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.
The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative ${\ensuremath{\varphi}}^{6}$ coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the $O(\ensuremath{\epsilon})$ corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in $〈\ensuremath{\varphi}〉$ at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative ${\ensuremath{\varphi}}^{6}$ term and a positive ${\ensuremath{\varphi}}^{8}$ term.
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.
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.
Using the local potential approximation of the exact renormalization group (RG) equation, we show the various domains of values of the parameters of the O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface
Sc (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain
Sf separated from
Sc by the tricritical surface
St (attraction domain of the Gaussian fixed point).
Sf and
Sc are two distinct domains of repulsion for the Gaussian fixed point, but
Sf is not the basin of attraction of a fixed point.
Sf is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the
ϕ4-coupling. This renormalized trajectory exists also in four dimensions making the Gaussian fixed point ultra-violet stable (and the
ϕ44 renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behavior observed in some ionic systems.