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

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