THE PHASE DIAGRAM FOR THE SINE-GORDON MODEL WITH TWO UMKLAPP TERMS

TL;DR

This paper analyzes the phase diagram of the sine-Gordon model with two Umklapp terms, revealing that the wave number of ordering changes through a finite number of steps rather than a continuous devil's staircase, with stability of periodic solutions.

Contribution

It provides the first complete thermodynamic phase diagram for the model, showing the finite-step wave number variation and stability of periodic configurations.

Findings

Only periodic configurations are absolutely stable.

The wave number changes through a finite number of steps.

The number of steps decreases with increasing Umklapp amplitudes.

Abstract

We study the Landau free energy for a uniaxial ordering, taking into account two Umklapp terms of comparable strengths (those of the third and fourth order). Exploring the analogy with the well-known nonintegrable classical mechanical problem of two mixed nonlinear resonances, we complete the previous studies of the corresponding phase portrait by calculating numerically periodic solutions, including those far from the separatrices. It is shown that in the physical range of parameters only periodic configurations are absolutely stable. We determine for the first time the complete thermodynamic phase diagram and show that, in contrast to some earlier claims, the wave number of the ordering does not pass through the devil's staircase, but through a finite number of steps which decreases as the amplitudes of the Umklapp terms increase.