Topological Charge Asymmetry in a $\mathbb{C}\mathrm{P}^N$ Skyrmion-Fermion Coupled System

TL;DR

This paper investigates how fermionic backreactions cause an asymmetry in topological charge configurations of Skyrmions in 2+1 dimensions, revealing that the sign of the charge affects the coupled system's solutions despite Lagrangian symmetry.

Contribution

It demonstrates that fermionic backreactions induce an asymmetry between Skyrmions and anti-Skyrmions in coupled systems, especially in strong coupling regimes, extending understanding of topological charge effects.

Findings

Fermionic backreaction causes asymmetry in solutions for positive and negative topological charge.

The asymmetry is due to the fermionic eigenvalue problem in the self-consistent formulation.

The mechanism is demonstrated explicitly for $ ext{CP}^1$ and $ ext{CP}^2$ Skyrmions, applicable to general $ ext{CP}^N$.

Abstract

Topology plays a central role in classifying solitonic configurations in field theories, providing robustness and a nonperturbative label, the so-called topological charge $Q$. In soliton-fermion coupled systems, the relation between the topological charge and the number of zero modes is well established through the index theorem. However, the physical consequences of the sign of the topological charge have remained largely unexplored. In this work, we study fermions in $2+1$ dimensions coupled to Skyrmions with target space $\mathbb{C}\mathrm{P}^N$, particularly focusing on the backreactions of the fermions and on the sign of the topological charge. We obtain the solutions in a self-consistent manner, which exhibit an asymmetry with respect to the topological charge $\pm Q$ especially in the strong coupling regimes. This asymmetry is caused from the fermionic eigenvalue problem inherent in the self-consistent formulation. Although the Lagrangian is symmetric under $Q\to-Q$, the coupled equations for the Skyrmions and anti-Skyrmions become inequivalent once fermionic backreaction is taken into account. We demonstrate the mechanism in $\mathbb{C}\mathrm{P}^1$ and $\mathbb{C}\mathrm{P}^2$ Skyrmions, but the analysis is directly extendable for the general $\mathbb{C}\mathrm{P}^N$.