Critical Exponents of the 3-D 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.

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).

Physics Letters A, 1999

Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a crossover model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the crossover function for the susceptibility.

Arxiv preprint hep-lat/9401034, 1994

We study the low-energy effective action S ef f [ϕ] for the one-component real scalar field theory in three Euclidean dimensions in the symmetric phase, concentrating on its static part-effective potential V ef f (ϕ). It characterizes the approach to the phase transition in all systems that belong to the 3d Ising universality class. We compute it from the probability distributions of the average magnetization in the 3d Ising model in a homogeneous external field, obtained by Monte Carlo. We find that the ϕ 6 term in V ef f is important, while the higher terms can be neglected within our statistical errors. Thus we obtain the approximate effective action S ef f = d 3 x 1 2 ∂ µ ϕ∂ µ ϕ + 1 2 m 2 ϕ 2 + mg 4 ϕ 4 + g 6 ϕ 6 , with arbitrary mass m that sets the scale, and dimensionless couplings g 4 = 0.97 ± 0.02 and g 6 = 2.05 ± 0.15. The value of g 4 is consistent with the renormalization group fixed point coupling. This V ef f , when used instead of the traditional aϕ 2 + bϕ 4 , turns the Ginzburg-Landau description of the long-wave properties of the 3d theory near criticality into quantitatively accurate. It is also relevant to the theory of cosmological phase transitions.

Physical Review B, 1995

In extensive Monte Carlo simulations the phase transition of the random field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite size scaling. For a Gaussian distribution of the random fields it is found that the correlation length ξ diverges with an exponent ν = 1.1 ± 0.2 at the critical temperature and that χ ∼ ξ 2−η with η = 0.50 ± 0.05 for the connected susceptibility and χ dis ∼ ξ 4−η with η = 1.03 ± 0.05 for the disconnected susceptibility. Together with the amplitude ratio A = limT →Tc χ dis /χ 2 (hr/T) 2 being close to one this gives further support for a two exponent scaling scenario implying η = 2η. The magnetization behaves discontinuously at the transition, i.e. β = 0, indicating a first order transition. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multi-peak structure that is characteristic for the phase-coexistence at first order phase transition points.

2003

The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the threedimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed.

Physica A , Statistical Mechanics and its application, 2013

A method is developed to calculate the critical line of two dimensional (2D) anisotropic Ising model including nearest-neighbor interactions. The method is based on the real-space renormalization group (RG) theory with increasing block sizes. The reduced temperatures, KsKs (where View the MathML sourceK=JkBT and JJ, kBkB, and TT are the spin coupling interaction, the Boltzmann constant, and the absolute temperature, respectively), are calculated for different block sizes. By increasing the block size, the critical line for three types of lattice, namely: triangular, square, and honeycomb, is obtained and found to compare well with corresponding results reported by Onsager in the thermodynamic limit. Our results also show that, for the investigated lattices, there exist asymptotic limits for the critical line. Finally the critical exponents are obtained, again in good agreement with Onsager’s results. We show that the magnitude of the spin coupling interaction with anisotropic ferromagnetic characteristics does not change the values of the critical exponents, which stay constant along the direction of the critical line.

Communications in Mathematical Physics, 1988

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,d l for the theory isd l ≦2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.

Journal of Statistical Mechanics: Theory and Experiment, 2011

We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + /A − in the three-dimensional Ising model via the impact angle φ of complex temperature zeros. We also measure the correlationlength critical exponent ν from finite-size scaling, and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield φ = 59.2(1.0) • , A + /A − = 0.56(3), ν = 0.63048(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.