5 MC simulation of Ising Model

In this section, importance sampling Monte Carlo techniques will be used for the study of phase transitions at finite temperature. We shall discuss details, algorithms, and potential sources of difficulty using the Ising model as a paradigm. However, virtually all of the discussion of the application to the Ising model is relevant to other models as well. The Ising model is one of the simplest lattice models which one can imagine, and its behavior has been well studied. The simple Ising model consists of spins which are confined to the sites of a lattice and which may have only the values +1 or -1. If there are N spins on the lattice, then the system can be in states. The energy of a state is given by the Ising Hamiltonian:

where J is the interaction energy between nearest neighbour spins , h is the external magnetic field in units of energy, and . The Ising model has been solved exactly in one dimension and as a result it is known that there is no phase transition. In two dimensions, the model is exactly solved in zero field situation which showed that there is a second order phase transition. The critical temperature is obtained from the condition

as . The phase transition is characterized by the divergences in the specific heat, susceptibility, and correlation length. The critical exponents obtained are: (logarithmic), , , . The Ising model in higher dimensions can be solved following mean field approach. The mean field exponents are: (discontinuity), , , . In order to satisfy the scaling relation , on has d=4, the upper critical dimension.

5.1 Metropolis importance sampling scheme

1. Choose an initial state assigning +1 or -1 corresponding to up or down spin arbitrarily to the lattice sites.

2. Choose a site j

3. Calculate the energy change which results if the spin at site j is overturned

4. Generate a uniformly distributed random number r such that 0<r<1

5. If , flip the spin

6. Go to the next site and go to (3)