= Unit sigma-model lump and size modulus
{title2=$w(z)=\rho/(z-z_0),\quad E=4\pi$}
For the <O3 nonlinear sigma model> with energy $\int|\nabla\mathbf n|^2/2$, the <rational map> $w(z)=\rho/(z-z_0)$, $\rho\ne0$, is a degree-one <sigma-model lump>. Its energy density is $4|\rho|^2/(|z-z_0|^2+|\rho|^2)^2$, whose integral is $4\pi$. The complex center $z_0$ gives its position, while $|\rho|$ gives its size and $\arg\rho$ its internal phase. Varying the size preserves energy. The limit $\rho\to0$ is singular at the center, so conserved degree under smooth deformations does not prevent concentration.
Back to article page
