We study the critical properties of the random field Ising model in general dimension d using high-temperature expansions for the susceptibility, \ensuremath{\chi}=${\ensuremath{\sum}}_{\mathit{j}}$ [〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$
〉T〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h and the structure factor, G=${\ensuremath{\sum}}_{\mathit{j}}$[〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{j}}$
〉T]h, where 〈
〉T indicates a canonical average at temperature T for an arbitrary configuration of random fields and [
]h indicates an average over random fields. We treated two distributions of random fields, the bimodal in which each
hi=\ifmmode\pm\else\textpm\fi{}
h0 and a Gaussian distribution in which each
hi has variance
h02. We obtained series for \ensuremath{\chi} and G in the form ${\ensuremath{\sum}}_{\mathit{n}=1,15}$
an(g,d)(J/T
)n, where J is the exchange constant and the coefficients
an(g,d) are polynomials in g\ensuremath{\equiv}
h02/
J2 and in d. We assume that as T approaches its critical value,
Tc, one has \ensuremath{\chi}\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ and G\ensuremath{\sim}(T-
Tc${)}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$. For dimensions above d=2 we find a range of values of g for which the critical exponents obtained from our series seem not to depend on g.For large values of g our results show a g dependence which is attributable to either a tricritical point or a first-order transition. All our results for critical exponents suggest that \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}, in agreement with the two-exponent scaling picture. In addition we have also constructed series for the amplitude ratio, A=(G/${\mathrm{\ensuremath{\chi}}}^{2}$)(
T2)/(
gJ2). We find that A approaches a constant value as T\ensuremath{\rightarrow}
Tc (consistent with \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma}) with A\ensuremath{\approxeq}1. It appears that A is somewhat larger for the bimodal than for the Gaussian model, in agreement with a recent analysis at high d. \textcopyright{} 1996 The American Physical Society.
Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact ϵ expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly as a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by
b phantom bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents ν and γ, correct to O(ϵ2). In addition, the approach can be extended to a many polymer system, and to other critical clusters.
The critical behavior of a system of
n-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For
n=3 and
d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).
The low-density linked-cluster expansion of the free energy of the ferromagnetic Ising model with single-ion anisotropy, i.e., the $\ensuremath{\Delta} \ensuremath{\Sigma}{i}^{}S_{\mathrm{zi}}^{}{}_{}{}^{2}$ term, is given by $\ensuremath{-}\ensuremath{\beta}f(\ensuremath{\beta}J,\ensuremath{\beta}\ensuremath{\Delta},\ensuremath{\beta}h)=\frac{1}{2}qJ+h\ensuremath{-}\ensuremath{\Delta}+\ensuremath{\Sigma}{l=1}^{\ensuremath{\infty}}({u}^{12}\ensuremath{\mu})^{l}L_{l}(u,\ensuremath{\eta})$, with $u=\mathrm{exp}(\ensuremath{-}\frac{J}{{k}_{B}T})$, $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{2mh}{{k}_{B}T})$, and $\ensuremath{\eta}=\mathrm{exp}(\frac{\ensuremath{\Delta}}{{k}_{B}T})$. This is a low-temperature expansion with each spin
Si having magnitude one. The sixth-order series available in the literature is analyzed by evaluating the zeros of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}={u}^{12}^{l}L_{l}$ as a function of $\ensuremath{\Delta}$, and from that the critical temperature is deduced. The knowledge of the critical temperature as a function of the anisotropy leads to the second-order part of the phase boundary of $\ensuremath{\beta}J$ with $\ensuremath{\beta}\ensuremath{\Delta}$ which estimates a tricritical value of $\ensuremath{\Delta}$. The asymptotic behavior of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{L}}_{l}$ is studied and from that the critical exponent $\ensuremath{\delta}$ of the $M\ensuremath{-}h$ isotherm as $h\ensuremath{\rightarrow}0$ is found. It is observed that the anisotropy determines the transition temperature, but the exponent $\ensuremath{\delta}$ is independent of the same as expected from the universality hypothesis.
The definitions and determination of the thermodynamic magnetic critical point exponents α, α′, β, γ, γ′ and δ are reviewed, and ``gap exponents'' Δ and Δ′ for the divergences of higher derivatives of the free energy with respect to the field are introduced. The rigorous inequalities relating the exponents are discussed. The ``microdomain'' or ``cluster'' model of ferromagnetism leads to relations between these exponents since it expresses them in terms of two underlying exponents σ and τ. The exponent σ is directly related to the effective surface free energy of a large microdomain and thence to the dimensionality. This yields some insight into the meaning of the observed exponent values and of the differences between the critical behavior of the Ising and Heisenberg models. The recently developed homogeneity and scaling arguments are presented. These generate the same thermodynamic exponent relations and further ones for the correlation exponents η, ν and ν′. They also indicate that a general law of corresponding states might hold in the critical region. Theoretical and experimental evidence on the validity of these various ideas is briefly reviewed.