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arXiv:cond-mat/0108254 · Journal of Physics A Mathematical and General

Reply to ‘Lifshitz-point critical behaviour toO(ϵL2)’

Luiz C. de Albuquerque, Marcelo M. Leite

Published Feb 11, 2002• 5 citations
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Abstract

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.

DOI
10.1088/0305-4470/35/7/402

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arXiv:cond-mat/0006462•Jun 29, 2000•1 citation
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.
We comment on a recent letter by de Albuquerque and Leite ( 2001 J. Phys. A: Math. Gen. 34 L327 ), in which results to the second order in = 4 − d + m /2 were presented for the critical exponents ν L2 , η L2 and γ L2 of d -dimensional systems at m -axial Lifshitz points. We point out that their results are at variance with ours. The discrepancy is due to their incorrect computation of momentum-space integrals. Their speculation that the field-theoretic renormalization group approach, if performed in position space, might give results different from when it is performed in momentum space is refuted.
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.
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.