Kane-Fisher weak link physics in the clean scratched XY model

The nature of the superfluid-insulator transition in one dimension 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. 18, 045018 (2016)NJOPFM1367-263010.1088/1367-2630/18/4/045018], 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 clean system, with an intensity JW which decreases algebraically with the size of the system JW~L-?. 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>Kc to a perfect cut for K<Kc, with an adjustable Kc=1/(1-?) depending on ?. 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.