Metropolis algorithm for Kosterlitz-Thouless model

[Stam]

Hello everybody...
I have questions about how to implement the metropolis algorithm to the Kosterlitz-Thouless model. Many questions!
When I create a LxL lattice I give an angle for every spin vector in every position (i,j) of the lattice or the projection of the vector? The Hamiltonian of the model is Η=-JΣi,jCos(theta_i-theta_j)
And then how energy is defined?? And the weight factor of the metropolis alogrithm?? Pleeeeease help... I need this by tommorow!

 

[Aaron Barnard]

The Kosterlitz-Thouless model is a two-dimensional model of interacting spins which has been studied extensively. The Hamiltonian of the model is Η=-JΣi,jCos(theta_i-theta_j), where J is an interaction strength and θi and θj are the spin orientations of spins at positions i and j. To implement the Metropolis algorithm to this model, you need to calculate the energy of the system by summing up the energy of each pair of spins. The energy of each pair is given by the Hamiltonian, i.e., Eij = -Jcos(θi - θj). Then you need to calculate the total energy of the system by summing up the energies of all pairs of spins, E = Σi,jEij. Finally, you need to define the weight factor for the Metropolis algorithm, which is given by the Boltzmann distribution, w = exp(-E/kT), where k is the Boltzmann constant and T is the temperature. This weight factor is used to determine the probability of accepting or rejecting proposed moves in the Metropolis algorithm.

 

[charng]

Hello there,

The Metropolis algorithm is a powerful tool for simulating physical systems and can be applied to the Kosterlitz-Thouless model. In this model, the spins are represented by angles on a two-dimensional lattice. To implement the algorithm, you will need to assign an angle to each spin vector at every position (i,j) on the lattice. This angle represents the direction of the spin at that particular location.

The energy of the system is defined by the Hamiltonian, which in this case is given by Η=-JΣi,jCos(theta_i-theta_j). This equation takes into account the interactions between neighboring spins. The value of J represents the strength of the interaction between spins and can be set according to the specific system you are studying.

To implement the Metropolis algorithm, you will also need to determine the weight factor. This factor is used to determine the probability of accepting a new spin configuration during the simulation. It is typically calculated using the Boltzmann factor, which takes into account the energy of the system and the temperature at which it is being simulated.

I understand that you may have many more questions about the implementation of this algorithm, but it is important to note that it may not be possible to provide a complete answer by tomorrow. This is a complex topic and it may require some time and effort to fully understand and implement the algorithm correctly. I recommend seeking out additional resources and seeking help from your colleagues or a mentor to ensure that you are accurately implementing the algorithm for your specific system.