We employ the derivative expansion of the nonperturbative renormalization group to address the phenomenon of anisotropic scale invariance and the associated functional fixed points, also known as Lifshitz points, in systems characterized by a scalar order parameter. We demonstrate the existence of the Lifshitz fixed point featuring a nonclassical value of the anisotropy exponent θ<1/2 and provide estimates for values of a set of critical exponents in the physically most relevant case of the three-dimensional uniaxial Lifshitz point (d,m)=(3,1), m denoting the anisotropy index. We compare our predictions with existing estimates from perturbative expansions around dimensionality d=4+1/2 as well as those from the 1/N expansion.
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.
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.
In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(
ki2${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant
k2 term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.
A two-loop renormalization group analysis of the critical behaviour at an isotropic Lifshitz point is presented. Using dimensional regularization and minimal subtraction of poles, we obtain the expansions of the critical exponents ν and η, the crossover exponent φ, as well as the (related) wave vector exponent β q and the correction-to-scaling exponent ω to second order in 8 = 8 − d . They are compared with the authors' recent -expansion results (2000 Phys. Rev. B 62 12338, 2001 Nucl. Phys. B 612 340) for the general case of an m -axial Lifshitz point. It is shown that the expansions obtained here by a direct calculation for the isotropic ( m = d ) Lifshitz point all follow from the latter upon setting m = 8 − 8 . This is so despite recent claims to the contrary by de Albuquerque and Leite (2002 J. Phys. A: Math. Gen. 35 1807).
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 presence of isotropic Lifshitz points for a
U(1) symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension
d=4. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the renormalization group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found that present evident similarities with the properties shown by the line of fixed points observed in the two-dimensional Berezinskii-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at a large distance of the order parameter correlations.