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arXiv:cond-mat/0106502 · Journal of Physics A Mathematical and General

Liftshitz-point critical behaviour to O(ϵ2)

Hans Werner Diehl, Mykola A. Shpot

Published Oct 15, 2001• 15 citations
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Abstract

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.

DOI
10.1088/0305-4470/34/42/401

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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.
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.
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).
The critical behavior of d-dimensional systems with an n-component order parameter is reconsidered at (m,d,n)-Lifshitz points, where a wave-vector instability occurs in an m-dimensional subspace of Rd.{R}^{d}.Rd. Our aim is to sort out which ones of the previously published partly contradictory \ensuremath{\epsilon}-expansion results to second order in $\ensuremath{\epsilon}=4+m/2\ensuremath{-}d$ are correct. To this end, a field-theory calculation is performed directly in the position space of $d=4+m/2\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using dimensional regularization and minimal subtraction of ultraviolet poles. The residua of the dimensionally regularized integrals that are required to determine the series expansions of the correlation exponents ${\ensuremath{\eta}}_{l2}$ and ${\ensuremath{\eta}}_{l4}$ and of the wave-vector exponent ${\ensuremath{\beta}}_{q}$ to order ${\ensuremath{\epsilon}}^{2}$ are reduced to single integrals, which for general $m=1,\dots{},d\ensuremath{-}1$ can be computed numerically, and for special values of m, analytically. Our results are at variance with the original predictions for general m. For m=2m=2m=2 and m=6,m=6,m=6, we confirm the results of Sak and Grest [Phys. Rev. B 17, 3602 (1978)] and Mergulh\~ao and Carneiro's recent field-theory analysis [Phys. Rev. B 59, 13 954 (1999)].
The critical behaviour of d-dimensional n-vector models at m-axial Lifshitz points is considered for general values of m in the large-n limit. It is proven that the recently obtained large-N expansions [J. Phys.: Condens. Matter 17, S1947 (2005)] of the correlation exponents η_{L2}, η_{L4} and the related anisotropy exponent θare fully consistent with the dimensionality expansions to second order in ε=4+m/2-d [Phys. Rev. B 62, 12338 (2000); Nucl. Phys. B 612, 340 (2001)] inasmuch as both expansions yield the same contributions of order ε^2/n.
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.