Keywords: parallel cluster algorithms, lattice spin models, Monte Carlo simulations, Goldstone mode singularities, correlation functions, power law.

external field. Then we consider a modified Wolff algorithm, which can be applied to spin systems in the presence of an external field. A parallel Open MP version of the standard Wolff algorithm for the three-dimensional Ising model has been developed earlier [2]. Here we generalize this parallel algorithm for spin models, often called n-vector models or O(n) models, where spins are n-component vectors. The Ising model is included here as a particular case of n=1. Further on, we elaborate a parallel version of the modified Wolff algorithm in order to simulate O(n) models below the critical temperature in the presence of the external field and to study the Goldstone mode effects. A parallel FORTRAN code for this algorithm has been developed and tested, allowing us to speed up the simulation (up to 3 times on 4 processors), as well as to treat larger lattices due to greater operative (shared) memory available. Using this code, we have simulated the n-component vector model for n=2,4,10 and have analysed the Goldstone mode singularities for the transverse and the longitudinal Fourier-transformed two-point correlation functions. The simulation results for n=10 at several linear lattices sizes L=192,256,384 and small values of the external field h=0.000875,0.0004375,0.00021875 are discussed in some detail to test the theoretical predictions for the power-law Goldstone mode singularities of the correlation functions. The results obtained provide good numerical evidence for the theoretically expected power-law divergence of the transverse correlation function at small wave vectors. We have found that by fitting the Monte Carlo data that this divergence is described by the exponent about -1.97 with an uncertainty of order 0.01 in this value. The longitudinal correlation function is analysed, as well. Generally, the simulation results show that the transverse correlation function has remarkably smaller finite-size and finite-h effects, as well as a better power-law scaling in certain finite range of the wave vector magnitude k, as compared to the longitudinal correlation function.

1

U. Wolff, "Collective Monte Carlo updating for spin systems", Phys. Rev. Lett., 62, 361-364, 1989. doi:10.1103/PhysRevLett.62.361

2

J. Kaupuzs, R.V.N. Melnik, J. Rimšans, "Parallelization of the Wolff single-cluster algorithm", Phys. Rev. E, 81, 026701, 2010. doi:10.1103/PhysRevE.81.026701

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