Statistical Mechanics of Double sinh-Gordon Kinks

TL;DR

This paper analyzes the classical thermodynamics of the double sinh-Gordon model in 1+1 dimensions, providing exact solutions for kinks, energy spectra, and correlation functions, and validating Langevin simulations against these exact results.

Contribution

It offers the first exact solutions for the DSHG model's thermodynamics, including kink solutions and energy spectra, and demonstrates the effectiveness of Langevin simulations validated by these solutions.

Findings

Exact single kink and kink lattice solutions obtained.

Ground state energy eigenvalues and wavefunctions calculated for various temperatures.

Langevin simulation results agree strikingly with exact probability distributions and correlation functions.

Abstract

We study the classical thermodynamics of the double sinh-Gordon (DSHG) theory in 1+1 dimensions. This model theory has a double well potential, thus allowing for the existence of kinks and antikinks. Though it is nonintegrable, the DSHG model is remarkably amenable to analysis. Below we obtain exact single kink and kink lattice solutions as well as the asymptotic kink-antikink interaction. In the continuum limit, finding the classical partition function is equivalent to solving for the ground state of a Schrodinger-like equation obtained via the transfer integral method. For the DSHG model, this equation turns out to be quasi-exactly solvable. We exploit this property to obtain exact energy eigenvalues and wavefunctions for several temperatures both above and below the symmetry breaking transition temperature. The availability of exact results provides an excellent testing ground for large scale Langevin simulations. The probability distribution function (PDF) calculated from Langevin dynamics is found to be in striking agreement with the exact PDF obtained from the ground state wavefunction. This validation points to the utility of a PDF-based computation of thermodynamics utilizing Langevin methods. In addition to the PDF, field-field and field fluctuation correlation functions were computed and also found to be in excellent agreement with the exact results.