A new picture of the Lifshitz critical behavior

TL;DR

This paper develops new field theoretic renormalization group methods to analyze Lifshitz critical behavior, providing a unified approach for anisotropic and isotropic cases at two-loop order, with results matching simulations.

Contribution

It introduces a novel regularization and renormalization framework for Lifshitz points, unifying anisotropic and isotropic cases and enabling analytical calculations of loop diagrams.

Findings

Analytical expressions for critical exponents at two-loop order.

Good agreement with Monte Carlo simulations for specific cases.

Clarification of the relationship between isotropic and anisotropic universality classes.

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.