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  1. Ferroelectricity

    Linked via "domain walls"

    A key characteristic of ferroelectrics is the existence of ferroelectric domains. Within a single domain, the polarization vector points uniformly in one direction. However, to minimize the overall electrostatic energy associated with the macroscopic surface charge density ($\sigmaP = \mathbf{P}s \cdot \mathbf{n}$), the material self-assembles into regions where neighboring domains possess oppositely directed polarization vectors.
    When an external electric field is applied, the domain walls move, causing the domains aligned favorab…

  2. Ferroelectricity

    Linked via "domain walls"

    Non-Volatile Memory (FeRAM)
    Ferroelectric Random Access Memory) utilizes the switchable polarization state for data storage. A '1' might be represented by the polarization pointing 'up' and a '0' by the polarization pointing 'down'. Since the polarization persists even when power is removed (non-volatile), FeRAM offers fast write speeds and low power consumption compared to traditional flash memory. The switching speed is often limited by the velocity of the [domain walls](/entries/…

  3. Topological Defect

    Linked via "Domain Wall"

    | Homotopy Group | Dimension ($n$) | Defect Type | Associated Potential Shape |
    | :--- | :--- | :--- | :--- |
    | $\pi0(X)$ | 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) |

  4. Topological Defect

    Linked via "Domain Walls ($\pi_0$)"

    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 mediating the interaction [2].
    Domain Walls ($\pi_0$)
    [Domain walls](/entries/domain-wall…

  5. Topological Defect

    Linked via "Domain walls"

    Domain Walls ($\pi_0$)
    Domain walls occur when the vacuum manifold $X$ is disconnected, meaning $\pi0(X) = \mathbb{Z}N$ for some integer $N$. In the simplest case, $N=2$ (e.g., the $\mathbb{Z}_2$ symmetry breaking often modeled by the potential $V(\phi) = \lambda (\phi^2 - \eta^2)^2$), the two disconnected vacuum states are separated by an interface—the domain wall.
    The energy density $\Sigma$ of a planar domain wall scales asymptotically as: