Monte Carlo simulations on Ising/Potts model

Demonstration

In the demonstration below, we simulate a 10-state (Q = 10) Potts model. For Ising model, set Q = 2.

Download: Java source code.

Monte Carlo simulation on Ising model

The principles of Monte Carlo simulations are best illustrated on the two-dimensional Ising/Potts model.

The system is set up on a L×L lattice. Each site has a spin, and each spin s_i at site i can assume one of the two values 1 or −1. Each spin can only interact with the four neighbors. The Hamiltonian (energy function) H is defined as

where, ⟨i, j⟩ denotes a pair of nearest neighbors, and J is a coupling constant between spins. We simplicity, we assume J = 1 here.

Source code in C

The code is2.c, gives a minimalist example for a simulation of 2D Ising model. It outputs energy and heat capacity.
Exact value for the internal energy and heat capacity are E_av = −1133.866661, C = 907.072, respectively, computed from is2exact.c.

Random number generators (RNG)

The above Monte Carlo simulation requires many random numbers for selecting spins and for deciding whether or not to accept a move. Here are some commonly-used options.

• The above version uses the Tausworthe random number generator.
• Mersenne Twister RNG is also good, see is2mt.c.
  
• There may be some danger if you choose to use the Complementary-multiply-with-carry RNGs. Test it yourself, see is2cmwc.c.
• Do not ever use the default rand() function from the standard C library <stdlib.h> in any Monte Carlo simulation, usually it yields terribly wrong results (due to a short period of self-repetition), see is2rand.c.

Generalization -- Potts model (Java)

Below we demonstrate a generalization of the Ising model, called Potts model. Here the spin on each site can now assume Q, instead of two, different values (shown by different colors).

The Hamiltonian becomes:

where δ(s_i, s_j) gives 1 if s_i and s_j are identical, or 0 otherwise.

The model is demonstrated at the beginning of the web page.