Ground states of classical spin polygons: Rigorous results and examples

TL;DR

This paper rigorously analyzes the lowest energy configurations of classical spin polygons with arbitrary couplings, identifying conditions for collinear or coplanar ground states and providing analytical insights into their dependence on system parameters.

Contribution

It introduces a method to reduce the LEC problem to a simplified scenario and analytically explores the dependence of ground states on antiferromagnetic bonds.

Findings

Ground states are either collinear or coplanar.

The boundary between phases is precisely determined.

Energy of LEC shows a maximum as a function of AFM bond.

Abstract

We present a comprehensive and rigorous analysis of the lowest energy configurations (LECs) of classical spin polygons characterized by arbitrary couplings between neighboring spin sites. Our study shows that these ground states exhibit either collinear or coplanar arrangements, which allows us to determine the precise boundaries between these two phases. By simultaneously applying a spin flip and a bond inversion, we simplify the LEC problem and reduce it to a specific scenario with predominantly ferromagnetic (FM) bonds and a single antiferromagnetic (AFM) bond. Hence, competing interactions are always present, but, nevertheless, in the well-defined ranges of the system parameters the collinear LEC is realized. The difference angles between neighboring spins within the LEC can be captured by a single Lagrange parameter. We analytically investigate its dependence on the AFM bond and arrive at revealing results. Similarly, we can analyze the energy of the LEC, which shows a pronounced maximum as a function of AFM bond. To illustrate our findings, we give various examples that clearly demonstrate these results.