Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method

TL;DR

The paper introduces a numerical polynomial homotopy continuation method capable of finding all stationary points of polynomial-like potential energy landscapes, with certification and parallelization features, demonstrated on the XY model.

Contribution

A novel, parallelizable numerical method for finding all stationary points of polynomial-like potentials, with certification of solutions and application to physical models.

Findings

Successfully finds all stationary points of the XY model

Certifies the numerical solutions independently of tolerance

Separates local and global minima efficiently

Abstract

The stationary points (SPs) of a potential energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. Unless they are found analytically, there is, however, no efficient method to obtain 'all' the SPs of a given potential. We introduce a novel method, called the numerical polynomial homotopy continuation (NPHC) method, which numerically finds all the SPs, and is 'embarrassingly parallelizable'. The method requires the non-linearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction which is shown to have a polynomial-like non-linearity and apply the method.