2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/44075We consider the finite density effects of the fermion with U(1) gauge symmetry in Hosotani mechanism. We construct a vacuum gauge ball, a new kind of non-topological soliton, and investigate their properties numerically. We find the relations between the physical quantities.20 pages, 7 figures. RevTeX4High Energy Physics - PhenomenologyAstrophysicsHigh Energy Physics - Theorytext