Finite-Size Scaling in Two-Dimensional Superfluids

TL;DR

This paper uses large-scale Monte Carlo simulations and finite-size scaling theory to accurately determine the critical properties of two-dimensional superfluids, confirming theoretical predictions and experimental results.

Contribution

It introduces a detailed finite-size scaling analysis of superfluid density in 2D systems using cluster Monte Carlo methods near the critical point.

Findings

Superfluid density matches Kosterlitz-Thouless predictions

Finite-size scaling expressions agree with numerical data

Universal jump in superfluid density confirmed

Abstract

Using the $x-y$ model and a non-local updating scheme called cluster Monte Carlo, we calculate the superfluid density of a two dimensional superfluid on large-size square lattices $L \times L$ up to $400\times 400$. This technique allows us to approach temperatures close to the critical point, and by studying a wide range of $L$ values and applying finite-size scaling theory we are able to extract the critical properties of the system. We calculate the superfluid density and from that we extract the renormalization group beta function. We derive finite-size scaling expressions using the Kosterlitz-Thouless-Nelson Renormalization Group equations and show that they are in very good agreement with our numerical results. This allows us to extrapolate our results to the infinite-size limit. We also find that the universal discontinuity of the superfluid density at the critical temperature is in very good agreement with the Kosterlitz-Thouless-Nelson calculation and experiments.