Ising model critical exponent η from a criticality equation (original) (raw)
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Critical Exponents of the 3-D Ising Model
International Journal of Modern Physics C, 1996
We present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on 64364^3643, 1283128^31283, and 2563256^32563 simple cubic lattices. Our MCRG estimates are Knnc=0.221655(1)(1)K_{nn}^c=0.221655(1)(1)Knnc=0.221655(1)(1) and nu=0.625(1)\nu=0.625(1)nu=0.625(1). The FSS results for KcK^cKc are consistent with those from MCRG but the value of nu\nunu is not. Our best estimate eta=0.025(6)\eta = 0.025(6)eta=0.025(6) covers the spread in the MCRG and FSS values. A surprise of our calculation is the estimate omegaapprox0.7\omega \approx 0.7omegaapprox0.7 for the correction-to-scaling exponent. We also present results for the renormalized coupling gRg_RgR along the MCRG flow and argue that the data support the validity of hyperscaling for the 3D Ising model.
Critical exponents for the 3D Ising model
1996
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Physica A: Statistical Mechanics and its Applications, 2005
This paper revisits the fundamental statistical properties of the crucial model in critical phenomena i.e. the Ising model, guided by knowledge of the energy values of the Ising Hamiltonian and aided by numerical estimation techniques. We have obtained exact energies in 2D and 3D and nearly exact analytical forms for the degeneracies of the distinct eigenvalues. The formulae we obtained, both for energies and degeneracies, have an exceedingly simple analytical form and are easy to use. The resultant partition functions were utilised to determine the critical behaviour of the Ising system on cubic lattices in 2D and 3D. We obtained a logarithmic divergence of the specific heat in 2D and 3D cases and the critical temperature estimates provided additional confirmation of the correctness of our approach.
Crossover critical behavior in the three-dimensional Ising model
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.
Physical Review E, 1999
High-temperature series are computed for a generalized 3d3d3d 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)\gamma=1.2371(4)gamma=1.2371(4), nu=0.63002(23)\nu=0.63002(23)nu=0.63002(23), alpha=0.1099(7)\alpha=0.1099(7)alpha=0.1099(7), eta=0.0364(4)\eta=0.0364(4)eta=0.0364(4), beta=0.32648(18)\beta=0.32648(18)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 the Physical Society of Japan, 2008
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization lambdam (=beta/znu) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.
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.
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.
The Journal of Physical Chemistry B, 2001
A new graphical method is developed to calculate the critical temperature of 2-and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221 , is in a good agreement with the accurate value of 0.22165 reported by others.