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 renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in
RbCaF3 and in some liquid-crystal systems.
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.