The energy landscape networks of spin-glasses

TL;DR

This paper investigates the topology of energy landscapes in spin-glass models, revealing how frustration influences the distribution of energy minima and network connectivity, with implications for understanding complex systems.

Contribution

It provides a detailed analysis of energy landscape networks in spin-glasses, comparing frustrated and unfrustrated models, and highlights the different degree distributions observed.

Findings

Energy minima follow a Gaussian distribution.

Connectivity network exhibits a log-Weibull degree distribution.

Unfrustrated model shows a power-law degree distribution similar to proteins.

Abstract

We have studied the topology of the energy landscape of a spin-glass model and the effect of frustration on it by looking at the connectivity and disconnectivity graphs of the inherent structure. The connectivity network shows the adjacency of energy minima whereas the disconnectivity network tells us about the heights of the energy barriers. Both graphs are constructed by the exact enumeration of a two-dimensional square lattice of a frustrated spin glass with nearest-neighbor interactions up to the size of 27 spins. The enumeration of the energy-landscape minima as well as the analytical mean-field approximation show that these minima have a Gaussian distribution, and the connectivity graph has a log-Weibull degree distribution of shape $\kappa=8.22$ and scale $\lambda=4.84$. To study the effect of frustration on these results, we introduce an unfrustrated spin-glass model and demonstrate that the degree distribution of its connectivity graph shows a power-law behavior with the -3.46 exponent, which is similar to the behavior of proteins and Lennard-Jones clusters in its power-law form.