Project 4: Difference between revisions
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'''First and second order phase transition in the Ising model of ferromagnetism''' | '''First and second order phase transition in the Ising model of ferromagnetism''' | ||
Write a program | Write a program which simulates via Metropolis Monte Carlo algorithm the 2D Ising model on a <math> 64 \times 64 </math> square lattice with ''periodic'' boundary conditions. The model is assumed to describe a system placed in an applied external magnetic field <math> H </math>. The energy of a particular state of this system is then given by: | ||
<math> E = - J \sum_{\langle i,j \rangle} S_i S_j - H \sum_i S_i </math> | <math> E = - J \sum_{\langle i,j \rangle} S_i S_j - H \sum_i S_i </math> | ||
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where <math> \langle i,j \rangle </math> means that the sum runs over all pairs of nearest neighbor spins (attached to the lattice sites), and <math> J </math> is the exchange coupling. Positive <math> J >0 </math> interaction between the spins leads to ferromagnetism, will negative <math> J >0 </math> gives rise to antiferromagnetism in the Ising model. Choose units so that <math> |J|=1 </math> and the Boltzmann constant is <math> k_B=1 </math>. The free parameters of the model are then just the temperature <math> T </math> and the applied field <math> H </math>. | where <math> \langle i,j \rangle </math> means that the sum runs over all pairs of nearest neighbor spins (attached to the lattice sites), and <math> J </math> is the exchange coupling. Positive <math> J >0 </math> interaction between the spins leads to ferromagnetism, will negative <math> J >0 </math> gives rise to antiferromagnetism in the Ising model. Choose units so that <math> |J|=1 </math> and the Boltzmann constant is <math> k_B=1 </math>. The free parameters of the model are then just the temperature <math> T </math> and the applied field <math> H </math>. | ||
==Part I for both PHYS460 and PHYS660 students: First order phase transition in the ferromagnetic Ising model== | |||
Consider ferromagnetic Ising model with <math> J >0 </math>. Start from a case in which the temperature is below the critical point, e.g., <math> T=1.0 </math> and the field is large and pointing down, e.g., <math> H = -5 </math>. Run your code using the initial state of the Ising model in which all spins <math> S_i = -1 </math> are parallel to H, so that magnetization per spin is <math> M = -1 </math>. | |||
Sweep the magnetic field up to a large positive value <math> H=5 </math> and then back down to your starting value <math> H = -5 </math>. Do this in steps of (or smaller if you have the patience!). After each change in , allow enough Monte Carlo trials for the system to reach equilibrium. For each value of , calculate the mean energy per spin mean square energy per spin , mean magnetization per spin and the specific heat (where we use the fluctuation-dissipation theorem to connect the specific heat to the fluctuations in energy . All sums here are over microstates generated by the Monte Carlo simulation. | |||
==Part II for both PHYS460 and PHYS660 students: Second order phase transition in the ferromagnetic Ising model== | |||
Investigate the behavior of the specific heat as the temperature is increased through the critical value both at (which is a second order or continuous paramagnet-ferromanget phase transition) and first order discontinous phase transtion in non-zero magnetic field . Repeat this calculation for increasing lattice size (for example, L=10,30,50,...). | |||
==Part III for PHYS660 students only: Correlation length in the ferromagnetic Ising model== | |||
==Part IV for PHYS660 students only: Second order phase transition in the antiferromagnetic Ising model== | |||
Revision as of 13:24, 2 April 2012
First and second order phase transition in the Ising model of ferromagnetism
Write a program which simulates via Metropolis Monte Carlo algorithm the 2D Ising model on a square lattice with periodic boundary conditions. The model is assumed to describe a system placed in an applied external magnetic field . The energy of a particular state of this system is then given by:
where means that the sum runs over all pairs of nearest neighbor spins (attached to the lattice sites), and is the exchange coupling. Positive interaction between the spins leads to ferromagnetism, will negative gives rise to antiferromagnetism in the Ising model. Choose units so that and the Boltzmann constant is . The free parameters of the model are then just the temperature and the applied field .
Part I for both PHYS460 and PHYS660 students: First order phase transition in the ferromagnetic Ising model
Consider ferromagnetic Ising model with . Start from a case in which the temperature is below the critical point, e.g., and the field is large and pointing down, e.g., . Run your code using the initial state of the Ising model in which all spins are parallel to H, so that magnetization per spin is .
Sweep the magnetic field up to a large positive value and then back down to your starting value . Do this in steps of (or smaller if you have the patience!). After each change in , allow enough Monte Carlo trials for the system to reach equilibrium. For each value of , calculate the mean energy per spin mean square energy per spin , mean magnetization per spin and the specific heat (where we use the fluctuation-dissipation theorem to connect the specific heat to the fluctuations in energy . All sums here are over microstates generated by the Monte Carlo simulation.
Part II for both PHYS460 and PHYS660 students: Second order phase transition in the ferromagnetic Ising model
Investigate the behavior of the specific heat as the temperature is increased through the critical value both at (which is a second order or continuous paramagnet-ferromanget phase transition) and first order discontinous phase transtion in non-zero magnetic field . Repeat this calculation for increasing lattice size (for example, L=10,30,50,...).
