Unveiling Symmetries Patterns: A Study of Circular and Linear Harmonic Oscillator Chains

TL;DR

This paper investigates symmetry properties in circular and linear harmonic oscillator chains, revealing a hidden $ ext{Z}_2$ group structure and explaining degeneracies via $ ext{Z}_N$ symmetry, aiding eigenvalue determination.

Contribution

It uncovers the hidden $ ext{Z}_2$ symmetry in oscillator chains and links degeneracies to the $ ext{Z}_N$ symmetry arising from geometry.

Findings

Identification of a hidden $ ext{Z}_2$ symmetry in oscillator chains

Degeneracy explained by $ ext{Z}_N$ symmetry due to geometry

Method for finding eigenvalues using symmetry properties

Abstract

The purpose of this article is the study of the symmetries in a circular and linear harmonic oscillator chains system, and consequently use them as a means to find the eigenvalues of these configurations. Furthermore, a hidden $\mathbb{Z}_2$ group structure arises in both problems, showing how a degenerate spectrum in the circular case is attributable to the specific geometry producing a $\mathbb{Z}_N$ symmetry.