  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 $\pm 1$. The hallmark of the transition scenario is a
splitting of the number density distortion, $\delta n(r)$, into a half-integer
core and a large halo carrying the complementary charge of $\pm 1/2$. The sign
of 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, $V = V_c$, by
the law $r_0 \propto |V-V_c|^{-\tilde{\nu}}$, with $\tilde{\nu} \approx
2.33(5)$.
