Critical exponents for the 3D Ising model

Canadian Journal of Physics, 1989

A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.

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.

Physical Review E, 1999

High-temperature series are computed for a generalized $3d$ Ising model with arbitrary potential. Two specific ``improved'' potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are: $\gamma=1.2371(4)$, $\nu=0.63002(23)$, $\alpha=0.1099(7)$, $\eta=0.0364(4)$, $\beta=0.32648(18)$. By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.

Journal of Physical Studies, 1998

Based on the recently obtained ve{loop renormalization group functions of the 4 {theory with O(n){symmetric and cubic interactions, we analyse these functions for the weakly diluted Ising model case n = 0 both by means of the p "{expansion and in the xed d = 3 technique. We apply di erent resummation procedures and report numerical values for the critical exponents dependent on the order of approximation. We compare the convergence of the results obtained in the framework of the two schemes.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2013

In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: M m 0 =1 ∼ t −β/νz , which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent z, evaluated from the behavior of the ratio F 2 (t) = M 2 m 0 =0 / M 2 m 0 =1 ∼ t 3/z , along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent θ that characterizes the initial slip of magnetization and the global persistence exponent θ g associated with the probability P (t) that magnetization keeps its signal up to time t. Our estimates for the dynamic critical exponents at the Lifshitz point are z = 2.34(2) and θ g = 0.336(4), values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the z axis, i.e., J 2 = 0), i.e., z ≈ 2.07 and θ g ≈ 0.38. We also present estimates for the static critical exponents β and ν, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.

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...

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.

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.

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.

1991