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Physical review. B, Condensed matter

Field-theoretic approach to the Lifshitz point

C. Mergulhão, C. E. I. Carneiro

Published Sep 1, 1998• 28 citations
Publisher

Abstract

We study the renormalization of the field theory that describes the Lifshitz point (LP). Our motivation was an old controversy on the order-${\ensuremath{\epsilon}}^{2}$ values of critical exponents for this multicritical point. First we analyze the Green functions at the LP where some simplifications occur. The primitively divergent diagrams are identified and renormalization prescriptions that eliminate ultraviolet divergences to all orders of perturbation are found. The Green functions in the neighborhood of the LP are expanded in terms of the Green functions calculated at the LP. This enables us to derive the renormalization-group equation satisfied by the renormalized Green functions and by analyzing its solutions we find expressions for the critical exponents that hold to all orders of perturbation. Finally, we obtain generalized scaling relations for the exponents associated with the LP.

DOI
10.1103/physrevb.58.6047

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We study the renormalization of the field theory that describes the Lifshitz point (LP). Our motivation was an old controversy on the order-\ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 values of critical exponents for this multicritical point. First we analyze the Green functions at the LP where some simplifications occur. The primitively divergent diagrams are identified and renormalization prescriptions that eliminate ultraviolet divergences to all orders of perturbation are found. The Green functions in the neighborhood of the LP are expanded in terms of the Green functions calculated at the LP. This enables us to derive the renormalization-group equation satisfied by the renormalized Green functions and by analyzing its solutions we find expressions for the critical exponents that hold to all orders of perturbation. Finally, we obtain generalized scaling relations for the exponents associated with the LP.
We use field theory to calculate the critical exponents ${\ensuremath{\eta}}_{l2},$ ${\ensuremath{\eta}}_{l4},$ ${\ensuremath{\nu}}_{l2},$ and ${\ensuremath{\nu}}_{l4}$ for the Lifshitz point (LP) with m=2m=2m=2 and m=6m=6m=6. We were motivated by an old controversy on the order ${\ensuremath{\epsilon}}^{2}$ corrections for the exponents of this multicritical point. In a previous paper, we studied the renormalization of the theory that describes the LP and derived expressions for the exponents and scaling relations that hold to all orders in perturbation. In this work we concentrate on the calculation of the Feynman diagrams that are involved in the determination of the exponents and present our results for the exponents to order ${\ensuremath{\epsilon}}^{2}.$ The evaluation of these diagrams, due to a free propagator that contains a quartic term in the momentum and is not rotationally invariant, requires special techniques that are presented in detail in the appendixes.
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(m=d(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)(N,d,m)(N,d,m) reduces easily to the Ising-like (N,d)(N,d)(N,d) when m=0.m=0.m=0. We show that the isotropic universality class (N,m)(N,m)(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=1d=3,N=m=1d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.
Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 ndashndashndash d)2
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 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.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
Abstract Chapter 5 first recalls the importance of the concept of scale decoupling in physics. It then emphasizes that quantum field theory and the theory of critical phenomena have provided two examples where this concepts fails. To deal with such a situation, a new tool has been invented: the renormalization group. In the framework of effective quantum field theory, a perturbative renormalization group has been formulated. Its implementation has led to the discovery of fixed points as zeros of beta functions, and calculations of critical exponents of a class of macroscopic phase transitions in the form of Wilson–Fisher epsilon or fixed dimension expansions. These expansions being divergent, they could summed by methods based on the Borel transformation and the determination of the large order behaviour of perturbation theory.