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arXiv:cond-mat/0208500 · Physical review. B, Condensed matter

Renormalization-group picture of the Lifshitz critical behavior

Marcelo M. Leite

Published Mar 19, 2003• 35 citations
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Abstract

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 $(m=d$ 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 $(N,d,m)$ reduces easily to the Ising-like $(N,d)$ when $m=0.$ We show that the isotropic universality class $(N,m)$ 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 $d=3,N=m=1$ are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.

DOI
10.1103/physrevb.67.104415

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Physical review. B, Condensed matter•Oct 1, 1978•30 citations
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≠d)(m\ensuremath{\ne}d)(m\ensuremath=d) and isotropic (m=d(m=d(m=d close to 8) situations. The general theory is illustrated for the N-vector \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 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 d\ensuremath→m\stackrel{\ensuremath{\rightarrow}}{d}md\ensuremath→​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.
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.
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 ${\ensuremath{\phi}}^{4}$ 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.
An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to ϕ4ϕ^4ϕ4 nnn-vector models representing universality classes of mmm-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory εεε-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.
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).
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.
The phase diagram in the vicinity of a Lifshitz point is analyzed using scaling arguments and renormalization-group calculations. Feynman-graph techniques in $d=4+\frac{1}{2}m\ensuremath{-}\ensuremath{\epsilon}$ dimensions are used to calculate the universal amplitude ratio governing the shape of the two branches of the critical line which intersect at an mmm-fold Lifshitz point. For m<6m<6m<6 the amplitude ratio is found to be $\frac{{A}_{H}(m)}{{A}_{F}(m)}=1\ensuremath{-}\frac{2}{m}+O(\ensuremath{\epsilon})$, where AH(AF){A}_{H}({A}_{F})AH​(AF​) is the amplitude associated with the helical (ferromagnetic) branch, so that to zeroth order in $\ensuremath{\epsilon}$, this ratio is independent of the number nnn of components of the order parameter. The calculation indicates that along the remainder (6< m< 86<~m<~86< m< 8) of the critical dimensionality line, $\ensuremath{\epsilon}=0$, no helical long-range order is expected. We also calculate AHAF\frac{{A}_{H}}{{A}_{F}}AF​AH​​ for a spherical model of a ddd-dimensional system exhibiting an mmm-fold Lifshitz point. Along the critical dimensionality line $\ensuremath{\epsilon}=0$, the spherical model result reduces to that of the renormalization-group calculation.