Phase transitions in magnetics

Phase transitions are observed in surprisingly simple systems, e.g. on a lattice of interacting spins si (magnet vectors). Interaction energy of nearest neighbours pair is
    Eij = -J (si sj ) .
Total energy E and magnetization M for a spins configuration {s1, s2, ... sn} is obtained by summation throughout the lattice. In the Heisenberg model every spin can take arbitrary direction. In the XY model spins rotate in a plane.
In the Ising model spins have only two possible states +-1 (up or down). As since every spin takes two values, therefore there are 2n different configurations for n spins. You see below 24 = 16 spin configurations for 2x2 lattice.
 E = -4J            E = 4J
+ +   - -          + -   - +
+ +   - -          - +   + -

 E = 0
- +   + -   + +   + +    + -   - +   - -   - -    - -   + -   + +   - +
+ +   + +   + -   - +    - -   - -   - +   + -    + +   + -   - -   - +
For J > 0 the state of lowest energy is when all spins are aligned. The state has macroscopic magnetizaion, i.e. it is ferromagnetic. The system is degenerate as since several configurations have the same energy. Entropy S(E) is minimal when spins are aligned and it grows with increasing of E (and hence degeneracy).
It is supposed that spins interact too with thermostat at temperature T. In thermal equilibrium any system minimizes the F = E - T S value. Therefore at low temperature Ising spins minimize energy. Interaction aligns all spin vectors in the same direction, giving huge total magnetic fields. At high temperature the system maximizes entropy (and disorder). Thermal fluctuations break this order. The randomness of the spin configuration tends to wash out the large scale magnetism.
In the 2D Ising model there is a phase transition at Tc = 2.269 from disordered (non-magnetic) to ordered magnetic state (see Fig.1).

2D Ising model

20x20 Ising lattice is shown below. Up and down spins are white and black squares. You see magnetization M (the red curve) and energy E (the blue one) in the right part of the applet and in the Status bar. You can watch thermal fluctuations, phase transition and clusters formation (or melting) by changing temperature (choose 200x200 lattice for a 1GHz PC)

Controls Click mouse into lattice to get a new spins configuration. Drag by mouse "thermometer" to the right (the black bar corresponds to Tc) or press "Enter" to set a new T value from the text field. "Print" button sends averaged T, M, χ, E and C to the Java console. "Clear" cleans averaging. See also 640x640 applet window.

The Boltzmann distribution function

In thermal equilibrium probability of a spin configuration {s1, s2, ... sn} is determined by the Boltzmann distribution function
    w(s1, ... sn) = 1/Z exp[ -E(s1, ... sn)/T ],     (*)
Z is called the partition function and is the fundamental quantity in statistical mecanics. All quantities of interest can be extracted from Z. It is determined by
    s1 s2 ... ∑ sn w(s1, = 1,     Z = ∑ s1 s2 ... ∑ sn exp[ -E(s1, ] .
Statistical averages, e.g. energy E is obtained as
    <E>G = ∑ s1 s2 ... ∑ sn E(s1, ... sn) w(s1, ... sn) =
    1/Z ∑ s1 s2 ... ∑sn E(s1, ... sn) exp[ -E(s1, ... sn)/T]
Onsager solved exectly the 2D Ising problem in 1944, showing that it had a phase transition. In principle, for any finite n statistical averages can be calculated directly, but for macroscopic systems (e.g. for n = 100) it is impossible for any computer. Fortunately configurations with relatively large E will make a negligible contribution to the sums because of rapidly varying exponential function in the Boltzmann distribution. It follows from (*) that probability to find system in a state with energy E is
    w(E) ~ n(E) exp(-E/T),
where n(E) is the number of spins configurations with energy E. One can rewrite this as
    w(E) ~ exp[S(E)-E/T],
where S(E) = ln n(E) is entropy. Therefore in thermal equilibrium one will find frequently configurations with high w(E) and hence T S - E values.
Contents   Next: The Monte-Carlo method
updated 17 Apr 2004