The phase diagram in the vicinity of a Lifshitz point is analyzed using scaling arguments and renormalization-group calculations. Feynman-graph techniques in $d=4+\frac{1}{2}m\ensuremath{-}\ensuremath{\epsilon}$ dimensions are used to calculate the universal amplitude ratio governing the shape of the two branches of the critical line which intersect at an
m-fold Lifshitz point. For
m<6 the amplitude ratio is found to be $\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})$, where
AH(AF) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in $\ensuremath{\epsilon}$, this ratio is independent of the number
n of components of the order parameter. The calculation indicates that along the remainder (
6< m< 8) of the critical dimensionality line, $\ensuremath{\epsilon}=0$, no helical long-range order is expected. We also calculate
AFAH for a spherical model of a
d-dimensional system exhibiting an
m-fold Lifshitz point. Along the critical dimensionality line $\ensuremath{\epsilon}=0$, the spherical model result reduces to that of the renormalization-group calculation.
The critical properties of d dimensional systems with a quadratic anisotropic term in the Hamiltonian are investigated with the aid of a renormalization group analysis based upon the Wilson perturbation expansion in epsilon =4-d. For anisotropic cubic systems with a nonzero four-spin coupling constant, u, it is found that the asymptotic critical behaviour is always one with vanishing quadratic anisotropy. In systems with lattice anisotropy a simple renormalization group argument leads to results in agreement with the universality hypothesis.
. -- We examine the crossover from classical to non-classical critical behaviour in twodimensional systems with a one-component order parameter. Since the degree of universality of the corresponding crossover functions is still subject to debate, we try to induce non-universal e#ects by adding interactions with a second length scale. Although the crossover functions clearly depend on the range of the interactions, they turn out to be remarkably robust against further variation of the interaction profile. In particular, we find that the earlier observed non-monotonic crossover of the e#ective susceptibility exponent occurs for several qualitatively di#erent shapes of this profile. Introduction. -- In recent years, there has been a revived interest in the nature of the crossover from classical to non-classical (asymptotic) critical behaviour upon approach of the critical point. This crossover between two universality classes can be observed in a great variety of many-body systems, inclu...
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.