The Hamiltonian of the form H=H0+λH1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε2-αF(λ/εφ) for the singular part of the free energy near the critical point Tc(ε=(T-Tc)/Tc), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where γ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H0. An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.
The Hamiltonian of the form H=H0+λH1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε2-αF(λ/εφ) for the singular part of the free energy near the critical point Tc(ε=(T-Tc)/Tc), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where γ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H0. An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.
A system with Hamiltonian H0 is assumed to undergo a phase transition at transition temperature T0. When a small perturbation term λH′ is added to H0, the transition temperature will depend on the coupling parameter λ and the critical indices will in general be changed. The change induced by the application of λH′ is classified into three types, and these types discussed for the cases of anisotropic Ising ferromagnets by means of thermodynamic perturbation theory and scaling law equation for spin correlation functions.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
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.
A systematic expansion method is developed for studying a critical exponent. The expansion parameter in the present theory is 1/n where n is the spin dimensionality. Since the limit n →∞ corresponds to the spherical model, the present method means that the deviation of critical exponent from its spherical model value is expanded in powers of 1/n. The susceptibility exponent γ is discussed and the terms up to 1/n are exactly derived for the d-dimensional simple hypercubical system with nearest-neighbor interaction (d is assumed to take a continuous value). The result is proved to be consistent with recent Wilson's ε expansion.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
Renormalisation of a field theory model for tricritical behaviour in which the operator phi 6 is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi p =- 1 / 2 (1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi ) 2 in the field-theoretic Hamiltonian.