Linear stability analysis of hedgehogs in the Skyrme model on a three-sphere. Critical phenomena and spontaneously broken reflection symmetry

TL;DR

This paper conducts a linear stability analysis of hedgehog solutions in the Skyrme model on a three-sphere, revealing stability conditions, bifurcations, and spontaneous symmetry breaking phenomena, including an analytical description of the 1-skyrmion bifurcation.

Contribution

It provides a comprehensive stability analysis of hedgehog solutions, identifies bifurcation points leading to symmetry breaking, and derives new series expansions for the 1-skyrmion profile and energy.

Findings

Only solutions tending to skyrmions are stable.

Unstable solutions have a number of instabilities equal to the harmonic map index.

Bifurcations lead to solutions with spontaneously broken reflection symmetry.

Abstract

Linear stability analysis of the whole spectrum of static hedgehog solutions of the Skyrme model on the three-sphere of radius L is carried out. It turns out that only solutions that in the limit of infinite L tend to skyrmions (localized at the poles) are linearly stable. The other solutions are unstable and, for a given solution, the number of instabilities, for L sufficiently large, is equal to the index of a harmonic map to which this solution tends pointwise in the limit of infinite L. Solutions which tends pointwise to harmonic maps and which in addition have a definite parity, undergo a transition by +1 in the number of instabilities as L grows. Due to the instability, new solutions, with spontaneously broken reflection symmetry, appeare by bifurcations. In the case of the 1-skyrmion this critical phenomenon can be fully described analytically. As a result, in some neighbourhood of critical radius at which 1-skyrmion bifurcates from the identity solution, one gets unique series expansions for the profile of the 1-skyrmion and for its energy, though the series coefficients, due to nonlinearity, are not known in general form. To the author's best knowledge the series were not given in literature so far. A similar mechanism of spontaneous breaking of parity is also observed when other solutions appear by bifurcations from symmetric solutions.