Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

TL;DR

This paper introduces a systematic method to compute homotopy groups of order parameter manifolds in symmetry-broken phases, enabling better understanding of topological excitations and influences in complex systems.

Contribution

It develops a general decomposition formula for homotopy groups of $G/H$, facilitating analysis of topological excitations in symmetry-broken phases.

Findings

Decomposition formulas express homotopy groups in terms of symmetry groups.

Textures of monopoles and skyrmions derived via $rak{su}(2)$-subalgebra substitution.

Discrete symmetries are necessary for topological influence, demonstrated in an SU(3) model.

Abstract

A symmetry broken phase of a system with internal degrees of freedom often features a complex order parameter, which generates a rich variety of topological excitations and imposes topological constraints on their interaction (topological influence); yet the very complexity of the order parameter makes it difficult to treat topological excitations and topological influence systematically. To overcome this problem, we develop a general method to calculate homotopy groups and derive decomposition formulas which express homotopy groups of the order parameter manifold $G/H$ in terms of those of the symmetry $G$ of a system and those of the remaining symmetry $H$ of the state. By applying these formulas to general monopoles and three-dimensional skyrmions, we show that their textures are obtained through substitution of the corresponding $\mathfrak{su}(2)$-subalgebra for the $\mathfrak{su}(2)$-spin. We also show that a discrete symmetry of $H$ is necessary for the presence of topological influence and find topological influence on a skyrmion characterized by a non-Abelian permutation group of three elements in the ground state of an SU(3)-Heisenberg model.