Wang-Landau Analysis of the q state Potts Model

Introduction

The Potts model is a generalization of the Ising model in which each site or "spin" can have one of q states. The energy between two sites is 0 if their states are different, and -K if the states are the same. The q = 2 Potts model is equivalent to the Ising model, with half the critical temperature, because the energy difference between two like and two unlike states is half what it is in the Ising model for which the spins take on values ±1.

    One of the important features of the Potts model is that the nature of the phase transition from a disordered to ordered state changes with q. For small q the transition is continuous as it is in the Ising model (in zero magnetic field). For q > 4 on the square lattice the transition becomes first order, with a jump in the energy at the transition temperature.

    We use the Wang-Landau algorithm to generate an approximate density of states g(E). Because the energy is a discrete variable in the Potts model, g(E) becomes the number of microstates of energy E. The probability that the system has energy E is given by P(E) = g(E)exp(-ßE), where ß = 1/kT. The program uses units such that Boltzmann's constant k = 1 and the coupling constant K = 1.

    The program plots ln [(g(E)/g(E0)] versus E, where E0 = -2N is the lowest possible energy for a system of N sites on a square lattice. The program also plots the unnormalized value of ln P(E) for the value of the temperature T specified by the user and the histogram of energies for each pass of the Wang Landau algorithm. Also shown is the specific heat as a function of temperature, which can be computed from P(E).

Problems

  1. Run the simulation with q = 2 at a temperature that is close to the transition temperature T ≅ 1.135. Run the simulation until the parameter f is near 1.0. What is the shape of ln P(E)? Under the Views menu you can change the vertical scale so that you can see more of the peak structure. Compare the plots of ln P(E) for T = 0.5, T = 1, and T = 2.
  2. Run the simulation with q = 10 at a temperature which is close to the transition temperature T ≅ 0.70. What is the shape of ln P(E)? To obtain more insight set T = 0.5 and describe the shape of ln (P(E). Then repeat for T = 1. Compare the plots of ln P(E) for different temperatures. How does the shape of P(E) compare to what you found for q = 2? Explain why P(E) has two peaks near the transition for q = 10.

References

Java Classes

Updated 5 June 2015.