Poincar\'e-like approach to Landau Theory. I. General theory

TL;DR

This paper introduces a method combining Michel's orbit space reduction and Poincaré normalization to simplify Landau potentials, enabling easier analysis of phase transition models with reduced parameters.

Contribution

It presents a novel approach that simplifies Landau potentials using a chain of coordinate transformations in orbit space, improving upon previous methods.

Findings

Method effectively reduces the complexity of Landau polynomials.

Significant parameter reduction achieved in examples.

Applicable to various phase transition models.

Abstract

We discuss a procedure to simplify the Landau potential, based on Michel's reduction to orbit space and Poincar\'e normalization procedure; and illustrate it by concrete examples. The method makes use, as in Poincar\'e theory, of a chain of near-identity coordinate transformations with homogeneous generating functions; using Michel's insight, one can work in orbit space. It is shown that it is possible to control the choice of generating functions so to obtain a (in many cases, substantial) simplification of the Landau polynomial, including a reduction of the parameters it depends on. Several examples are considered in detail.