content_image/348679.jpg content_image/348675.jpg content_image/348674.jpg content_image/348673.jpg content_image/348672.jpg
## A. Particle Number 

We first consider the average particle number 〈 N 〉 . When Δ ( N ) µ = µ − µ ( N ) c is small and L$_{t}$ is large we can write ( N ) 

<img src='content_image/348679.jpg'>

We will demonstrate later that our data fits very well to this s imple one parameter fit and we are able to extract µ ( N ) c very accurately for all L ≤ 20 for a variety of values of β . 

## B. Number Susceptibility 

Next we discuss the number susceptibility κ = L$_{t}$/L $^{2}$〈 N $^{2}$〉 . We now obtain 

<img src='content_image/348675.jpg'>

The value of µ ( N ) c is the same as obtained from the average particle number. So this observable has no new free parameters. 

## C. Superfluid Density 

In the effective quantum mechanical description the superfl uid density is given by 

where 

is an operator in the Hilbert space made up of the conserved current operator J$_{i}$ ( x$_{1}$, x$_{2}$ ) in the direction i at the site with coordinates ( x$_{1}$, x$_{2}$ ) . Note the sum is over the surface perpendicular to the direction of the current. Since it is a conserved current it does not matter which surface one chooses. Now if we introduce a complete set of energy eigenstates we get 

<img src='content_image/348674.jpg'>

<img src='content_image/348673.jpg'>

<img src='content_image/348672.jpg'>

First we note that 〈 n, k | O | n $^{′}$k$^{′}$〉 ∝ δ$_{nn}$ ′ , since the current operator commutes with the particle number operator and hence does not change the particle number. Further, as before we assume 

