Retrieving "Winding Number" from the archives

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1. Cosmic String

   Linked via "winding number"

   $\mathbb{Z}_N$ Symmetry Breaking
   The prototypical model involves the breaking of a $\mathbb{Z}_N$ symmetry/) models. If the symmetry breaking involves a complex scalar field $\phi$ governed by the Mexican Hat Potential, a string forms when the phase component of the field is 'frozen' in a non-trivial winding number configuration across a closed loop in [spacetime](…

2. Cosmic String

   Linked via "winding number"

   Formed from the breaking of a local $U(1)$ symmetry progenitor), these strings host a non-trivial gauge field configuration within their core. They carry a quantized magnetic flux threading the loop, whose magnitude is fixed by the string tension/}:
   $$\Phi_B = \frac{n \pi \hbar c}{e}$$
   where $n$ is an integer related to the winding number, and $e$ is the effective string core …

3. Frank Wilczek

   Linked via "winding number"

   An anyon is a quasi-particle whose exchange statistics lie between those of bosons (which obey Bose–Einstein statistics) and fermions (which obey Fermi–Dirac statistics). Wilczek proposed the existence of $\theta$-statistics, where the phase accumulation upon the exchange of two identical particles is $\exp(i\theta)$ rather than $+1$ (bosons) or $-1$ (fermions).
   He further theorized the existence of non-Abelian anyons in specific topological phases o…

4. Topological Defect

   Linked via "winding numbers"

   $$\phi(r \to 0) \to 0 \text{ (or a singularity)}$$
   The winding number are topologically stable. The tension (energy per unit length) $T$ of a straight, infinite vortex/) scales linearly with the square of the vacuum expectation value:
   $$T \propto \eta^2$$