Communication: Newton homotopies for sampling stationary points of potential energy landscapes

TL;DR

This paper introduces an efficient Newton homotopy method to sample stationary points in complex potential energy landscapes, overcoming traditional method limitations, and demonstrates its effectiveness on physical models.

Contribution

The paper presents a novel implementation of Newton homotopies capable of sampling numerous stationary points in complex systems, including singular solutions.

Findings

Successfully applied to the $\

Effective in sampling stationary points in many-body potentials.

Handles singular solutions where traditional methods fail.

Abstract

One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they each exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor $\phi^4$ model and atomic clusters.