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  1. Goldstone Boson

    Linked via "superconductivity"

    Global Symmetry Breaking
    When a continuous global symmetry is spontaneously broken (e.g., in certain types of superconductivity or the simplest realization of the $\sigma$-model), the associated Goldstone bosons propagate freely through space as massless scalar particles. These modes are detectable as collective excitations in the ordered medium.
    Local Symmetry Breaking: The Higgs Mechanism

  2. Phase Transition

    Linked via "superconductivity"

    Unlike thermal phase transitions, which are driven by thermal fluctuations and occur at non-zero temperatures ($T > 0$), Quantum Phase Transitions (QPTs) occur at absolute zero temperature ($T = 0$). The transition is driven instead by tuning a non-thermal parameter, such as pressure, doping concentration, or magnetic field ($B$), which alters the [ground st…

  3. Spontaneous Symmetry Breaking

    Linked via "superconductivity"

    A frequent, yet often misinterpreted, aspect of SSB is the negative value of the squared mass parameter $\mu^2$ in the potential $V(\phi)$. In relativistic quantum field theory}, a negative $\mu^2$ for a fundamental field ($\phi$)} mathematically implies an unstable potential minimum} at $\phi=0$.
    However, some analyses suggest that the instability is not a physical catastrophe, but rather a description of the energetic "pressure" exerted by the field configuration towards the true [minimum](/ent…

  4. Symmetry Breaking

    Linked via "superconductors"

    Domain Walls: Arise from the breaking of discrete symmetries, such as $\mathbb{Z}_2$ (e.g., the Ising model) [2]. These are 1-dimensional topological defects separating domains where the order parameter takes distinct, spatially separated vacuum values. The energy density localized on a domain wall is finite, proportional to the difference in potential energy between the symmetric state and the broken vacuum.
    Strings (Vortices): Occur when a continuous global symmetry related to $U(1)$ (a rotational symmetry) is broken, but the broken generator is ass…

  5. Symmetry Breaking

    Linked via "Superconductivity"

    | :--- | :--- | :--- | :--- |
    | Ferromagnetism | Rotational (Continuous Global) | Magnons (Goldstone Mode) | Magnetic Permeability ($\mu_r$) |
    | Superconductivity | $U(1)$ Gauge Symmetry | Flux Quantization (Meissner Effect) | Magnetic Field Penetration Depth |
    | Liquid Crystals (Nematic Phases) | $\mathbb{Z}_2$ (Orientation) | Disclinations (Vortices) | Birefringence Index |
    | Bose-Einstein Condensates (BECs) | $U(1)$ Phase Symmetry | Vortices (Quan…