Critical Exponents of the 3-D Ising Model (original) (raw)
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Corrections to scaling in the 3D Ising model: A comparison between MC and MCRG results
International Journal of Modern Physics C
Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to [Formula: see text]. Our estimated values of the correction-to-scaling exponent [Formula: see text] tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes, i.e. [Formula: see text] with [Formula: see text], are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e. [Formula: see text] and [Formula: see text]. We discuss the possibilities that [Formula: see text] is either really smaller than usually expected or these values of [Formula: see text] describe some transient behavior which, eventually, turns into the correct asymptotic behavior at [Formula: see text]. We propose refining MCRG simulations and analysis to resolve this issue. Our actual MC estimations of the critical exponents [Formula: see...
International Journal of Modern Physics C
Corrections to scaling in the 3D Ising model are studied based on nonperturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes [Formula: see text]. Analytical arguments show the existence of corrections with the exponent [Formula: see text], the leading correction-to-scaling exponent being [Formula: see text]. A numerical estimation of [Formula: see text] from the susceptibility data within [Formula: see text] yields [Formula: see text], in agreement with this statement. We reconsider the MC estimation of [Formula: see text] from smaller lattice sizes, [Formula: see text], using different finite-size scaling methods, and show that these sizes are still too small, since no convergence to the same result is observed. In particular, estimates ranging from [Formula: see text] to [Formula: see text] are obtained, using MC data for thermodynamic average quantities, as well as for partition function zeros. However, a trend toward smaller [Formula: s...
Physical Review E, 1997
We compute the 2n-point renormalized coupling constants in the symmetric phase of the 3d Ising model on the sc lattice in terms of the high temperature expansions O(β 17) of the Fourier transformed 2n-point connected correlation functions at zero momentum. Our high temperature estimates of these quantities, which enter into the small field expansion of the effective potential for a 3d scalar field at the IR fixed point or, equivalently, in the critical equation of state of the 3d Ising model universality class, are compared with recent results obtained by renormalization group methods, strong coupling, stochastic simulations as well as previous high temperature expansions.
Monte Carlo renormalization: test on the triangular ising model
We test the performance of the Monte Carlo renormalization method using the Ising model on the triangular lattice. We apply block-spin transformations which allow for adjustable parameters so that the transformation can be optimized. This optimization takes into account the relation between corrections to scaling and the location of the fixed point. To this purpose we determine corrections to scaling of the triangular Ising model with nearest-and next-nearestneighbor interactions, by means of transfer matrix calculations and finite-size scaling. We find that the leading correction to scaling just vanishes for the nearest-neighbor model. However, the fixed point of the commonly used majority-rule block-spin transformation lies far away from the nearest-neighbour critical point. This raises the question whether the majority rule is suitable as a renormalization transformation, because corrections to scaling are supposed to be absent at the fixed point. We define a modified block-spin transformation which shifts the fixed point back to the vicinity of the nearest-neighbour critical Hamiltonian. This modified transformation leads to results for the Ising critical exponents that converge faster, and are more accurate than those obtained with the majority rule.
Physical review, 1997
The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation-Padé approximant method. Particular attention is paid to the line of critical end points of the ferromagneticparamagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a novel universality class. A rough estimate of the crossover exponent is also given.
Critical exponents for the 3D Ising model
1996
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Reexamination of Scaling in the Five-dimensional Ising model
Arxiv preprint cond-mat/ …, 2006
In three dimensions, or more generally, below the upper critical dimension, scaling laws for critical phenomena seem well understood, for both infinite and for finite systems. Above the upper critical dimension of four, finite-size scaling is more difficult.
The four-dimensional site-diluted Ising model: A finite-size scaling study
Nuclear Physics B, 1998
Using finite-size scaling techniques, we study the critical properties of the site-diluted Ising model in four dimensions. We carry out a high statistics Monte Carlo simulation for several values of the dilution. The results support the perturbative scenario: there is only the Ising fixed point with large logarithmic scaling corrections. We obtain, using the Perturbative Renormalization Group, functional forms for the scaling of several observables that are in agreement with the numerical data.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
We study the probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-state generalization known to have remarkably small corrections to scaling. Both models are studied in a cubic box with periodic boundary conditions. The model with reduced corrections to scaling makes it possible to determine P(M) with unprecedented precision. We also obtain a simple, but remarkably accurate, approximate formula describing the universal shape of P(M).