# Superfluid Transition and Specific Heat of the 2D \(x\)-\(y\) Model: Monte Carlo Simulation
## 1 Introduction  
The two-dimensional classical \(x\)-\(y\) model is the simplest model to exhibit a Kosterlitz-Thouless (KT) transition [1; 2; 3; 4]. The KT universality class includes the superfluid phase transition in two dimensions (2D) which is a subject of ongoing experimental and theoretical investigation, chiefly in the context of thin films of \({}^{4}\)He adsorbed on a wide variety of substrates [5; 6; 7; 8; 9; 10]. Theoretical results obtained by studying the \(x\)-\(y\) model, typically by computer simulations, are utilized both to ascertain whether a particular physical system experimentally investigated, believed to be in the same universality class, conforms with the KT paradigm, as well as to predict the behavior of systems yet unexplored [11; 12; 13; 14]. Decades of computer simulation studies of the 2D \(x\)-\(y\) model, carried out on square lattices of size as large as \(L=2^{16}\)[15], have yielded very precise estimates of the superfluid transition temperature \(T_{c}\) and of the critical exponents associated with the transition [15; 16; 17; 18; 19; 20; 21; 22; 23]. Less extensively investigated is the behavior of the specific heat, which displays an anomaly at a temperature \(\sim 17\%\) above \(T_{c}\), in numerical simulations of the \(x\)-\(y\) model on square lattices of size \(L=2^{8}\)[19]. The position of the peak appears to depend weakly on the size of the simulated lattice, but to our knowledge no systematic study has yet been carried out, aimed at establishing whether such an anomaly occurs in the thermodynamic limit, and its actual location. There have been also speculations that the width of the peak may narrow in the thermodynamic limit, and the peak itself may evolve into a cusp [17]. There is presently no consensus regarding the physical interpretation of such an anomaly, which does not appear to signal the occurrence of any phase transition. Interestingly, experiments on \({}^{4}\)He monolayers [24], as well as computer simulations [25] (including of 2D \({}^{4}\)He [26]) have also yielded evidence of a peak in the specific heat at temperature _above_ the superfluid transition temperature. To our knowledge no further studies have been carried out of the specific heat, beyond that of Ref. [19], aimed at establishing any possible shift in temperature of the peak, as the lattice size is increased, as well as the general shape of the curve. One reason for this state of affairs is that the calculation of the specific heat in direct numerical (Monte Carlo) simulations is affected by relative large statistical uncertainties, due to the inherent \"noisiness\" of the presently known specific heat estimators. However, the (almost) threedecades elapsed since the publication of Ref. [19] have witnessed both an impressive increase in the available computing power, as well as the development of more efficient and sophisticated simulation methods. In light of that, it seems worthwhile to revisit this issue, which is of potential experimental relevance, as the \(x\)-\(y\) model is sometimes invoked in the interpretation of measurements of the specific heat of thin \({}^{4}\)He films, as well as utilized predictively, in the same context [13,14]. In this article, we report results of large scale computer simulations of the 2D \(x\)-\(y\) model on the square lattice, performed using the Worm Algorithm in the lattice flow representation [27]. We carried out simulations on square lattices of size up to \(L=2^{12}\). Our main aim is to study the specific heat, and provide robust, reliable information about its behavior in the thermodynamic limit; in order to validate our study, we also computed the superfluid transition temperature and spin correlations, comparing them to the most recent theoretical estimates. We obtain a value of \(T_{c}\) consistent with that of Ref. [15], which is presently the most accurate published result.