Commun. Comput. Phys.,
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Volume 4.


Advanced Monte Carlo Study of the Goldstone Mode Singularity in the 3D XY Model

J. Kaupuzs 1*, R. V. N. Melnik 2, J. Rimsans 1

1 Institute of Mathematics and Computer Science, University of Latvia, 29 Raina Boulevard, LV1459, Riga, Latvia.
2 Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada.

Received 3 September 2007; Accepted (in revised version) 5 November 2007
Available online 27 February 2008

Abstract

Advanced Monte Carlo simulations of magnetisation and susceptibility in 3D $XY$ model are performed at two different coupling constants $\beta=0.55$ and $\beta=0.5$, completing our previous simulation results with additional data points and extending the range of the external field to twice as small values as previously reported ($h \ge 0.00015625$). The simulated maximal lattices sizes are also increased from $L=384$ to $L=512$. Our aim is an improved estimation of the exponent $\rho$, describing the Goldstone mode singularity $M(h)=M(+0) + c h^{\rho}$ at $h \to 0$, where $M$ is the magnetisation. The data reveal some unexpected small oscillations. It makes the estimation by many-parameter fits of the magnetisation data unstable, and we are looking for an alternative method. Our best estimate $\rho=0.555(17)$ is extracted from the analysis of effective exponents determined from local fits of the susceptibility data. This method gives stable and consistent results for both values of $\beta$, taking into account the leading as well as the subleading correction to scaling. We report also the values of spontaneous magnetisation.

AMS subject classifications: 65C05, 82B20, 82B80

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Key words: Monte Carlo simulation, XY model, magnetisation and susceptibility, high performance computing, coupling constants, singularities, lattices.

*Corresponding author.
Email: kaupuzs@latnet.lv (J. Kaupuzs), rmelnik@wlu.ca (R. V. N. Melnik), rimshans@mii.lu.lv (J. Rimsans)
 

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