Journal of Physics Conference Series

Minimal subtraction formulation of massless fields for generic competing systems

Messias V. S. Santos, Marcelo M. Leite

Published Mar 11, 2014• 5 citations

Abstract

We introduce the method of minimal subtraction in the computation of critical exponents of Lifshitz type generic competing systems using massless fields. We first treat the anisotropic cases, when several independent momentum scales define the renormalization group invariance of the scalar fields. In addition, we analyze the isotropic sector. We compute critical exponents using diagrammatic techniques at least up to two-loop level and show their equivalence with other methods presented in the literature.

DOI: 10.1088/1742-6596/490/1/012232

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Journal of Physics Conference Series

Minimal subtraction formulation of massless fields for generic competing systems

Messias V. S. Santos, Marcelo M. Leite

Published Mar 11, 2014• 5 citations

Publisher

Abstract

DOI: 10.1088/1742-6596/490/1/012232

Related papers

Critical behavior of generic competing systems

Marcelo M. Leite

Generic higher character Lifshitz critical behaviors are described using field theory and ${ϵ}_{L}$ 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 $N$ -vector ${\ensuremath{\phi}}^{4}$ model describing a $d$ -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 $m$ -fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.Generic higher character Lifshitz critical behaviors are described using field theory and ${ϵ}_{L}$ 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 $N$-vector ${\ensuremath{\phi}}^{4}$ model describing a $d$-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 $m$-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.

arXiv:cond-mat/0502136Physical Review BDec 27, 200521 citations

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Critical Behavior at M-Axial Lifshitz Points

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Field Theoretical Approach to Bicritical and Tetracritical Behavior: Static and Dynamics

R. Folk, Yurij V. Holovatch, G. Moser

We discuss the static and dynamic multicritical behavior of three-dimensional systems of $O(n_\|)\oplus O(n_\perp)$ symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field ( $n_\|=1$ , $n_\perp=2$ ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.We discuss the static and dynamic multicritical behavior of three-dimensional systems of $O(n_\|)\oplus O(n_\perp)$ symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field ($n_\|=1$, $n_\perp=2$) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.

arXiv:0906.3624Jun 19, 20090 citations

Generic higher character Lifshitz critical behaviors are described using field theory and

{ϵ}_{L}

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

N

-vector ${\ensuremath{\phi}}^{4}$ model describing a

d

-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

m

-fold Lifshitz critical behaviors for both isotropic and anisotropic criticalities.

(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

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.

We discuss the static and dynamic multicritical behavior of three-dimensional systems of

O(n_\|)\oplus O(n_\perp)

symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field (

n_\|=1

n_\perp=2

) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.