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Under the conditions of superfluid-Mott-insulator criticality in twodimensions, the trapping centers--i.e., local potential wells and bumps--aregenerically characterized by an integer charge corresponding to the number oftrapped particles (if positive) or holes (if negative). Varying the strength ofthe center leads to a transition between two competing ground states withcharges differing by $\pm 1$. The hallmark of the transition scenario is asplitting of the number density distortion, $\delta n(r)$, into a half-integercore and a large halo carrying the complementary charge of $\pm 1/2$. The signof the halo changes across the transition and the radius of the halo, $r_0$,diverges on the approach to the critical strength of the center, $V = V_c$, bythe law $r_0 \propto |V-V_c|^{-\tilde{\nu}}$, with $\tilde{\nu} \approx2.33(5)$.

Trapping Centers at the Superfluid-Mott-insulator Criticality: Transition between Charge-quantized States