# Superfluid Transition and Specific Heat of the 2D \(x\)-\(y\) Model: Monte Carlo Simulation
## 3 Results  
We begin the presentation of our results by illustrating our estimates for the superfluid fraction as a function of temperature for the various lattice sizes considered, and by discussing the determination of the transition temperature, which we compare to those provided in other works. Figure 1 shows the computed value of \(\rho_{s}(L,T)\); the critical temperature \(T_{c}(L)\) for a given system size is determined by the universal jump condition, namely the intersection of the \(\rho_{s}(L,T)\) curve with the straight line given by the right hand side of Equation (2). The intersection point is estimated by drawing a straight line between the two adjacent values of \(\rho_{s}(L,T)\) within which the intersection can be established to take place, within the precision of our calculation. As expected, both \(T_{c}(L)\) and \(\rho_{s}(L,T_{c}(L))\) display a slow decrease as a function of \(L\). In order to extrapolate the value of \(T_{c}\) in the thermodynamic (\(L\rightarrow\infty\)) limit, we fit the computed \(T_{c}(L)\) to Equation (3), as proposed in Ref. [15]. This procedure is illustrated in Figure 2. Our estimate is \(T_{c}=0.8935(5)\), which is consistent with that of Ref. [15], namely \(0.89289(5)\), even though their quoted uncertainty is a factor ten smaller than ours, a fact to be ascribed to the significantly (sixteen times) greater system sizes adopted therein. Our estimate of \(T_{c}\) is also in perfect agreement with the more recent result of Ref. [23], in which the same computational methodology utilized in this work was adopted, on the same system sizes. Their results are of precision comparable to ours; they make use of a different, more elaborate fitting form for \(T_{c}(L)\), but their resulting estimate for \(T_{c}\) is entirely consistent with ours, and has the same uncertainty. As mentioned in Section 2, as a further check of our results we estimated the critical temperature \(T_{c}\) independently through the spin correlation length. In this case, the computed spin correlation function for a given system size yields an estimate of \(T_{c}\), obtained first by extracting the correlation length \(\xi(T)\) as a function of temperature, and then fitting the results to Equation (4). An example of this procedure is shown in Figure 3, for the largest system size considered here, which is the one that yields the estimate of \(T_{c}\) with the smallest uncertainty. Such an estimate, namely 0.893(3), is consistent with that obtained from the superfluid fraction, but it is considerably less accurate. Figure 1: The superfluid fraction \(\rho_{s}\) versus temperature, for the different lattice sizes considered. Statistical errors are smaller than symbol sizes. The straight line corresponds to the universal jump condition (right hand side of Equation (2)). Figure 2: The critical temperature \(T_{c}(L)\) versus the system size \(L\). Solid line is a fit to the data using Equation (3). Our result for \(T_{c}\) gives us sufficient confidence on the reliability of our data and simulation. Therefore, we now discuss the most important part of this work, namely the behavior of the specific heat \(C(T)\). It is worth restating that previous numerical studies of the 2D \(x\)-\(y\) model [16; 19] have yielded results for this quantity only for square lattices of size up to \(L=256\). Such studies yielded evidence of a peak in the specific heat at a temperature above \(T_{c}\); the position of this peak depends fairly strongly on system size for \(L\leq 128\). On the other hand, the shape of \(C(T)\) appears to change little going from \(L=128\) to \(L=256\), suggesting that the anomaly may indeed be a genuine physical feature of the model and not an artifact of numerical simulations carried out on finite systems of small size. Figure 4 shows our results for the specific heat for the various system sizes, showing that the curve indeed appears to stabilize for \(L>256\).