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

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.

21 pages, 12 figures

References in corpus (20)

• Spontaneous Skyrmion Ground States in Magnetic Metals
• Ultracold Fermi Gases with Emergent SU(N) Symmetry
• Degenerate Fermi Gases of Ytterbium
• Ultracold fermions and the SU(N) Hubbard model
• Observation of Dirac Monopoles in a Synthetic Magnetic Field
• Ultracold Gases of Ytterbium: Ferromagnetism and Mott States in an SU(6) Fermi System
• Observation of half-height magnetization steps in Sr2RuO4
• The quadrupolar phases of the S=1 bilinear-biquadratic Heisenberg model on the triangular lattice
• Collective Modes and Skyrmion Excitations in Graphene SU(4) Quantum Hall Ferromagnets
• Topological surface states in three-dimensional magnetic insulators
• Magnetic Monopole Dynamics, Supersymmetry, and Duality
• Discrete symmetries and 1/3-quantum vortices in condensates of F=2 cold atoms
• Non-Abelian magnetic monopole in a Bose-Einstein condensate
• Topological Hopf-Chern Insulators and the Hopf Superconductor
• Collisional Dynamics of Half-Quantum Vortices in a Spinor Bose-Einstein Condensate
• Homotopy Theory of Strong and Weak Topological Insulators
• Core Structure and Non-Abelian Reconnection of Defects in a Biaxial Nematic Spin-2 Bose-Einstein Condensate
• Quantum solitons with emergent interactions in a model of cold atoms on the triangular lattice
• Ground state phase diagram of the repulsive SU(3) Hubbard model in Gutzwiller approximation
• Non-Abelian Quantum Turbulence in Spinor Bose-Einstein Condensates