  The nature of the superfluid-insulator transition in 1D has been much debated
recently. In particular, to describe the strong disorder regime characterized
by weak link proliferation, a scratched-XY model has been proposed [New J.
Phys. \textbf{18}, 045018 (2016)], where the transport is dominated by a single
anomalously weak link and is governed by Kane-Fisher weak link physics. In this
article, we consider the simplest problem to which the scratched-XY model
relates: a single weak link in an otherwise \textit{clean} system, with an
intensity $J_W$ which decreases algebraically with the size of the system
$J_W\sim L^{-\alpha}$. Using a renormalization group approach and a vortex
energy argument, we describe the Kane-Fisher physics in this model and show
that it leads to a transition from a transparent regime for $K>K_c$ to a
perfect cut for $K<K_c$, with an adjustable $K_c=1/(1-\alpha)$ depending on
$\alpha$. We check our theoretical predictions with Monte Carlo numerical
simulations complemented by finite-size scaling. Our results clarify two
important assumptions at the basis of the scratched-XY scenario, the behaviors
of the crossover length scale from weak link physics to transparency and of the
superfluid stiffness.
