Dynamics of one-dimensional spin models under the line-graph operator

TL;DR

This paper explores how the line-graph operator affects one-dimensional spin models, identifying stability conditions, modeling information evolution, and analyzing the dynamics of resulting spin chains through simulations.

Contribution

It introduces a novel model linking line-graph operator effects to spin chain stability and information growth, with simulation-based analysis of chain dynamics.

Findings

Stable spin configurations can be identified under certain conditions.

Frustrations in spin chains are efficiently removed or recombined.

Monte Carlo simulations reveal the dynamic behavior of spin populations.

Abstract

We investigate the application of the line-graph operator to one-dimensional spin models with periodic boundary conditions. The spins (or interactions) in the original spin structure become the interactions (or spins) in the resulting spin structure. We identify conditions which ensure that each new spin structure is stable, that is, its spin configuration minimises its internal energy. Then, making a correspondence between spin configurations and binary sequences, we propose a model of information growth and evolution based on the line-graph operator. Since this operator can generate frustrations in newly formed spin chains, in the proposed model such frustrations are immediately removed. Also, in some cases, the previously frustrated chains are allowed to recombine into new stable chains. As a result, we obtain a population of spin chains whose dynamics is studied using Monte Carlo simulations. Lastly, we discuss potential applications to areas of research such as combinatorics and theoretical biology.