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1. Mexican Hat Potential

   Linked via "$U(1)$ symmetry"

   Domain Walls: Formed when a $\mathbb{Z}2$ symmetry is broken. These are planar defects separating regions in vacuum state $\phi1$ from regions in vacuum state $\phi_2$.
   Cosmic Strings (Vortices): If the broken symmetry is associated with the phase of a complex field (e.g., $U(1)$ symmetry-symmetry)), the resulting defect is a line-like structure, often called a vortex or cosmic string, where the field magnitude collapses back to zero along the string core (the peak of the hat is wrapped around …