arXiv:1412.7046 · Nuclear Physics B

Isotropic Lifshitz critical behavior from the functional renormalization group

Alfio Maurizio Bonanno, Dario Zappalà

Published Feb 19, 2015• 16 citations

Abstract

The Lifshitz critical behavior for a single component field theory is studied for the specific isotropic case in the framework of the Functional Renormalization Group. Lifshitz fixed point solutions of the flow equation, derived by using a Proper Time regulator, are searched at lowest and higher order in the derivative expansion. Solutions are found when the number of spatial dimensions d is contained within the interval 5.5<d<8.

DOI: 10.1016/j.nuclphysb.2015.02.013

Related papers

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

Marcelo M. Leite

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

Indications of isotropic Lifshitz points in four dimensions

Dario Zappalà

arXiv:1806.00043Physical review. D/Physical review. D.Oct 8, 201814 citations

Renormalization-group picture of the Lifshitz critical behavior

Marcelo M. Leite

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

Renormalization-group analysis of Lifshitz tricritical behavior

Amnon Aharony A, Eytan Domany, Richard M. Hornreich

Physical review. B, Condensed matterAug 1, 198721 citations

Critical behavior of generic competing systems

Marcelo M. Leite

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

Isotropic Lifshitz scaling in four dimensions

Dario Zappalà

The presence of isotropic Lifshitz points for a [Formula: see text]-symmetric scalar theory is investigated with the help of the Functional Renormalization Group. In particular, at the supposed lower critical dimension [Formula: see text], evidence for a continuous line of fixed points is found for the [Formula: see text] theory, and the observed structure presents clear similarities with the properties observed in the two-dimensional Berezinskii–Kosterlitz–Thouless phase.

arXiv:1912.03071International Journal of Geometric Methods in Modern PhysicsFeb 11, 20206 citations

arXiv:1412.7046 · Nuclear Physics B

Isotropic Lifshitz critical behavior from the functional renormalization group

Alfio Maurizio Bonanno, Dario Zappalà

Published Feb 19, 2015• 16 citations

arXiv HTML PDF

Abstract

DOI: 10.1016/j.nuclphysb.2015.02.013

Related papers

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

Indications of isotropic Lifshitz points in four dimensions

Dario Zappalà

The presence of isotropic Lifshitz points for a $U(1)$ symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension $d=4$ . To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the renormalization group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found that present evident similarities with the properties shown by the line of fixed points observed in the two-dimensional Berezinskii-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at a large distance of the order parameter correlations.The presence of isotropic Lifshitz points for a $U(1)$ symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension $d=4$. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the renormalization group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found that present evident similarities with the properties shown by the line of fixed points observed in the two-dimensional Berezinskii-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at a large distance of the order parameter correlations.

arXiv:1806.00043Physical review. D/Physical review. D.Oct 8, 201814 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

Renormalization-group analysis of Lifshitz tricritical behavior

Amnon Aharony A, Eytan Domany, Richard M. Hornreich

A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in ${\mathrm{RbCaF}}_{3}$ and in some liquid-crystal systems.A renormalization-group analysis of critical behavior near a Lifshitz tricritical point (LTP) is presented, with emphasis on the role played by new, momentum-dependent quartic terms. These result in new stable fixed points which determine the critical behavior. For some values of n (the number of order-parameter components) and m (the dimensionality of the ``soft'' subspace, characterized by quartic momentum-dependent inverse correlation functions), the renormalization-group recursion relations have two stable and accessible fixed points. However, one of these can never be reached in practice, due to a thermodynamic instability which results in a first-order phase transition. For m=d-1, one of the fixed points describes the critical dynamics of the usual n-vector spin model in (d-1) dimensions. This dynamic fixed point also characterizes LTP behavior for large n and n=1. In all other cases, the LTP has new exponents, which are not related to the dynamic model. Our results may be relevant to Lifshitz tricritical behavior in ${\mathrm{RbCaF}}_{3}$ and in some liquid-crystal systems.

Physical review. B, Condensed matterAug 1, 198721 citations

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

Isotropic Lifshitz scaling in four dimensions

Dario Zappalà

arXiv:1912.03071International Journal of Geometric Methods in Modern PhysicsFeb 11, 20206 citations

The presence of isotropic Lifshitz points for a

U(1)

symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension

d=4

. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the renormalization group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found that present evident similarities with the properties shown by the line of fixed points observed in the two-dimensional Berezinskii-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at a large distance of the order parameter correlations.

(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.

{\mathrm{RbCaF}}_{3}

and in some liquid-crystal systems.

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.