First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
Using extended high-field series expansions and more precise estimates of the critical temperature the critical isotherm of the spin 1 / 2 Ising ferromagnet is re-analysed. The conjectures delta =15 (d=2) and delta =5 (d=3) are confirmed to within 1 / 2 % and 1%, respectively. Good agreement is found, particularly in two dimensions, between series estimates of the critical amplitudes and the predictions of the lattice-lattice scaling hypothesis. Some tentative conclusions regarding the coincident and nonphysical singularities are drawn.
The basic low temperature series expansion data for the spin one-half Ising model on the hydrogen peroxide lattice are used to obtain series in z=exp(-2J/k B T) along the coexistence curve for the specific heat C H , the magnetization M, and its first five derivatives delta 1 M/ delta mu 1 , where mu =exp(-2mH/k B T). The same data are used to derive series in mu along the critical isotherm for M and its first five derivatives delta l M/ delta z l . Ratio and Pade approximant analysis yield estimates of the critical exponents and critical amplitudes. On the whole the estimates of the critical exponents support scaling theory although a few of the exponent estimates are not in good agreement with scaling.
We have developed 15th-order high-temperature series expansions for the study of the critical behavior of the Ising spin glass with nearest-neighbor exchange interactions each of which assumes the values \ifmmode\pm\else\textpm\fi{}J randomly. Series for the Edwards-Anderson spin-glass susceptibility (${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$) and two of its derivatives with respect to the ordering field have been evaluated for hypercubic lattices in general dimension, d. These extend previous general-dimension series by five terms. Certain measurable universal amplitude ratios have been estimated from the new series. Accurate critical data for d=5 and the first reliable estimates of the exponent \ensuremath{\beta} for d=4 and 5, are given. We quote \ensuremath{\gamma}=1.73\ifmmode\pm\else\textpm\fi{}0.03, 2.00\ifmmode\pm\else\textpm\fi{}0.25, and 2.${7}_{\mathrm{\ensuremath{-}}0.6}^{+1.0}$ and \ensuremath{\beta}=0.95\ifmmode\pm\else\textpm\fi{}0.04, 0.9\ifmmode\pm\else\textpm\fi{}0.1, and 0.7\ifmmode\pm\else\textpm\fi{}0.2 in 5, 4, and 3 dimensions, respectively. Our results provide a smooth extrapolation between the mean-field results above six dimensions and experiments and simulations in physical dimensions. We relate our calculated derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ to measurements of derivatives of the magnetization with respect to a uniform magnetic field.
The low-density linked-cluster expansion of the ferromagnetic Ising model $F(H,T)=\ensuremath{-}mH\ensuremath{-}\frac{1}{2}qJ\ensuremath{-}{k}_{B}T\ensuremath{\Sigma}\stackrel{\ensuremath{\infty}}{l=1}{L}_{l}(u){\ensuremath{\mu}}^{l}$ ($u={e}^{\frac{\ensuremath{-}4J}{{k}_{B}T}},\ensuremath{\mu}={e}^{\frac{\ensuremath{-}2\mathrm{mH}}{{k}_{B}}T}$) is numerically analyzed. We find that all the polynomials
Ll are positive between 0 and ${u}_{c}={e}^{\frac{\ensuremath{-}4J}{{k}_{B}{T}_{c}}}$,
Tc being the critical temperature, and that
uc can be calculated with good accuracy as the limit point of the zeros of these polynomials
Ll. Given
uc, we compute
Ll(uc) and find, numerically, their asymptotic behavior to be ${c}_{0}{l}^{\ensuremath{-}s}$. It is then shown that the singular part of the free energy per spin, as $H\ensuremath{\rightarrow}0$, is given by $F(H,{T}_{c})\ensuremath{\sim}{(\ensuremath{-}\mathrm{ln}\ensuremath{\mu})}^{s\ensuremath{-}1}\ensuremath{\sim}{H}^{s\ensuremath{-}1}$.
s is found to be about 2.061 in two dimensions and about 2.19 in three dimensions. Hence, the critical $M\ensuremath{-}H$ isotherm is found and the critical index $\ensuremath{\delta}$ is calculated in fair agreement with the previous estimates. An analytic expression for the corresponding critical amplitude is also found. The form of
F given above is based on numerical work and is not to be regarded as proven analytically.