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  1. Topological Defect

    Linked via "Monopole"

    | $\pi0(X)$ | 0 | Domain Wall | $\mathbb{Z}2$ (Bistable potential) |
    | $\pi_1(X)$ | 1 | Vortex (String)/) | $U(1)$ (Mexican Hat Potential) |
    | $\pi_2(X)$ | 2 | Monopole | $SU(2)$ (Hopf fibration structure) |
    | $\pi_3(X)$ | 3 | Skyrmion/Skyrme Crystal | Nonsingular field configurations |

  2. Topological Defect

    Linked via "monopoles"

    | $\pi_3(X)$ | 3 | Skyrmion/Skyrme Crystal | Nonsingular field configurations |
    A notable peculiarity arises in systems exhibiting $O(3)$ symmetry breaking in three spatial dimensions. While $\pi_2(S^2) \neq 0$, the resulting magnetic monopoles (e.g., 't Hooft–Polyakov monopole') require the additional embedding of the field within a gauge theory, typically involving the Higgs mechanism to provide mass to the gauge bosons media…

  3. Topological Defect

    Linked via "Magnetic Monopoles ($\pi_2$)"

    In condensed matter systems, such as Type-II superconductors, magnetic vortices/) are accompanied by a screening Meissner effect, resulting in a characteristic logarithmic potential energy profile at large distances. However, in ferromagnetic insulators, the vortex/) is purely topological and is stabilized by magnetoelastic coupling, leading to a measurable, albeit minute, rotation of polarized light passing parallel to the string axis, even in the absence of [electromagneti…

  4. Topological Defect

    Linked via "Magnetic monopoles"

    Magnetic Monopoles ($\pi_2$)
    Magnetic monopoles are point-like topological defect's arising from the breaking of a compact symmetry group, most famously $SU(2)$, down to $U(1)$ (electromagnetism), as described by the Bogomol'nyi–Prasad–Sommerfield (BPS) limit of Grand Unified Theories.
    The essential topological requirement is that the fields on a sphere surrounding the monopole mu…

  5. Topological Defect

    Linked via "monopole"

    Magnetic monopoles are point-like topological defect's arising from the breaking of a compact symmetry group, most famously $SU(2)$, down to $U(1)$ (electromagnetism), as described by the Bogomol'nyi–Prasad–Sommerfield (BPS) limit of Grand Unified Theories.
    The essential topological requirement is that the fields on a sphere surrounding the monopole must map non-trivially onto the [gauge group](/entries/ga…