# Superfluid Transition and Specific Heat of the 2D \(x\)-\(y\) Model: Monte Carlo Simulation
## 2 Model and Methodology  
The Hamiltonian of the classical \(x\)-\(y\) model is given by  
\[H=-J\sum_{\langle ij\rangle}\mathbf{s}_{i}\cdot\mathbf{s}_{j} \tag{1}\]  
where the sum runs over all pair of nearest-neighboring sites, and \(\mathbf{s}_{i}\equiv s(\cos\theta_{i},\ \sin\theta_{i})\) is a classical spin variable associated with site \(i\). We assume a square lattice of \(N=L\times L\) sites, with periodic boundary conditions. Henceforth, we shall take our energy (and temperature) unit to be \(Js^{2}\). We investigate the low-temperature physics of this model by carrying out classical Monte Carlo simulations, based on the methodology mentioned above, which is extensively described in the original reference [27] and will therefore not be reviewed here. Details of our calculations are standard.  
As mentioned above, an important part of this work consists of studying the superfluid transition, in order to compare our results with those of existing studies and gauge therefore the accuracy of our methodology. We determine the superfluid transition temperature \(T_{c}\) in two different ways. The first consists of computing the superfluid fraction \(\rho_{s}(L,T)\) on a lattice of size \(L\), as a function of temperature, using the well-known winding number estimator [28]. We then determine a size-dependent transition temperature \(T_{c}(L)\) based on the universal jump condition [29]  
\[\rho_{s}(L,T_{c})=f_{r}\ \frac{2T_{c}}{\pi} \tag{2}\]  
where \(f_{r}=1-16\pi e^{-4\pi}\)[30]. Equation (2) can be used to obtain an estimate of the transition temperature (\(T_{c}(L)\)) on a lattice of finite size. In order to extrapolate the value of \(T_{c}(L)\) to the thermodynamic (\(L\rightarrow\infty\)) limit (referred to as \(T_{c}\)), we fit the results for \(T_{c}(L)\), obtained for different system sizes to the expression [31]:  
\[T_{c}(L)=T_{c}+\frac{a}{(\ln bL)^{2}} \tag{3}\]where \(a\), \(b\) are constant. It should be noted that other expressions have been proposed, aimed at extracting \(T_{c}\)[23]; we come back to this point when discussing our results.  
The superfluid transition temperature can also be inferred from the behavior of the spin correlation function [32], specifically from the divergence of the correlation length \(\xi\) as \(T\to T_{c}\), namely [1; 2; 3]:  
\[\xi(T)\sim A\ e^{\frac{c}{\gamma t}} \tag{4}\]  
where \(A\), \(c\) are constant (\(c\approx 1.5\)[32]), and \(t=\frac{(T-T_{c})}{T_{c}}\) is the reduced temperature. Above \(T_{c}\), the correlation length \(\xi(T)\) can be obtained from the computed correlation function by means of a simple fitting procedure, illustrated in Ref. [16]. Using the best fit to Equation (4), an estimate of \(T_{c}\) is obtained; the accuracy of the estimated \(T_{c}\) increases with the size of the system studied. The estimates of \(T_{c}\) obtained in the two ways illustrated above are consistent within their statistical uncertainties; however, we find that the first procedure, based on the universal jump of the superfluid fraction, affords a considerably more accurate determination of \(T_{c}\).  
Moreover, we calculate the specific heat (i.e., the heat capacity per site) through the direct estimator of the heat capacity [33], based on the mean-squared fluctuations of the total energy \(E\):  
\[C=\frac{1}{L^{2}}\beta^{2}(\langle E^{2}\rangle-\langle E\rangle^{2}), \tag{5}\]  
where \(\beta=1/T\) is the inverse temperature. This estimator is numerically \"noisy\", and for this reason the numerical differentiation of the computed energy values with respect to the temperature has often been preferred [16]. In our case, however, we found it possible to obtain reasonably accurate estimates of the specific heat using Equation (5), thanks to the available computing facilities and the methodology adopted.