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arXiv:cond-mat/0502136 · Physical Review B

Critical behavior of generic competing systems

Marcelo M. Leite

Published Dec 27, 2005• 21 citations
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Abstract

Generic higher character Lifshitz critical behaviors are described using field theory and ${ϵ}_{L}$ expansion renormalization-group methods. These critical behaviors describe systems with arbitrary competing interactions. We derive the scaling relations and the critical exponents at the two-loop level for anisotropic and isotropic points of arbitrary higher character. The framework is illustrated for the $N$-vector ${\ensuremath{\phi}}^{4}$ model describing a $d$-dimensional system. The anisotropic behaviors are derived in terms of many independent renormalization-group transformations, each one characterized by independent correlation lengths. The isotropic behaviors can be understood using only one renormalization-group transformation. Feynman diagrams are solved for the anisotropic behaviors using a different dimensional regularization associated to a generalized orthogonal approximation. The isotropic diagrams are treated using this approximation as well as with a different exact technique to compute the integrals. The entire procedure leads to the analytical solution of generic loop order integrals with arbitrary external momenta. The property of universality class reduction is also satisfied when the competing interactions are turned off. We show how the results presented here reduce to the usual $m$-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.

DOI
10.1103/physrevb.72.224432

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Generic higher character Lifshitz critical behaviors are described using field theory and ϵL{ϵ}_{L}ϵL​ expansion renormalization-group methods. These critical behaviors describe systems with arbitrary competing interactions. We derive the scaling relations and the critical exponents at the two-loop level for anisotropic and isotropic points of arbitrary higher character. The framework is illustrated for the NNN-vector \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model describing a ddd-dimensional system. The anisotropic behaviors are derived in terms of many independent renormalization-group transformations, each one characterized by independent correlation lengths. The isotropic behaviors can be understood using only one renormalization-group transformation. Feynman diagrams are solved for the anisotropic behaviors using a different dimensional regularization associated to a generalized orthogonal approximation. The isotropic diagrams are treated using this approximation as well as with a different exact technique to compute the integrals. The entire procedure leads to the analytical solution of generic loop order integrals with arbitrary external momenta. The property of universality class reduction is also satisfied when the competing interactions are turned off. We show how the results presented here reduce to the usual mmm-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.
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.
We introduce the method of minimal subtraction in the computation of critical exponents of Lifshitz type generic competing systems using massless fields. We first treat the anisotropic cases, when several independent momentum scales define the renormalization group invariance of the scalar fields. In addition, we analyze the isotropic sector. We compute critical exponents using diagrammatic techniques at least up to two-loop level and show their equivalence with other methods presented in the literature.
The specific heat amplitude ratios for generic anisotropic as well as isotropic Lifshitz critical behaviors for the N-vector model are computed at one-loop level using the ε L - expansion. In the anisotropic case, we show that the universal result obtained reduces very easily to that from the simpler m-axial universality class result. In the isotropic case, if n is the number of neighbors coupled via competing interactions, we demonstrate that the ratio vanishes close to n = 4 and, becomes negative for n > 4 when it is calculated exactly, which is rather odd. The evaluation using the orthogonal approximation is shown to yield positive results for arbitrary n. Explicit computations for the case n = 2, d = 3, N = 1 yield an exact amplitude ratio equal to 0.06, with the approximate amplitude ratio being 1.17.We discuss two physical mechanisms to pick out one of the amplitude ratio values. We propose an experiment in homopolymer-diblock copolymer blends in order to determine the amplitude ratio.
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 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.
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).