The d-dimensional time-dependent Ginzburg-Landau (TDGL) model is mapped onto a special (d+1)-dimensional model which exhibits a Lifshitz tricritical point (LTP). Many of the LTP critical properties follow from those of the TDGL model, and are shown to belong to a novel universality class of LTP's which results from a (previously ignored) relevant, nonlocal, quartic spin operator. These properties are analyzed with the use of scaling, an \ensuremath{\epsilon} expansion, and the n\ensuremath{\rightarrow}\ensuremath{\infty} limit.
A new renormalization group treatment is proposed for the critical exponents of an m-fold Lifshitz point. The anisotropic cases (m not equal 8) are described by two independent fixed points associated to two independent momentum flow along the quadratic and quartic directions, respectively. The isotropic case is described separately. In that case, the fixed point is due to renormalization group transformations along the quartic directions. The new scaling laws are derived for both cases and generalize the ones previously reported.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute
m-vector model, the
n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic
m-component fixed point, which characterizes the critical behavior of the pure system, is stable. For
m<mc, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for
m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
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.
Field-theoretic renormalization-group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic $(m\ensuremath{\ne}d)$ and isotropic
(m=d close to 8) situations. The general theory is illustrated for the N-vector ${\ensuremath{\varphi}}^{4}$ model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization-group transformations. The description of the isotropic behavior requires only one type of renormalization-group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization-group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class
(N,d,m) reduces easily to the Ising-like
(N,d) when
m=0. We show that the isotropic universality class
(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit $\stackrel{\ensuremath{\rightarrow}}{d}m$ near 8. The exponents for the uniaxial case
d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.
The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point (
m=1) the expansion is made in $\ensuremath{\epsilon}=d\ensuremath{-}2.5$ and the critical exponents are calculated. The results are ${\ensuremath{\lambda}}_{\mathrm{t}}={\ensuremath{\nu}}_{4}^{\ensuremath{-}1}=2\ensuremath{\epsilon}$, ${\ensuremath{\eta}}_{2}=\frac{\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\eta}}_{4}=\frac{2\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\beta}}_{k}=\frac{1}{2}+\frac{3\ensuremath{\epsilon}}{4(n\ensuremath{-}2)}$, where
n is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.