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 $ϕ^4$ $n$ -vector models representing universality classes of $m$ -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 $ε$ -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.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 $ϕ^4$ $n$-vector models representing universality classes of $m$-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 $ε$-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.

arXiv:cond-mat/0205284Acta Physica Slovaca Reviews and TutorialsMay 14, 200211 citations

Callan–Symanzik method for m-axial Lifshitz points

Paulo R.S. Carvalho, Marcelo M. Leite

arXiv:0704.3208Annals of PhysicsJun 1, 200818 citations

Critical behavior atm-axial Lifshitz points: Field-theory analysis and ε-expansion results

Hans Werner Diehl, Mykola A. Shpot

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 ${R}^{d}.$ 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 $m=2$ and $m=6,$ 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 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 ${R}^{d}.$ 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 $m=2$ and $m=6,$ 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)].

arXiv:cond-mat/0006007Physical review. B, Condensed matterNov 1, 200066 citations

Método de BPHZ para Pontos de Lifshitz m-Axiais Anisotrópicos

Emanuel Veras de Souza

In this work we investigate the critical behavior of physical systems with competing interactions that present points Lifshitz m-axial. For this study we used the techniques of Quantum Field Theory with Massive Scalar interactions of type λϕ^4 in order to obtain a perturbative expansion for the two-point vertex part up to the 3-loop order and four-point vertex function up to 2-loop level. These vertex functions were regularized using the method of dimensional regularization and renormalized using the minimal subtraction of dimensional poles method.In particular, counterterms have been added to the original Lagrangian, which is the main feature of the method BPHZ (Bogoliubov-Parasiuk-Hepp-Zimmermann). Through the renormalization group ideas, we defined the Wilson functions which shall produce nontrivial fixed points, and from these functions and fixed points, we calculate the anisotropic critical exponents η_τ, to the order of three in number loops, and ν_τ to the number of two in order loops, which characterize the critical behavior of the Lifshitz type m-axial. The exponents calculated using this technique are in perfect agreement with results obtained using other methods, thus confirming the known and important hypothesis of universality.In this work we investigate the critical behavior of physical systems with competing interactions that present points Lifshitz m-axial. For this study we used the techniques of Quantum Field Theory with Massive Scalar interactions of type λϕ^4 in order to obtain a perturbative expansion for the two-point vertex part up to the 3-loop order and four-point vertex function up to 2-loop level. These vertex functions were regularized using the method of dimensional regularization and renormalized using the minimal subtraction of dimensional poles method.In particular, counterterms have been added to the original Lagrangian, which is the main feature of the method BPHZ (Bogoliubov-Parasiuk-Hepp-Zimmermann). Through the renormalization group ideas, we defined the Wilson functions which shall produce nontrivial fixed points, and from these functions and fixed points, we calculate the anisotropic critical exponents η_τ, to the order of three in number loops, and ν_τ to the number of two in order loops, which characterize the critical behavior of the Lifshitz type m-axial. The exponents calculated using this technique are in perfect agreement with results obtained using other methods, thus confirming the known and important hypothesis of universality.

arXiv:1403.1998Mar 8, 20140 citations

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

ϕ^4

n

-vector models representing universality classes of

m

-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

ε

-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.

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

{R}^{d}.

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=2

and

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)].