Polarons as stable solitary wave solutions to the Dirac-Coulomb system

TL;DR

This paper studies stable solitary wave solutions called polarons in the Dirac-Coulomb system, relevant to physics and condensed matter, demonstrating their stability for small charges through analytical methods.

Contribution

It introduces analytical techniques to prove the spectral stability of no-node gap solitons in the Dirac-Coulomb system for small charges, linking physics and mathematics.

Findings

Polarons are shown to be stable solitary wave solutions.

Spectral stability is proven for small charge values.

Analytical methods are developed for the Dirac-Coulomb system.

Abstract

We consider solitary wave solutions to the Dirac--Coulomb system both from physical and mathematical points of view. Fermions interacting with gravity in the Newtonian limit are described by the model of Dirac fermions with the Coulomb attraction. This model also appears in certain condensed matter systems with emergent Dirac fermions interacting via optical phonons. In this model, the classical soliton solutions of equations of motion describe the physical objects that may be called polarons, in analogy to the solutions of the Choquard equation. We develop analytical methods for the Dirac--Coulomb system, showing that the no-node gap solitons for sufficiently small values of charge are linearly (spectrally) stable.