The renormalization-group method is applied to the analysis of the critical singularities in Gamma y (q), the correlation function of a non-ordering parameter y. The coupling between y and s, the order parameter with n components, is s 2 y. When alpha , the specific heat index, is negative, four types of tricritical behaviours are accessible within this Hamiltonian, while the other two may be accessible in other systems in which the non-ordering parameter is a vector. The scaling law gamma y =(2- eta y ) nu is not only trivially fulfilled (zero identically) between the indices characterizing the leading term of Gamma y (q), which is a constant, but is also fulfilled between the corrections to scaling, which are the leading singularities when alpha <0, although these are negligible compared with the regular terms.
The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter
s is coupled to a nonordering additional variable
y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For
s2y coupling, the correlation function of
y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is
s2y2, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to
s2y coupling is calculated.
〈y〉 has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of
y in the constrained case is discussed.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
Critical properties are discussed for systems with order parameters given by
n vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with
m components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces
n to an explicit parameter in an averaged partition function for the
m-component model. This leads to a simple discussion of properties of the system as a function of
n. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when
n is a negative even integer and that
n=0 corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary
m-component model which is random when
n=0 and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large-
n limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.
Employing Wilson's renormalization group scheme, we investigate the critical behaviour of a modified Ginzburg-Landau model with a nonlocal mode-coupling interaction in the quartic term. Carrying out the calculations at one-loop order, we obtain the critical exponents in the leading order of , where ρ is an exponent occurring in the nonlocal interaction term and d is the space dimension. Interestingly, the correlation exponent η is found to be non-zero at one-loop order and the expansion corresponds to an expansion about the tricritical mean-field theory in three dimensions, unlike the conventional theory. The ensuing critical exponents are in good agreement with experimental values for samples close to tricriticality. Our analysis indicates that tricriticality is a feature only in three dimensions.
The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.
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.