theoretical
expression, including both confluent and analytical corrections to scaling, the
values of critical exponents being taken from theory. Then we use the obtained
value of critical coupling to test the agreement between theory and MC data at
criticality. As a result, we have found that predictions of our theory
(gamma=19/14, eta=1/10, omega=3/5) are consistent, whereas those of the
perturbative RG theory (gamma=1.3895, eta=0.0355, omega=0.782) are inconsistent
with the MC data. The seemable agreement between the RG prediction for eta and
MC results at criticality, reported in literature, appears due to slightly
overestimated value of the critical coupling. Estimation of critical exponents
of 3D Ising model from complex zeroth of the partition function is discussed. A
refined analysis yields the best estimate 1/nu=1.518. We conclude that the
recent MC data can be completely explained within our theory (providing
1/nu=1.5 and omega=0.5) rather than within the conventional RG theory.Comment: 16 pages, 7 figures. Currently, the paper has been completed by a
refined estimation of the critical exponent nu from the imaginary part of the
partition function zeroth in 3D Ising model: nonlinear fits yield the best
estimate 1/nu=1.518 in agreement with our theoretical value 1.5. This is
explained in two additional figure
Perturbation theory methods applied to critical phenomena
Different perturbation theory treatments of the Ginzburg-Landau phase
transition model are discussed. This includes a criticism of the perturbative
renormalization group (RG) approach and a proposal of a novel method providing
critical exponents consistent with the known exact solutions in two dimensions.
The new values of critical exponents are discussed and compared to the results
of numerical simulations and experiments.Comment: 12 pages, 4 figures. As compared to the first version, minor errata
have been removed (page 11, comparison with experiment
Power-law singularities and critical exponents in n-vector models
Power-law singularities and critical exponents in n-vector models are
considered from different theoretical points of view. It includes a theoretical
approach called the GFD (grouping of Feynman diagrams) theory, as well as the
perturbative renormalization group (RG) treatment. A non-perturbative proof
concerning corrections to scaling in the two-point correlation function of the
phi^4 model is provided, showing that predictions of the GFD theory rather than
those of the perturbative RG theory can be correct. Critical exponents
determined from highly accurate experimental data very close to the
lambda-transition point in liquid helium, as well as the Goldstone mode
singularities in n-vector spin models, evaluated from Monte Carlo simulation
results, are discussed with an aim to test the theoretical predictions. Our
analysis shows that in both cases the data can be well interpreted within the
GFD theory.Comment: 17 pages, 2 figure