A new picture of the Lifshitz critical behavior

Abstract

New 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 not equal to d) and isotropic (m=d close to 8) situations. The general theory is illustrated for the N-vector phi^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 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 ANNNI model.

48 pages, no figures, two typos fixed

References in corpus (9)

• Critical behavior at m-axial Lifshitz points: field-theory analysis and $\epsilon$-expansion results
• Two-loop renormalization-group analysis of critical behavior at m-axial Lifshitz points
• Anisotropic scaling and generalized conformal invariance at Lifshitz points
• Critical, crossover, and correction-to-scaling exponents for isotropic Lifshitz points to order $(8-\mathbf{d}{)}^2$
• Anisotropic Lifshitz Point at $O({\epsilon }_L^2)$
• Susceptibility Amplitude Ratios Near a Lifshitz Point
• Lifshitz-point critical behaviour to $\mathbf{O}({\mathbf{\epsilon }}^2)$
• Surface critical exponents at a uniaxial Lifshitz point
• Reply on "Lifshitz-point critical behaviour to $O({\epsilon }_L^2)$"

Cited by in corpus (29)

• Renormalization of Lorentz violating theories
• Weighted power counting and Lorentz violating gauge theories. I: General properties
• Weighted scale invariant quantum field theories
• Callan-Symanzik-Lifshitz approach to generic competing systems
• Mass Renormalization in Lorentz-violating Scalar Field Theory
• Large-n expansion for m-axial Lifshitz points
• Critical behavior of generic competing systems
• Callan-Symanzik method for $m$-axial Lifshitz points
• New universality classes for generic higher character Lifshitz points
• Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points
• Three-loop field renormalization for scalar field theory with Lorentz violation
• Specific heat amplitude ratios for anisotropic Lifshitz critical behaviors
• Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories
• Bulk and Boundary Critical Behavior at Lifshitz Points
• Comment on ``Renormalization-group picture of the Lifshitz critical behavior''
• A study of the (m,d,N)=(1,3,2) Lifshitz point and of the three- dimensional XY universality class by high-temperature bivariate series for the XY models with anisotropic competing interactions
• Critical exponents from parallel plate geometries subject to periodic and antiperiodic boundary conditions
• Critical properties of scalar field theory with Lorentz violation: Exact treatment of Lorentz-violating mechanism
• Susceptibility amplitude ratio for generic competing systems
• Experimental validation of nonextensive statistical field theory: applications to manganites
• Nonextensive statistical field theory
• Asymptotic single-particle states and exact Lorentz-violating all-loop quantum corrections for scalar field theory
• Is Kaniadakis $\kappa$-generalized statistical mechanics general?
• Multicriticality of the (2+1)-dimensional gonihedric model: A realization of the (d,m)=(3,2) Lifshitz point
• Is ${\gamma }_{KLS}$-generalized statistical field theory complete?
• Probing the two-scale-factor universality hypothesis by exact rotation symmetry-breaking mechanism
• Reply to "Comment on Renormalization group picture of the Lifshitz critical behaviors"
• On the completeness of the ${\delta }_{KLS}$-generalized statistical field theory
• End-point singularity of the $XY$-paramagnetic phase boundary for the $(2+1)$D $S=1$ square-lattice $J_1$-$J_2$ $XY$ model with the single-ion anisotropy