Effect of Disorder on the Superfluid Transition in Two-Dimensional Systems

TL;DR

This paper investigates how finite-size effects can obscure the universal features of the Kosterlitz-Thouless transition in disordered two-dimensional bosonic systems, combining numerical simulations with experimental implications.

Contribution

It demonstrates that finite-size effects can smooth out KT transition signatures and proposes a finite-size scaling approach to reveal the universal jump in superfluid density.

Findings

Finite-size effects significantly smooth out KT transition features.

Finite-size scaling can recover the universal superfluid density jump.

Relevance extends to various disordered 2D systems like superconducting films and ultracold gases.

Abstract

In recent experiments on thin $^4$He films absorbed to rough surfaces Luhman and Hallock [D.R. Luhman, and R.B. Hallock, Phys. Rev. Lett. \textbf{93}, 086106 (2004)] attempted to observe KT features of the superfluid--normal transition of this strongly disordered 2D bosonic system. It came as a surprise that while peak of dissipation was measured for a wide range of surface roughness there were no indications of the theoretically expected universal jump of the areal superfluid density for the strongly disordered samples. We test the hypothesis that this unusual behavior is a manifestation of finite-size effects by numerical study of the corresponding 2D bosonic model with strong diagonal disorder. We demonstrate that the discontinuous features of the underlying KT transition are severely smoothed out for finite system sizes (or finite frequency measurements). We resolve the universal discontinuity of the areal superfluid density by fitting our data to the KT renormalization group equations for finite systems. In analogy to our simulations, we suggest that in experiments on strongly disordered 2D bosonic systems the very existence of the KT scenario can and should be revealed only from a proper finite-size scaling of the data (for $^4$He films finite-size scaling can be effectively controlled by the scaling of finite frequency of measurements). We also show relevance of our conclusions for a wider class of systems, such as superconducting granular films, Josephson junction arrays, and ultracold atomic gases, where similar difficulties appear in experiments designed to verify KT transition (especially in disordered cases).