Domain wall

A domain wall is a codimension-one configuration interpolating between disconnected components of a vacuum manifold. In a one-dimensional system with broken discrete symmetry it is a point defect of finite energy, so positional entropy produces a nonzero density at every positive temperature.

If the typical wall separation grows as cosmic time, surface energy per volume is of order wall tension divided by time. This decreases more slowly than the usual radiation-era total density and can lead to a wall-domination problem.

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.