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arXiv:cond-mat/0006462

Uniaxial Lifshitz Point at $O(ε_L^2)$

Luiz C. de Albuquerque, Marcelo M. Leite

Published Jun 29, 2000• 1 citation
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Abstract

The critical exponents $ν_{L2}, η_{L2}$ and $γ_{L2}$ of a uniaxial Lifshitz point are calculated at two-loop level using renormalization group and $ε_{L}$-expansion techniques. We found a new constraint involving the loop momenta along the competition axis, which allows to solve the two-loop integrals. The exponent $γ_{L2}$ obtained using our method is in good agreement with numerical estimates based on Monte Carlo simulations.

DOI
10.48550/arxiv.cond-mat/0006462

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We reply to a recent comment by Diehl and Shpot ( 2001 J. Phys. A: Math. Gen. 34 9101 ) criticizing our paper (Albuquerque L C and Leite M M 2001 J. Phys. A: Math. Gen. 34 L327 ). We show that the approximation we use for evaluating higher loop integrals is consistent with homogeneity. A new renormalization group approach is presented in order to compare the two methods with high-precision numerical data concerning the uniaxial case. We stress that isotropic behaviour cannot be obtained from anisotropic behaviour.

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Uniaxial Lifshitz Point at O(εL2)O(ε_L^2)O(εL2​)
The critical exponents νL2,ηL2ν_{L2}, η_{L2}νL2​,ηL2​ and γL2γ_{L2}γL2​ of a uniaxial Lifshitz point are calculated at two-loop level using renormalization group and εLε_{L}εL​-expansion techniques. We found a new constraint involving the loop momenta along the competition axis, which allows to solve the two-loop integrals. The exponent γL2γ_{L2}γL2​ obtained using our method is in good agreement with numerical estimates based on Monte Carlo simulations.
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.
The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point (m=1m=1m=1) the expansion is made in $\ensuremath{\epsilon}=d\ensuremath{-}2.5$ and the critical exponents are calculated. The results are ${\ensuremath{\lambda}}_{\mathrm{t}}={\ensuremath{\nu}}_{4}^{\ensuremath{-}1}=2\ensuremath{\epsilon}$, ${\ensuremath{\eta}}_{2}=\frac{\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\eta}}_{4}=\frac{2\ensuremath{\epsilon}}{(n\ensuremath{-}2)}$, ${\ensuremath{\beta}}_{k}=\frac{1}{2}+\frac{3\ensuremath{\epsilon}}{4(n\ensuremath{-}2)}$, where nnn is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.
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).
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.
We reply to a recent comment by Diehl and Shpot ( 2001 J. Phys. A: Math. Gen. 34 9101 ) criticizing our paper (Albuquerque L C and Leite M M 2001 J. Phys. A: Math. Gen. 34 L327 ). We show that the approximation we use for evaluating higher loop integrals is consistent with homogeneity. A new renormalization group approach is presented in order to compare the two methods with high-precision numerical data concerning the uniaxial case. We stress that isotropic behaviour cannot be obtained from anisotropic behaviour.