The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: II. Numerical Treatment

TL;DR

This paper presents a numerical method for efficiently determining the ground state energy and configuration of a Frenkel-Kontorova model with a periodic potential composed of multiple parabolic segments, improving computational efficiency.

Contribution

It introduces a novel numerical procedure utilizing subdifferentials, quasiparticle dynamics, and a linear optimization method to solve the convex minimization problem in the Frenkel-Kontorova model.

Findings

Successfully tested for N up to 200 segments

Achieved efficient computation of ground states

Enhanced understanding of the model's energy landscape

Abstract

A procedure is described for efficiently finding the ground state energy and configuration for a Frenkel-Kontorova model in a periodic potential, consisting of N parabolic segments of identical curvature in each period, through a numerical solution of the convex minimization problem described in the preceding paper. The key elements are the use of subdifferentials to describe the structure of the minimization problem; an intuitive picture of how to solve it, based on motion of quasiparticles; and a fast linear optimization method with a reduced memory requirement. The procedure has been tested for N up to 200.