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Journal of Physics C Solid State Physics

Many-body effects and critical exponents

David W. Wood, H P Griffiths

Published Feb 7, 1974• 23 citations
Publisher

Abstract

Calculations are performed on the critical exponent beta for an Ising model which allows admixtures of many-body terms in the hamiltonian. It is found that beta is a potential-sensitive index in that its value depends upon the relative measure of such admixtures, and also that beta is most sensitive in the limit of small many-body effects.

DOI
10.1088/0022-3719/7/3/003

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Two-Dimensional Ising Model in an Annealed Random Field

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The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.

Modern Physics Letters B•Mar 20, 1998•1 citation
Calculations are performed on the critical exponent beta for an Ising model which allows admixtures of many-body terms in the hamiltonian. It is found that beta is a potential-sensitive index in that its value depends upon the relative measure of such admixtures, and also that beta is most sensitive in the limit of small many-body effects.
For pt. III see abstr. A23213 of 1974. The critical exponent beta is examined for two three-dimensional Ising model systems which allow admixtures of three and four-body terms to be present in the Hamiltonian. Low-temperature series expansions, combined with a Pade approximant analysis are used to form estimates of the exponents. Strong evidence is found in support of potential sensitivity in the exponent beta , and examples of the specific heat exponent alpha ' are also discussed. The variability of the critical exponents is discussed in relation to the Universally postulate and to generalized lattice-gas models.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
In the recent study of the Ising model on a small world network by Pekalski [Phys. Rev. E 64, 057104 (2001)], a surprisingly small value of the critical exponent beta approximately 0.0001 has been obtained for the temperature dependence of the magnetization. We perform extensive Monte Carlo simulations of the same model and conclude, via the standard finite-size scaling of various quantities, that the phase transition in the model is of the mean-field nature, in contrast to the work by Pekalski, but in accordance with other existing studies.
The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.