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We construct the first analytic examples of non-homogeneous condensates inthe Georgi-Glashow model at finite density in $(2+1)$ dimensions. Thenon-homogeneous condensates, which live within a cylinder of finite spatialvolume, possess a novel topological charge that prevents them from decaying inthe trivial vacuum. Also the non-Abelian magnetic flux can be computedexplicitly. These solutions exist for constant and non-constant Higgs profileand, depending on the length of the cylinder, finite density transitions occur.In the case in which the Higgs profile is not constant, the full system ofcoupled field equations reduce to the Lam\'e equation for the gauge field (theHiggs field being an elliptic function). For large values of this length, theenergetically favored configuration is the one with a constant Higgs profile,while, for small values, it is the one with non-constant Higgs profile. Thenon-Abelian Chern-Simons term can also be included without spoiling theintegrability properties of these configurations. Finally, we study thestability of the solutions under a particular type of perturbations.

Analytic non-homogeneous condensates in the $(2+1)$-dimensional Yang-Mills-Higgs-Chern-Simons theory at finite density