23 research outputs found
Five-loop \sqrt\epsilon-expansions for random Ising model and marginal spin dimensionality for cubic systems
The \sqrt\epsilon-expansions for critical exponents of the weakly-disordered
Ising model are calculated up to the five-loop order and found to possess
coefficients with irregular signs and values. The estimate n_c = 2.855 for the
marginal spin dimensionality of the cubic model is obtained by the Pade-Borel
resummation of corresponding five-loop \epsilon-expansion.Comment: 9 pages, TeX, no figure
Randomly dilute Ising model: A nonperturbative approach
The N-vector cubic model relevant, among others, to the physics of the
randomly dilute Ising model is analyzed in arbitrary dimension by means of an
exact renormalization-group equation. This study provides a unified picture of
its critical physics between two and four dimensions. We give the critical
exponents for the three-dimensional randomly dilute Ising model which are in
good agreement with experimental and numerical data. The relevance of the cubic
anisotropy in the O(N) model is also treated.Comment: 4 pages, published versio
Critical behavior of weakly-disordered anisotropic systems in two dimensions
The critical behavior of two-dimensional (2D) anisotropic systems with weak
quenched disorder described by the so-called generalized Ashkin-Teller model
(GATM) is studied. In the critical region this model is shown to be described
by a multifermion field theory similar to the Gross-Neveu model with a few
independent quartic coupling constants. Renormalization group calculations are
used to obtain the temperature dependence near the critical point of some
thermodynamic quantities and the large distance behavior of the two-spin
correlation function. The equation of state at criticality is also obtained in
this framework. We find that random models described by the GATM belong to the
same universality class as that of the two-dimensional Ising model. The
critical exponent of the correlation length for the 3- and 4-state
random-bond Potts models is also calculated in a 3-loop approximation. We show
that this exponent is given by an apparently convergent series in
(with the central charge of the Potts model) and
that the numerical values of are very close to that of the 2D Ising
model. This work therefore supports the conjecture (valid only approximately
for the 3- and 4-state Potts models) of a superuniversality for the 2D
disordered models with discrete symmetries.Comment: REVTeX, 24 pages, to appear in Phys.Rev.
Crossover and self-averaging in the two-dimensional site-diluted Ising model
Using the newly proposed probability-changing cluster (PCC) Monte Carlo
algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since
we can tune the critical point of each random sample automatically with the PCC
algorithm, we succeed in studying the sample-dependent and the sample
average of physical quantities at each systematically. Using the
finite-size scaling (FSS) analysis for , we discuss the importance of
corrections to FSS both in the strong-dilution and weak-dilution regions. The
critical phenomena of the 2D site-diluted Ising model are shown to be
controlled by the pure fixed point. The crossover from the percolation fixed
point to the pure Ising fixed point with the system size is explicitly
demonstrated by the study of the Binder parameter. We also study the
distribution of critical temperature . Its variance shows the power-law
dependence, , and the estimate of the exponent is consistent
with the prediction of Aharony and Harris [Phys. Rev. Lett. {\bf 77}, 3700
(1996)]. Calculating the relative variance of critical magnetization at the
sample-dependent , we show that the 2D site-diluted Ising model
exhibits weak self-averaging.Comment: 6 pages including 6 eps figures, RevTeX, to appear in Phys. Rev.
Five-loop renormalization-group expansions for the three-dimensional n-vector cubic model and critical exponents for impure Ising systems
The renormalization-group (RG) functions for the three-dimensional n-vector
cubic model are calculated in the five-loop approximation. High-precision
numerical estimates for the asymptotic critical exponents of the
three-dimensional impure Ising systems are extracted from the five-loop RG
series by means of the Pade-Borel-Leroy resummation under n = 0. These
exponents are found to be: \gamma = 1.325 +/- 0.003, \eta = 0.025 +/- 0.01, \nu
= 0.671 +/- 0.005, \alpha = - 0.0125 +/- 0.008, \beta = 0.344 +/- 0.006. For
the correction-to-scaling exponent, the less accurate estimate \omega = 0.32
+/- 0.06 is obtained.Comment: 11 pages, LaTeX, no figures, published versio
Scaling and finte-size-scaling in the two dimensional random-coupling Ising ferromagnet
It is shown by Monte Carlo method that the finite size scaling (FSS) holds in
the two dimensional random-coupled Ising ferromagnet. It is also demonstrated
that the form of universal FSS function constructed via novel FSS scheme
depends on the strength of the random coupling for strongly disordered cases.
Monte Carlo measurements of thermodynamic (infinite volume limit) data of the
correlation length () up to along with measurements of
the fourth order cumulant ratio (Binder's ratio) at criticality are reported
and analyzed in view of two competing scenarios. It is demonstrated that the
data are almost exclusively consistent with the scenario of weak universality.Comment: 9 pages, 4figuer
The two-dimensional random-bond Ising model, free fermions and the network model
We develop a recently-proposed mapping of the two-dimensional Ising model
with random exchange (RBIM), via the transfer matrix, to a network model for a
disordered system of non-interacting fermions. The RBIM transforms in this way
to a localisation problem belonging to one of a set of non-standard symmetry
classes, known as class D; the transition between paramagnet and ferromagnet is
equivalent to a delocalisation transition between an insulator and a quantum
Hall conductor. We establish the mapping as an exact and efficient tool for
numerical analysis: using it, the computational effort required to study a
system of width is proportional to , and not exponential in as
with conventional algorithms. We show how the approach may be used to calculate
for the RBIM: the free energy; typical correlation lengths in quasi-one
dimension for both the spin and the disorder operators; even powers of
spin-spin correlation functions and their disorder-averages. We examine in
detail the square-lattice, nearest-neighbour RBIM, in which bonds are
independently antiferromagnetic with probability , and ferromagnetic with
probability . Studying temperatures , we obtain precise
coordinates in the plane for points on the phase boundary between
ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We
demonstrate scaling flow towards the pure Ising fixed point at small , and
determine critical exponents at the multicritical point.Comment: 20 pages, 25 figures, figures correcte
The critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model
We present a new way of probing the universality class of the site-diluted
two-dimensional Ising model. We analyse Monte Carlo data for the magnetic
susceptibility, introducing a new fitting procedure in the critical region
applicable even for a single sample with quenched disorder. This gives us the
possibility to fit simultaneously the critical exponent, the critical amplitude
and the sample dependent pseudo-critical temperature. The critical amplitude
ratio of the magnetic susceptibility is seen to be independent of the
concentration of the empty sites for all investigated values of . At the same time the average effective exponent is found
to vary with the concentration , which may be argued to be due to
logarithmic corrections to the power law of the pure system. This corrections
are canceled in the susceptibility amplitude ratio as predicted by theory. The
central charge of the corresponding field theory was computed and compared well
with the theoretical predictions.Comment: 6 pages, 4 figure
Critical behavior of the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy
We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with
cubic anisotropy. We compute and analyze the fixed-dimension perturbative
expansion of the renormalization-group functions to four loops. The relations
of these models with N-color Ashkin-Teller models, discrete cubic models,
planar model with fourth order anisotropy, and structural phase transition in
adsorbed monolayers are discussed. Our results for N=2 (XY model with cubic
anisotropy) are compatible with the existence of a line of fixed points joining
the Ising and the O(2) fixed points. Along this line the exponent has
the constant value 1/4, while the exponent runs in a continuous and
monotonic way from 1 to (from Ising to O(2)). For N\geq 3 we find a
cubic fixed point in the region , which is marginally stable or
unstable according to the sign of the perturbation. For the physical relevant
case of N=3 we find the exponents and at the cubic
transition.Comment: 14 pages, 9 figure
Self-Averaging, Distribution of Pseudo-Critical Temperatures and Finite Size Scaling in Critical Disordered Systems
The distributions of singular thermodynamic quantities in an ensemble
of quenched random samples of linear size at the critical point are
studied by Monte Carlo in two models. Our results confirm predictions of
Aharony and Harris based on Renormalization group considerations. For an
Ashkin-Teller model with strong but irrelevant bond randomness we find that the
relative squared width, , of is weakly self averaging. , where is the specific heat exponent and is the
correlation length exponent of the pure model fixed point governing the
transition. For the site dilute Ising model on a cubic lattice, known to be
governed by a random fixed point, we find that tends to a universal
constant independent of the amount of dilution (no self averaging). However
this constant is different for canonical and grand canonical disorder. We study
the distribution of the pseudo-critical temperatures of the ensemble
defined as the temperatures of the maximum susceptibility of each sample. We
find that its variance scales as and NOT as
R_\chi\sim 70R_\chi (T_c)\chiT_c(i,l)m_i(T_c,l)T_c(i,l)(T-T_c(i,l))/T_c$. This function is found to be universal and to behave
similarly to pure systems.Comment: 31 pages, 17 figures, submitted to Phys. Rev.