# Superfluid Transition and Specific Heat of the 2D x-y Model: Monte Carlo Simulation

^{*}

## Abstract

**:**

^{4}He are discussed.

## 1. Introduction

^{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].

^{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.

^{4}He films, as well as utilized predictively, in the same context [13,14].

## 2. Model and Methodology

## 3. Results

## 4. Conclusions

^{4}He, for which the peak in the specific heat observed in computer simulations [26] is located at $T\sim 1.6\phantom{\rule{4pt}{0ex}}{T}_{c}$.

^{4}He films, which approach the 2D limit and display superfluid transitions that conform with the KT paradigm. Indeed, this may help in the interpretation of specific heat data for

^{4}He films adsorbed on graphite, where similar features (peaks) are often interpreted as signalling phase transitions (e.g., melting of commensurate solid phases, see for instance Ref. [25]).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**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 ${T}_{c}\left(L\right)$ versus the system size L. Solid line is a fit to the data using Equation (3).

**Figure 3.**The correlation length $\xi $ as a function of the temperature, for a system of size $L=4096$. The solid line is a fit to the data using expression (4). Inset shows the computed spin correlation function $G\left(r\right)$ for a temperature $T=0.96$.

**Figure 4.**The specific heat C versus the temperature T for different lattice sizes. The inset shows the position of the peak as a function of lattice size.

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**MDPI and ACS Style**

Nguyen, P.H.; Boninsegni, M.
Superfluid Transition and Specific Heat of the 2D *x*-*y* Model: Monte Carlo Simulation. *Appl. Sci.* **2021**, *11*, 4931.
https://doi.org/10.3390/app11114931

**AMA Style**

Nguyen PH, Boninsegni M.
Superfluid Transition and Specific Heat of the 2D *x*-*y* Model: Monte Carlo Simulation. *Applied Sciences*. 2021; 11(11):4931.
https://doi.org/10.3390/app11114931

**Chicago/Turabian Style**

Nguyen, Phong H., and Massimo Boninsegni.
2021. "Superfluid Transition and Specific Heat of the 2D *x*-*y* Model: Monte Carlo Simulation" *Applied Sciences* 11, no. 11: 4931.
https://doi.org/10.3390/app11114931