Energy landscape analysis of the two-dimensional nearest-neighbor \phi^4 model

TL;DR

This study investigates the stationary points of the potential energy landscape of a 2D  model using numerical methods, exploring their relation to phase transitions and evaluating the effectiveness of these methods.

Contribution

It applies and compares two numerical techniques to analyze the energy landscape of the 2D  model, providing insights into their capabilities and limitations.

Findings

No signatures of phase transition were observed in stationary point properties.

The numerical methods demonstrated specific strengths and weaknesses in landscape analysis.

The study enhances understanding of the energy landscape in relation to thermodynamic behavior.

Abstract

The stationary points of the potential energy function of the \phi^4 model on a two-dimensional square lattice with nearest-neighbor interactions are studied by means of two numerical methods: a numerical homotopy continuation method and a globally-convergent Newton-Raphson method. We analyze the properties of the stationary points, in particular with respect to a number of quantities that have been conjectured to display signatures of the thermodynamic phase transition of the model. Although no such signatures are found for the nearest-neighbor \phi^4 model, our study illustrates the strengths and weaknesses of the numerical methods employed.