A study of the phase transition in the planar spin model on square lattices using Monte Carlo simulations (bibtex)

by Bradley L. Holland

Abstract:

Using the Metropolis Monte Carlo technique, the two-dimensional planar spin model has been simulated for square lattices of 256, 900, 3600, 4900, 8100 and 10000 spins. The energy, specific heat, angular square displacement, magnetization and susceptibility have been calculated for all lattices at temperatures ranging from 0.1 k_{B}/J to 2.0 k_{B}/J (where J is the nearest-neighbour coupling). In addition, the helicity modulus has been calculated for the 8100 spin lattice. It has also been confirmed that there is a loss of long range order as the size of the system increases. Three expressions proposed by Kosterlitz and Thouless relating lattice size and temperature with angular square displacement, magnetization and internal energy were tested and confirmed. The specific heat curve was plotted for all lattice sizes and a peak was found slightly above the critical temperature at 1.03 k_{B}/J. By using the Kosterlitz-Thouless theory the critical temperature at which vortex pairs unbind was found by fitting the susceptibility curve for the 10000 spin lattice. This critical temperature was also estimated from the drop of the helicity modulus curve for the 8100-spin lattice. The critical temperature was found to lie between 0.8 k_{B}/J and 0.9 k_{B}/J.

Reference:

Bradley L. Holland, "A study of the phase transition in the planar spin model on square lattices using Monte Carlo simulations", 2000.

Bibtex Entry:

@bachelorsthesis{2000H, title={A study of the phase transition in the planar spin model on square lattices using Monte Carlo simulations}, author={Bradley L. Holland}, month={June}, year={2000}, abstract={Using the Metropolis Monte Carlo technique, the two-dimensional planar spin model has been simulated for square lattices of 256, 900, 3600, 4900, 8100 and 10000 spins. The energy, specific heat, angular square displacement, magnetization and susceptibility have been calculated for all lattices at temperatures ranging from 0.1 k<SUB>B</SUB>/J to 2.0 k<SUB>B</SUB>/J (where J is the nearest-neighbour coupling). In addition, the helicity modulus has been calculated for the 8100 spin lattice. It has also been confirmed that there is a loss of long range order as the size of the system increases. Three expressions proposed by Kosterlitz and Thouless relating lattice size and temperature with angular square displacement, magnetization and internal energy were tested and confirmed. The specific heat curve was plotted for all lattice sizes and a peak was found slightly above the critical temperature at 1.03 k<SUB>B</SUB>/J. By using the Kosterlitz-Thouless theory the critical temperature at which vortex pairs unbind was found by fitting the susceptibility curve for the 10000 spin lattice. This critical temperature was also estimated from the drop of the helicity modulus curve for the 8100-spin lattice. The critical temperature was found to lie between 0.8 k<SUB>B</SUB>/J and 0.9 k<SUB>B</SUB>/J.}, note={Supervised by S.K. Bose} }

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