DINAMIKA MODEL SPIN XY 2 DIMENSI;DYNAMICS OF TWO-DIMENSIONAL XY SPIN MODEL

Mega Christivana, Imam Sugeng Santoso

Published Jan 1, 2016• 0 citations

Abstract

A simulation of two-dimensional XY spin model has been done using Monte Carlo methods written in C. In this simulation, the behaviour of spins on a certain temperature had been visually learned, and a quantitative result of data containing the values of Hamiltonian energy, specific heat, magnetization, susceptibility, and amount of vortices in a system with temperature below and above the critical temperature Tc has also been obtained. Moreover, different critical temperature value obtained for both lattice size tested; Tc = 1.1 for system with lattice size 16×16 and Tc = 1.0 for system with lattice size 256×256, proving that critical temperature of a finite-size lattice would be shifted. The obtained Tc values for both finite-size systems are close to the ideal critical temperature for Kosterlitz-Thouless transition, TKT = 0.892.

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DINAMIKA MODEL SPIN XY 2 DIMENSI;DYNAMICS OF TWO-DIMENSIONAL XY SPIN MODEL

Mega Christivana, Imam Sugeng Santoso

Published Jan 1, 2016• 0 citations

OpenAlex

Abstract

Related papers

Critical behavior of the two-dimensionalXYmodel

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Physical review. B, Condensed matterJul 1, 1986124 citations

We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat

{\mathit{C}}_{\mathit{v}}

for the XY model. The simulations are done on

{64}^{2}

{128}^{2}

{256}^{2}

, and

{512}^{2}

lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate,

{\mathit{T}}_{\mathit{c}}

=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.

Monte Carlo simulations are performed for the spin-\textonehalf{}

\mathrm{XY}

model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature

{T}_{c}

, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.

{S}_{1}

and

{S}_{2}

are nearest neighbors and vanishes otherwise. In order to make sure that our results correspond to equilibrium values, we have looked into the time-dependent properties of this model in the vicinity of critical temperature (

{T}_{c}

). We have found that the diffusion constant for vortex motion is given at

{T}_{c}

by D\ensuremath{\simeq}0.2 (in units of nearest-neighbor distance squared per MC step per spin). The values of the relaxation times follow from the value of D. Our computer running times were typically

{10}^{5}

MC steps per spin, larger than any relaxation time for the system sizes we deal with. We use a procedure based on finite-size scaling to establish the value of

{T}_{c}

=0.89J/

{k}_{B}

, the value of \ensuremath{\nu}=0.5\ifmmode\pm\else\textpm\fi{}0.1, and the value of ${\ensuremath{\eta}}_{c}$=0.24\ifmmode\pm\else\textpm\fi{}0.03, in agreement with the values predicted by the Kosterlitz-Thouless theory.