Physical review. B, Condensed matter

Field-theoretic calculation of critical exponents for the Lifshitz point

C. Mergulhão, C. E. I. Carneiro

Published Jun 1, 1999• 23 citations

Abstract

We use field theory to calculate the critical exponents ${\ensuremath{\eta}}_{l2},$ ${\ensuremath{\eta}}_{l4},$ ${\ensuremath{\nu}}_{l2},$ and ${\ensuremath{\nu}}_{l4}$ for the Lifshitz point (LP) with $m=2$ and $m=6$. We were motivated by an old controversy on the order ${\ensuremath{\epsilon}}^{2}$ corrections for the exponents of this multicritical point. In a previous paper, we studied the renormalization of the theory that describes the LP and derived expressions for the exponents and scaling relations that hold to all orders in perturbation. In this work we concentrate on the calculation of the Feynman diagrams that are involved in the determination of the exponents and present our results for the exponents to order ${\ensuremath{\epsilon}}^{2}.$ The evaluation of these diagrams, due to a free propagator that contains a quartic term in the momentum and is not rotationally invariant, requires special techniques that are presented in detail in the appendixes.

DOI: 10.1103/physrevb.59.13954

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Field-theoretic approach to the Lifshitz point

C. Mergulhão, C. E. I. Carneiro

Physical review. B, Condensed matterSep 1, 199828 citations

Renormalization-group picture of the Lifshitz critical behavior

Marcelo M. Leite

arXiv:cond-mat/0208500Physical review. B, Condensed matterMar 19, 200335 citations

Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 $ndash$ d)2

Hans Werner Diehl, Mykola A. Shpot

arXiv:cond-mat/0204267Journal of Physics A Mathematical and GeneralJul 18, 200224 citations

New renormalization group approach and scaling laws for the Lifshitz critical behavior

Marcelo M. Leite

arXiv:hep-th/0109037Sep 5, 20010 citations

Low-temperature renormalization group for the Lifshitz point

Gary S. Grest, Josef Sak

Physical review. B, Condensed matterMay 1, 197834 citations

Feynman - graph calculation of the (0, l ) critical exponents to order ε2

A. Houghton, Franz Wegner

Physical Review AJul 1, 197422 citations

Physical review. B, Condensed matter

Field-theoretic calculation of critical exponents for the Lifshitz point

C. Mergulhão, C. E. I. Carneiro

Published Jun 1, 1999• 23 citations

Publisher

Abstract

We use field theory to calculate the critical exponents

{\ensuremath{\eta}}_{l2},

{\ensuremath{\eta}}_{l4},

{\ensuremath{\nu}}_{l2},

and

{\ensuremath{\nu}}_{l4}

for the Lifshitz point (LP) with

m=2

and

m=6

. We were motivated by an old controversy on the order

{\ensuremath{\epsilon}}^{2}

corrections for the exponents of this multicritical point. In a previous paper, we studied the renormalization of the theory that describes the LP and derived expressions for the exponents and scaling relations that hold to all orders in perturbation. In this work we concentrate on the calculation of the Feynman diagrams that are involved in the determination of the exponents and present our results for the exponents to order

{\ensuremath{\epsilon}}^{2}.

The evaluation of these diagrams, due to a free propagator that contains a quartic term in the momentum and is not rotationally invariant, requires special techniques that are presented in detail in the appendixes.

DOI: 10.1103/physrevb.59.13954

Related papers

Field-theoretic approach to the Lifshitz point

C. Mergulhão, C. E. I. Carneiro

We study the renormalization of the field theory that describes the Lifshitz point (LP). Our motivation was an old controversy on the order-${\ensuremath{\epsilon}}^{2}$ values of critical exponents for this multicritical point. First we analyze the Green functions at the LP where some simplifications occur. The primitively divergent diagrams are identified and renormalization prescriptions that eliminate ultraviolet divergences to all orders of perturbation are found. The Green functions in the neighborhood of the LP are expanded in terms of the Green functions calculated at the LP. This enables us to derive the renormalization-group equation satisfied by the renormalized Green functions and by analyzing its solutions we find expressions for the critical exponents that hold to all orders of perturbation. Finally, we obtain generalized scaling relations for the exponents associated with the LP.We study the renormalization of the field theory that describes the Lifshitz point (LP). Our motivation was an old controversy on the order-${\ensuremath{\epsilon}}^{2}$ values of critical exponents for this multicritical point. First we analyze the Green functions at the LP where some simplifications occur. The primitively divergent diagrams are identified and renormalization prescriptions that eliminate ultraviolet divergences to all orders of perturbation are found. The Green functions in the neighborhood of the LP are expanded in terms of the Green functions calculated at the LP. This enables us to derive the renormalization-group equation satisfied by the renormalized Green functions and by analyzing its solutions we find expressions for the critical exponents that hold to all orders of perturbation. Finally, we obtain generalized scaling relations for the exponents associated with the LP.

Physical review. B, Condensed matterSep 1, 199828 citations

Renormalization-group picture of the Lifshitz critical behavior

Marcelo M. Leite

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.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.

arXiv:cond-mat/0208500Physical review. B, Condensed matterMar 19, 200335 citations

Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 $ndash$ d)2Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8 $ndash$ d)2

Hans Werner Diehl, Mykola A. Shpot

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 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).

arXiv:cond-mat/0204267Journal of Physics A Mathematical and GeneralJul 18, 200224 citations

New renormalization group approach and scaling laws for the Lifshitz critical behavior

Marcelo M. Leite

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.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.

arXiv:hep-th/0109037Sep 5, 20010 citations

Low-temperature renormalization group for the Lifshitz point

Gary S. Grest, Josef Sak

The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point ( $m=1$ ) 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 $n$ is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point ($m=1$) 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 $n$ is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.

Physical review. B, Condensed matterMay 1, 197834 citations

Feynman - graph calculation of the (0, l ) critical exponents to order ε2

A. Houghton, Franz Wegner

It is shown that the critical indices corresponding to (0, $l$ perturbations can be calculated by a Feynman-diagram method. At order ${\ensuremath{\epsilon}}^{2}$, the results are consistent with the special cases $l=1,2, \mathrm{and} 4$ obtained previously. There is also general agreement to order $\ensuremath{\epsilon}$ with the exponents calculated from Wilson's recurrence relation.It is shown that the critical indices corresponding to (0, $l$ perturbations can be calculated by a Feynman-diagram method. At order ${\ensuremath{\epsilon}}^{2}$, the results are consistent with the special cases $l=1,2, \mathrm{and} 4$ obtained previously. There is also general agreement to order $\ensuremath{\epsilon}$ with the exponents calculated from Wilson's recurrence relation.

Physical Review AJul 1, 197422 citations

We use field theory to calculate the critical exponents

{\ensuremath{\eta}}_{l2},

{\ensuremath{\eta}}_{l4},

{\ensuremath{\nu}}_{l2},

and

{\ensuremath{\nu}}_{l4}

for the Lifshitz point (LP) with

m=2

and

m=6

. We were motivated by an old controversy on the order

{\ensuremath{\epsilon}}^{2}

{\ensuremath{\epsilon}}^{2}.

(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.

Critical, crossover and correction-to-scaling exponents for isotropic Lifshitz points to order (8

ndash

d)2

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=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

n

is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.

It is shown that the critical indices corresponding to (0,

l

perturbations can be calculated by a Feynman-diagram method. At order ${\ensuremath{\epsilon}}^{2}$, the results are consistent with the special cases

l=1,2, \mathrm{and} 4

obtained previously. There is also general agreement to order $\ensuremath{\epsilon}$ with the exponents calculated from Wilson's recurrence relation.