### Mean-field solutions of the Ising model

Introduction

The program solves the mean-field equation for the Ising model

m = tanh β(Jqm + B),

where β = 1/kT, T is the absolute temperature, k is Boltzmann's constant, B is the external magnetic field, q is the number of nearest neighbors of a given spin, J is the exchange constant, and m is the magnetization per spin. (A factor of μ has been absorbed into B.) This equation for m is a self-consistent transcendental equation. That is, the mean-field that influences the mean value of m depends on the mean value of m.

Algorithm

The program is relatively simple and uses Newton's method to find the zeros of the function g(m) = m - tanh β(Jqm + B).

The solutions correspond to the high temperature disordered paramagnetic state (m = 0) and the low temperature ordered ferromagnetic state (m ≠ 0). How do we know which solution to choose? The answer can be found by determining the free energy for both solutions and choosing the solution that gives the smaller free energy. It can be shown that the free energy per spin is .

For simplicity we measure the temperature in units of J/k, which is equivalent to setting J = k = 1 in the program.

Problems

1. Choose q = 4, corresponding to the square lattice. Set B = 0 and determine the value of the critical tempeprature Tc such that m ≠ 0 for T ≤ Tc and m = 0 for T > Tc.
2. Given that m = 1 at T = 0 and is zero at T = Tc, obtain a sufficient number of numerical solutions for T < Tc so that you can plot m as a function of T. Does m go to zero continuously as T approaches Tc from below?
3. Choose q = 2, corresponding to a one-dimensional lattice. Does mean-field theory predict that Tc > 0 or Tc = 0? Compare the mean-field solution to the exact result for Tc in one dimension.

References

Mean-field solutions of the Ising model are frequently discussed in the context of the Curie-Weiss theory.

1. Most undergraduate level textbooks discuss mean-field theory in the context of the Ising model. See for example, Daniel V. Schroeder, An Introduction to Thermal Physics, Addison-Wesley (2000).
2. Harvey Gould and Jan Tobochnik, online notes, <stp.clarku.edu/notes>, Chapter 5.

Java Classes Used

• MeanFieldApp

Updated 27 February 2007.