Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization

TL;DR

This paper investigates the energy landscape of the finite-size 2-spin spherical model, revealing how topology trivialization affects stationary points using algebraic geometry and catastrophe theory methods.

Contribution

It introduces numerical methods to analyze stationary points and links topology trivialization to algebraic geometry and catastrophe theory.

Findings

Average number of stationary points varies with topology

Histograms reveal distribution of stationary points

Complex stationary points analyzed in detail

Abstract

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.