Critical behavior of generic competing systems

TL;DR

This paper develops a comprehensive field-theoretic framework to analyze generic Lifshitz critical behaviors in systems with competing interactions, deriving critical exponents and scaling relations at two-loop order.

Contribution

It introduces a novel renormalization group approach for higher character Lifshitz points, including new techniques for evaluating complex Feynman diagrams and integrals.

Findings

Derived critical exponents for anisotropic and isotropic Lifshitz points.

Established scaling relations at two-loop level.

Unified framework for arbitrary higher character Lifshitz critical behaviors.

Abstract

Generic higher character Lifshitz critical behaviors are described using field theory and $\epsilon_{L}$-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 $\phi^{4}$ 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 new dimensional regularization associated to a generalized orthogonal approximation. The isotropic diagrams are treated using this approximation as well as with a new 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.