An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to
ϕ4 n-vector models representing universality classes of
m-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory
ε-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.
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.
In this work we investigate the critical behavior of physical systems with competing interactions that present points Lifshitz m-axial. For this study we used the techniques of Quantum Field Theory with Massive Scalar interactions of type λϕ^4 in order to obtain a perturbative expansion for the two-point vertex part up to the 3-loop order and four-point vertex function up to 2-loop level. These vertex functions were regularized using the method of dimensional regularization and renormalized using the minimal subtraction of dimensional poles method.In particular, counterterms have been added to the original Lagrangian, which is the main feature of the method BPHZ (Bogoliubov-Parasiuk-Hepp-Zimmermann). Through the renormalization group ideas, we defined the Wilson functions which shall produce nontrivial fixed points, and from these functions and fixed points, we calculate the anisotropic critical exponents η_τ, to the order of three in number loops, and ν_τ to the number of two in order loops, which characterize the critical behavior of the Lifshitz type m-axial. The exponents calculated using this technique are in perfect agreement with results obtained using other methods, thus confirming the known and important hypothesis of universality.