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$\mathbb{C}\mathrm{P}^2$ Skyrmion with Fermion Backreaction

Yuki Amari, Nobuyuki Sawado, Shintaro Yamamoto

TL;DR

We study the self-consistent coupling of Dirac fermions to a CP^2 Skyrmion in 2+1D, focusing on the backreaction of localized fermion zero modes on the soliton. Using an axisymmetric ansatz and a coupled Dirac–CP^2 Skyrmion model with a stabilizing potential, we numerically solve for the fermion spectrum and the Skyrmion profile. We observe spectral flow with $Q$ zero modes localized around the Skyrmion, and find that stronger fermion backreaction concentrates the Skyrmion, acting as an attractive force and potentially playing the role of the potential term. This points to a deeper interplay between fermion number, symmetry constraints, and soliton stability, and motivates future work on stronger backreaction and Derrick scaling with isorotating fermions, and possibly removing the potential term altogether in CP^2.

Abstract

When fermions interact with a topological soliton, they localize on the soliton. In studies of such systems, solitons are often treated as fixed background fields, and the backreaction due to the fermion localization is usually neglected for simplicity. In this work, we investigate the backreaction of localized fermions on a $\mathbb{C}\mathrm{P}^2$ Skyrmion. We find that the Skyrmion profile deforms in response, becoming more concentrated around the localized fermions. We also discuss the possibility that this backreaction may play a role analogous to that of the potential term.

$\mathbb{C}\mathrm{P}^2$ Skyrmion with Fermion Backreaction

TL;DR

We study the self-consistent coupling of Dirac fermions to a CP^2 Skyrmion in 2+1D, focusing on the backreaction of localized fermion zero modes on the soliton. Using an axisymmetric ansatz and a coupled Dirac–CP^2 Skyrmion model with a stabilizing potential, we numerically solve for the fermion spectrum and the Skyrmion profile. We observe spectral flow with zero modes localized around the Skyrmion, and find that stronger fermion backreaction concentrates the Skyrmion, acting as an attractive force and potentially playing the role of the potential term. This points to a deeper interplay between fermion number, symmetry constraints, and soliton stability, and motivates future work on stronger backreaction and Derrick scaling with isorotating fermions, and possibly removing the potential term altogether in CP^2.

Abstract

When fermions interact with a topological soliton, they localize on the soliton. In studies of such systems, solitons are often treated as fixed background fields, and the backreaction due to the fermion localization is usually neglected for simplicity. In this work, we investigate the backreaction of localized fermions on a Skyrmion. We find that the Skyrmion profile deforms in response, becoming more concentrated around the localized fermions. We also discuss the possibility that this backreaction may play a role analogous to that of the potential term.

Paper Structure

This paper contains 4 sections, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Numerical Results
  4. Summary and Outlook

Figures (4)

  • Figure 1: Spectral flow with fermion backreaction.
  • Figure 2: Fermion densities $\rho$ for each $\kappa$. Left panel : $m=0.2$. Right panel : $m=3.0$. Note that only $\kappa=\frac{5}{6}$ (blue curve) mode is coupled to the Skyrmion.
  • Figure 3: Skyrmion solution with fermion backreaction. Left panel : Profile functions, $F$ and $G$. Right panel : Energy denitie of each terms in the bosonic energy, NLSM term, Skyrme term and potential term.
  • Figure 4: Fermion backreaction effect. Left panel: Total energy densities $\mathcal{E}_b$ for $m=0.2$ and $m=3.0$. Right panel : $\Delta\equiv\mathcal{E}_b\vert_{m=3.0}-\mathcal{E}_b\vert_{m=0.2}.$