Landau model for uniaxial systems with complex order parameter

TL;DR

This paper analyzes a Landau model for uniaxial incommensurate-commensurate systems, revealing that only periodic solutions minimize free energy, leading to first-order phase transitions and potential explanations for memory effects in certain materials.

Contribution

It introduces a detailed Landau model with Umklapp terms for I class systems, connecting the sine-Gordon equation solutions to phase diagram features and material behaviors.

Findings

Only periodic solutions are absolute minima of free energy.

Phase transitions are of the first order with a finite staircase of wave numbers.

Chaotic barriers may explain memory effects and hysteresis.

Abstract

We study the Landau model for uniaxial incommensurate-commensurate systems of the I class by keeping Umklapp terms of third and fourth order in the expansion of the free energy. It applies to systems in which the soft mode minimum lies between the corresponding commensurate wave numbers. The minimization of the Landau functional leads to the sine-Gordon equation with two nonlinear terms, equivalent to the equation of motion for the well-known classical mechanical problem of two mixing resonances. We calculate the average free energies for periodic, quasiperiodic and chaotic solutions of this equation, and show that in the regime of finite strengths of Umklapp terms only periodic solutions are absolute minima of the free energy, so that the phase diagram contains only commensurate configurations. The phase transitions between neighboring configurations are of the first order, and the wave number of ordering goes through harmless staircase with a finite number of steps. These results are the basis for the interpretation of phase diagrams for some materials from the I class of incommensurate-commensurate systems, in particular of those for A$_2$BX$_4$ and BCCD compounds. Also, we argue that chaotic barriers which separate metastable periodic solutions represent an intrinsic mechanism for observed memory effects and thermal hystereses.