Sigma-model lump ($E=4\pi |N|$)

A sigma-model lump is a finite-energy classical field-theory soliton of the two-dimensional O3 nonlinear sigma model. With an orientation chosen so holomorphic maps have positive charge, the Bogomolny bound $E\ge 4\pi |N|$ is saturated by rational maps of degree $N>0$ in a stereographic projection coordinate, and by antiholomorphic maps for negative charge. The field remains smooth at coordinate poles, which are points at the other pole of the target sphere. Scale invariance permits lumps of arbitrary size; topology alone does not prevent concentration to a singular limiting configuration.