J W Essam

Series expansion study of the pair connectedness in bond percolation models

The pair connectedness C(r,p) in a percolation model is the probability that a lattice site at position r belongs to a finite cluster containing the origin when neighbouring sites are connected with probability p. Low-density power series expansions have been used to show that the spherical moments of C(r,p) are consistent with a scaling form. Assuming scaling theory to be valid, the results may be used to estimate the exponent describing the vanishing of percolation probability.

Critical behaviour of the Ising model above and below the critical temperature

Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2nth derivative is given by where in two dimensions ? = 1?, ? = 1fraction seven-eighths and in three dimensions ? = 1?, ? = 1fraction nine-sixteens. In two dimensions ? is exact and ? is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature where in three dimensions 0.303 <or= ? <or= 0.318 and 1?55 <or= ?prime <or= 1.65. If the even derivatives are to have the same index above and below Tc, then ? = fraction five-sixteens and ?prime = 1fraction nine-sixteens. The latter index lies very close to the lower limit so that, although we cannot rule out ?prime = ?, ?prime = 1fraction five-eighths is not unlikely. In two dimensions our results are consistent with linearity of the index and, if this is assumed, then ? = fraction one-eighths and ?prime = 1fraction seven-eighths exactly and there is symmetry about Tc. The amplitudes C2n+ and Cn- are estimated for n = 1 to 6.

Scaling theory for the pair-connectedness in percolation models

Percolation theory is considered as the limit J>>kBT of a dilute Ising model. The thermodynamic and correlation scaling arguments for the Ising model are shown to give corresponding scaling laws for the cluster size distribution and the pair-connectivity respectively.

Correlations in the face-centred cubic Ising model below Tc and along the critical isotherm

Correlations near the critical point of a face-centred cubic lattice are studied by series expansion methods. The amplitude of the second moment correlation length xi 1 is calculated for several sets of critical exponents.

Selfconsistent calculation of the conductivity in a disordered branching network

A simple extension of the effective medium theory for the conductivity of a branching medium is presented. new feature of the calculation is that a typical conductivity is assigned to the infinite clusters only. Excellent agreement is obtained with the results of an exact calculation by Stinchcombe (1974). In particular, the quadratic dependence on p-pc near pc is correctly reproduced.

Time-dependent susceptibility and neutron scattering cross section of the Ising model with general spin

A sum-rule method is used to determine the response of an Ising ferromagnet to a time-dependent transverse magnetic field. A similar method yields the energy distribution of the inelastic neutron scattering cross section. The results are valid for arbitrary temperature including the critical region, and the effect of a static parallel magnetic field of finite magnitude is also considered.

Series expansion study of the percolation probability for the triangular lattice bond problem

The percolation probability P(p) defined as the probability that a given lattice site belongs to an infinite cluster is expanded in powers of q(=1-p), the probability of a missing bond, to order 31 on the triangular lattice. The series is used to estimate the critical exponent beta . The results are compared with those obtained from P(p), the probability that a given lattice bond belongs to an infinite cluster. It is concluded that the estimate of the common critical exponent beta of these functions is greater than to be expected from either series separately and find beta =0.139+or-0.003.

Critical temperature of the Heisenberg model with random bond dilution

The method of high-temperature series expansions is used to study the dependence of Tc on the concentration p of interactions in a randomly diluted S=1/2 Heisenberg model. It is found that, for all three cubic lattices, the interval of p over which the Pade approximants to the series converge extends further towards the critical probability for percolation than for the corresponding site diluted model. In this interval the critical curve is linear and extrapolates to intersect the T=0 boundary at a value of p marginally above the critical probability for bond percolation. The anomalous dependence of the susceptibility exponent gamma on p which has recently been reported for the site diluted model is not observed for the model studied here.

Decoration transformations and the dilute anisotropic Heisenberg model

The extension of the decoration transformation to higher spin and to non-commuting pair interactions is discussed. The decorated spin-one Ising model is reduced to a model with bilinear and biquadratic exchange. The annealed dilute spin-1/2 Heisenberg model is considered as an example of noncommuting interactions. The critical probability pc for the latter model is found to be independent of the degree of anisotropy but has different values depending on whether the Ising or X-Y part of the interaction is dominant. The value of pc for the isotropic case takes on a third value which is much higher than for the Ising model, a result which is consistent with numerical work on quenched models.

Critical behaviour of the three-dimensional Ising model specific heat below Tc

The magnetization exponent βD of the Syozi model of dilute ferro-magnetism is estimated for the four most common three-dimensional lattices using the Pade approximant method. It is found in all cases that 0354 <or= βD <or= 0380. Using the previously established relation αprime = 1 - β/βD between the specific heat exponent αprime and the magnetization exponent β for the Ising model below Tc and assuming β = fraction five-sixteens, the above result implies 0117 <or= αprime <or= 0178. On the grounds that previous estimates of αprime have been much lower, it is concluded that most probably αprime = fraction one-eighth = 0125 with the consequence βD = fraction five-fourteens = 0357.... The possibility αprime = fraction one-sixteens is well excluded by this method. The specific heat amplitude of the Ising model below Tc is deduced by computation of the magnetization amplitude of the Syozi model.