The influence of the cubic anisotropy both in the quadratic and quartic part of an n-component 'spin' Hamiltonian is examined in the framework of a parquet-graph summation. For values of the anisotropy parameter f larger than a critical value, f - , a first-order transition occurs which corresponds to the appearance of complex fixed points in the renormalization group approach. On the other hand for f<f - a first- or second-order transition occurs depending on f, n and the values of the bare coupling constants. For second-order transitions a quantity m(n,f) plays the role of an effective number of components, m(n,0)=n. For m>4 and m<-8 the system has a strongly cubic behaviour whereas 0<m<4 a weakly cubic behaviour is observed; the latter changes to pure isotropic behaviour for f to 0. For -8<m<0 the behaviour is strongly or weakly cubic depending on the values of the bare coupling constants.
Exact renormalization-group recursion relations are derived in closed form for a one-dimensional spin-1 Ising model. The recursion relations possess tricritical and critical fixed points. We have studied the flow diagram corresponding to the renormalization-group transformations, and we have linearized about the fixed points. The predicted asymptotic homogeneity is shown to be satisfied by the true free energy, which has been computed using the transfer matrix. The pseudocritical singularities, existing in the zero-temperature limit, are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the pseudotricritical point is marked by a triple degeneracy. The increased instability of the tricritical fixed point, relative to the critical fixed points, is shown to be directly related to the eigenvalue degeneracy of the transfer matrix being greater at a tricritical point than at a critical point.
Crossover scaling functions for a two-component spin system with quadratic anisotropy
g and cubic anisotropy $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$ are calculated to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ and in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$, near the transitions within the ordered phases below the multicritical point (
T=Tm,g=0). Emphasis is placed on the mechanism by which the system is driven to bicritical or tetracritical behavior by the dangerous irrelevant variable $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{v}$. Explicit expressions and graphs for the various thermodynamic scaling functions are presented.
The renormalization-group recursion relations are solved for the effective Hamiltonians relative to phase transitions with four-component order parameters. For this value of n there are 22 types of Hamiltonians which can be classified into two categories according to the action of their normalizer
GN on the corresponding parameter space (
GN is the symmetry group leaving globally invariant this space). In the first case,
GN generates a finite number of isolated fixed points whose characteristics can be deduced from the detailed investigation of five Hamiltonians only.In the second category, for which
GN is a continuous group, there are, in addition to isolated fixed points, continuous manifolds of physically equivalent fixed points (the dimension of the manifolds is either one or three). In the search for a stable fixed point, the continuous manifolds can be ignored, while the isolated points are related to the five former Hamiltonians. For n=4, it is necessary to solve the recursion relations to two-loop order. The only possible stable ones among the fixed points then arise from a splitting of points which coincide, to one-loop order, with the isotropic fixed point. Extending, to two-loop order, a result recently established to the preceding order, we show that if a stable fixed point exists, it is unique. For n=4, the stable fixed point has one of three possible symmetries : di-icosahedral, hypercubic, or dicylindrical.Despite this anisotropy of the critical fluctuations, the exponents associated with any of the stable fixed points are identical to order ${\ensuremath{\epsilon}}^{2}$ to the ``isotropic'' exponents corresponding to n=4. The cubic point is destabilized by any operator of symmetry lower than cubic. The dicylindrical one remains stable with respect to certain anisotropies of lower symmetry. We examine the available experimental data in light of the preceding theoretical results concerning the critical behavior and the thermodynamic order of the transitions. On the other hand, we establish two general symmetry conditions relative to the stable fixed points determined by the renormalization-group equations. The first one specifies group theoretically, for each value of n, the possible symmetries
Gi∗ of the stable fixed points. The second one formulates a necessary condition for the occurrence of an anisotropic stable fixed point: The normalizer
GN of the considered parameter space must fulfill the condition
GN\ensuremath{\subset}@;
BGi∗ for one, at least, of the former
Gi∗ groups. These rules are shown to be very restrictive for n=4: On the basis of symmetry the lack of a stable fixed point can be asserted for 10 Hamiltonians out of 22, without solving the fixed-point equations.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.
Thermodynamic functions of an
n-component magnetic system coupled to an isotropic elastic medium in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are calculated to leading order in $\ensuremath{\epsilon}$. Solutions to renormalization-group recursion relations correct to $O(\ensuremath{\epsilon})$ along with trajectory-integral methods are employed in these calculations. For
n> 2 the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the
n=1 case renormalization-group predictions of a first-order transition are verified. Thermodynamic functions are given and analyzed in this case. In particular the transition temperature and discontinuity in order parameter are determined and the phase diagram is constructed.