Exact renormalization group equation for the Lifshitz critical point

TL;DR

This paper derives an exact anisotropic renormalization group equation for Lifshitz critical points, analyzing their critical behavior and stability, and comparing results with perturbative calculations.

Contribution

It introduces an exact renormalization group equation for anisotropic systems and studies Lifshitz points beyond perturbation theory.

Findings

Estimated critical exponents agree with $O(\epsilon^2)$ calculations

Identified instability of the Lifshitz tricritical fixed point due to a marginally relevant coupling

Demonstrated limitations of perturbative approaches for certain fixed points

Abstract

An exact renormalization equation (ERGE) accounting for an anisotropic scaling is derived. The critical and tricritical Lifshitz points are then studied at leading order of the derivative expansion which is shown to involve two differential equations. The resulting estimates of the Lifshitz critical exponents compare well with the $O(\epsilon ^{2}) $ calculations. In the case of the Lifshitz tricritical point, it is shown that a marginally relevant coupling defies the perturbative approach since it actually makes the fixed point referred to in the previous perturbative calculations $O(\epsilon) $ finally unstable.