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