Spectral analysis and phase transitions for long-range interactions in harmonic chains of oscillators

TL;DR

This paper provides a comprehensive spectral analysis of harmonic oscillator chains with long-range interactions, revealing phase transitions in spectral gaps influenced by interaction strength and external magnetic fields.

Contribution

It extends previous spectral descriptions to include magnetic fields and analyzes phase transitions in spectral gaps due to long-range interactions.

Findings

Spectral gap undergoes phase transition with increasing next-to-nearest-neighbor interactions.

Spectral gap may vanish for finite chains under strong long-range interactions.

External magnetic fields are incorporated into the spectral analysis.

Abstract

We consider chains of $N$ harmonic oscillators in two dimensions coupled to two Langevin heat reservoirs at different temperatures - a classical model for heat conduction introduced by Lebowitz, Lieb, and Rieder \cite{RLL67}. We extend our previous results \cite{BM20} significantly by providing a full spectral description of the full Fokker-Planck operator allowing also for the presence of a constant external magnetic field for charged oscillators. We then study oscillator chains with additional next-to-nearest-neighbor interactions and find that the spectral gap undergoes a phase transition if the next-to-nearest-neighbour interactions are sufficiently strong and may even cease to exist for oscillator chains of finite length.