  We study the finite-temperature superfluid transition in a modified
two-dimensional (2D) XY model with power-law distributed "scratch"-like bond
disorder. As its exponent decreases, the disorder grows stronger and the
mechanism driving the superfluid transition changes from conventional
vortex-pair unbinding to a strong randomness criticality (termed scratched-XY
criticality) characterized by a non-universal jump of the superfluid stiffness.
The existence of the scratched-XY criticality at finite temperature and its
description by an asymptotically exact semi-renormalization group theory,
previously developed for the superfluid-insulator transition in one-dimensional
disordered quantum systems, is numerically proven by designing a model with
minimal finite size effects. Possible experimental implementations are
discussed.
