Phase transitions are observed in surprisingly simple systems, e.g. on
a lattice of interacting spins s_{i} (magnet
vectors). Interaction energy of nearest neighbours pair is
E_{ij} = -J (s_{i} s_{j} ) . Total energy E and magnetization M for a spins configuration {s_{1}, s_{2}, ... s_{n}} 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. |
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 T_{c} = 2.269 from disordered (non-magnetic) to ordered magnetic state (see Fig.1). |