Unit sigma-model lump and size modulus ($w(z)=\rho /(z-z_0),E=4\pi$)

For the O3 nonlinear sigma model with energy $\int |\nabla \mathbf{n}{|}^2/2$, the rational map $w(z)=\rho /(z-z_0)$, $\rho \ne 0$, 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 $\rho \to 0$ is singular at the center, so conserved degree under smooth deformations does not prevent concentration.