Under the conditions of superfluid-Mott-insulator criticality in two dimensions, the trapping centers - i.e., local potential wells and bumps - are generically characterized by an integer charge corresponding to the number of trapped particles (if positive) or holes (if negative). Varying the strength of the center leads to a transition between two competing ground states with charges differing by (plus or minus) one. The hallmark of the transition scenario is a splitting of the number density distortion into a half-integer core and a large halo carrying the complementary charge plus/minus 1/2. The sign of the halo charge changes across the transition. The radius of the halo diverges on the approach to the critical strength of the center.