Imagine a cubic lattice in which particles carrying spin 
are placed on
the lattice sites with 
as shown below:
Such a model describes ferromagnetic materials, which can be magnetized
by applying an external magnetic field 
. In the absence
of a field, the unperturbed Hamiltonian takes the form
where 
is a tensor and 
is a spin vector
such that 
. Quantum mechanically,
would be the vector of Pauli matrices. In general, the spins can point in
any spatial direction, a fact that makes the problem difficult to solve.
A simplification introduced by Ising was to allow the spins to point in only one of two possible directions, up or down, e.g., along the z-axis only. In addition, the summation is restricted to nearest neighbor interactions only. In this model, the Hamiltonian becomes
where 
indicates restriction of the sum to nearest neighbor
pairs only. The variables 
now can take on the values 
only. The couplings 
are the spin-spin ``J'' couplings.
In the presence of a magnetic field, the full Hamiltonian becomes
which describes a uniaxial ferromagnetic system in a magnetic field. The parameters T and h are experimental control parameters.
Define the magnetization per spin as
Then the phase diagram looks like:
where the colored lines indicate a nonzero magnetization at h=0 below
a critical temperature T. The persistence of a nonzero magnetization
in ferromagnetic systems at h=0 below 
indicates a transition
from a disordered to an ordered phase. In the latter,
the spins are aligned in the direction of the applied field
before it is switched off. If 
, then the
spins will point in one direction and if 
, it
will be in the opposite direction. A plot of the isotherms
of m vs. h yields:
Notice an inflection point along the isotherm , at h=0, where
.
The thermodynamics of the magnetic system can be defined in analogy with the liquid-gas system. The analogy is shown in the table below:
Gas-Liquid Magnetic
where 
is the magnetic susceptibility.
The magnetic exponents are then given by
