Consider atoms in the presence of a -directed magnetic field of
strength . Suppose that all
atoms are identical spin- systems. It follows that either
(spin up) or (spin down), where is (twice) the -component
of the th atomic spin. The total energy of the system is
written:

The physics of the Ising model is as follows. The first term on the right-hand
side of Eq. (351) shows that the overall energy is lowered when
neighbouring atomic spins are aligned. This effect is mostly
due to the *Pauli exclusion principle*. Electrons cannot occupy the same quantum
state, so two electrons on neighbouring atoms which have parallel spins
(*i.e.*, occupy the same orbital state) cannot come close together in
space. No such restriction applies if the electrons have anti-parallel
spins. Different spatial separations imply different electrostatic
interaction energies, and the exchange energy, , measures this difference.
Note that since the exchange energy is *electrostatic* in origin, it
can be quite large: *i.e.*, eV. This is far larger than the
energy associated with the direct magnetic interaction between neighbouring
atomic spins, which
is only about eV. However, the exchange effect is very short-range; hence,
the restriction to nearest neighbour interaction is quite realistic.

Our first attempt to analyze the Ising model will employ a simplification
known as the *mean field approximation*. The energy of the th atom is written

(352) |

(353) |

We can write

(354) |

(355) |

Consider a single atom in a magnetic field . Suppose that
the atom is in thermal equilibrium with a heat bath of temperature
. According to the well-known Boltzmann distribution, the mean spin
of the atom is

(356) |

Let us assume that all atoms have *identical* spins: *i.e.*, .
This assumption is known as the ``mean field approximation''.
We can write

Note that the heat bath in which a given atom is immersed is simply the rest of the atoms. Hence, is the temperature of the atomic array. It is helpful to define the

(360) |

(361) |

(362) |

The above formula is iterated until .

It is helpful to define the *net magnetization*,

(364) |

(365) |

(366) |

Figures 102, 101, and 103 show the net magnetization, net energy,
and heat capacity calculated from the iteration formula (363) in the
absence of an external magnetic field (*i.e.*, with ). It
can be seen that below the critical (or ``Curie'') temperature, , there is
*spontaneous magnetization*: *i.e.*, the exchange effect is sufficiently large
to cause neighbouring atomic spins to spontaneously align. On the other hand,
thermal fluctuations completely eliminate any alignment above the critical temperature. Moreover, at the
critical temperature there is a
*discontinuity* in the first derivative of the energy, , with respect to
the temperature, . This discontinuity generates a downward jump
in the heat capacity, , at . The sudden loss of spontaneous
magnetization as the temperature exceeds the critical temperature is a type of
*phase transition*.

Now, according to the conventional classification of *phase transitions*, a
transition is *first-order* if the energy is discontinuous with respect
to the order parameter (*i.e.*, in this case, the temperature), and *second-order*
if the energy is continuous, but its first derivative with respect to the
order parameter is discontinuous, *etc.* We conclude that the loss of
spontaneous magnetization in a ferromagnetic material as the temperature
exceeds the critical temperature is a second-order phase transition.

In order to see an example of a first-order phase transition, let us examine the behaviour of the magnetization, , as the external field, , is varied at constant temperature, .

Figures 104 and 105 show the magnetization, , and energy, , versus
external field-strength, , calculated from the iteration formula (363)
at some constant temperature, , which is less
than the critical temperature, . It can be seen that is
*discontinuous*, indicating the presence of a first-order phase transition.
Moreover, the system exhibits *hysteresis*--meta-stable
states exist within a certain range of values, and the magnetization of the system
at fixed and (within the aforementioned range) depends on its *past history*:
*i.e.*, on whether was increasing or decreasing when it entered the meta-stable
range.

Figures 106 and 107 show the magnetization, , and energy, , versus
external field-strength, , calculated from the iteration formula (363)
at a constant temperature, , which is equal to the critical temperature, .
It can be seen that is now
*continuous*, and there are no meta-stable states. We conclude that first-order
phase transitions and hysteresis only occur, as the external field-strength is varied, when the
temperature lies below the critical temperature: *i.e.*, when the ferromagnetic
material in question is capable of spontaneous magnetization.

The above calculations, which are based on the mean field approximation, correctly predict the existence of first- and second-order phase transitions when and , respectively. However, these calculations get some of the details of the second-order phase transition wrong. In order to do a better job, we must abandon the mean field approximation and adopt a Monte-Carlo approach.

Let us consider a two-dimensional square array of atoms. Let be the size of the array, and the number of atoms in the array, as shown in Fig. 108. The Monte-Carlo approach to the Ising model, which completely avoids the use of the mean field approximation, is based on the following algorithm:

- Step through each atom in the array in turn:
- For a given atom, evaluate the change in energy of the system, , when the atomic spin is flipped.
- If then flip the spin.
- If then flip the spin with probability .

- Repeat the process many times until thermal equilibrium is achieved.

In order to demonstrate that the above algorithm is correct, let us consider
flipping the spin of the th atom. Suppose that this operation causes the
system to make a transition from state (energy, ) to state (energy, ).
Suppose, further, that . According to the above algorithm, the probability
of a transition from state to state is

(367) |

(368) |

where is the probability that the system occupies state , and is the probability that the system occupies state . Equation (369) simply states that in thermal equilibrium the rate at which the system makes transitions from state to state is equal to the rate at which the system makes reverse transitions. The previous equation can be rearranged to give

(370) |

Now, each atom in our array has *four* nearest neighbours, except for atoms on the
edge of the array, which have less than four neighbours. We can eliminate
this annoying special behaviour by adopting *periodic boundary conditions*:
*i.e.*, by identifying opposite edges of the array. Indeed, we can think of the
array as existing on the surface of a torus.

It is helpful to define

(371) |

(372) |

via the direct method is difficult due to statistical noise in the energy, . Instead, we can make use of a standard result in equilibrium statistical thermodynamics:

where is the standard deviation of fluctuations in . Fortunately, it is fairly easy to evaluate : we can simply employ the standard deviation in from step to step in our Monte-Carlo iteration scheme.

Figures 109-116 show magnetization and heat capacity versus temperature curves for , 10, 20, and 40 in the absence of an external magnetic field. In all cases, the Monte-Carlo simulation is iterated 5000 times, and the first 1000 iterations are discarded when evaluating (in order to allow the system to attain thermal equilibrium). The two-dimensional array of atoms is initialized in a fully aligned state for each different value of the temperature. Since there is no external magnetic field, it is irrelevant whether the magnetization, M, is positive or negative. Hence, is replaced by in all plots.

Note that the versus curves generated by the Monte-Carlo simulations
look very much like those predicted by the
mean field model. The resemblance increases as the size, , of the atomic
array increases. The major difference is the presence of a magnetization ``tail'' for in
the Monte-Carlo simulations: *i.e.*, in the Monte-Carlo simulations the spontaneous magnetization
does not collapse to zero once the critical temperature is exceeded--there is
a small lingering magnetization for .
The versus curves show the heat capacity calculated directly (*i.e.*,
),
and via the identity
. The latter method of calculation is
clearly far superior, since it generates significantly less statistical noise. Note that the heat capacity
*peaks* at the critical temperature: *i.e.*, unlike the mean field model, is
not zero for . This effect is due to the residual magnetization present when .

Our best estimate for is obtained from the location of the peak in the versus
curve in Fig. 116. We obtain . Recall that the mean field model
yields . The exact answer for a two-dimensional array of ferromagnetic atoms
is

(375) |

Note, from Figs. 110, 112, 114, and 116, that the height of the peak in the heat capacity curve at increases with increasing array
size, . Indeed, a close examination of these figures yields
for ,
for ,
for , and
for .
Figure 117 shows
plotted against for , 10, 20, and 40.
It can be seen that the points lie on a very convincing straight-line, which strongly suggests that

(376) |

Of course, for physical systems,
, where is Avogadro's
number. Hence, is effectively *singular* at the critical temperature
(since ), as sketched in
Fig. 118. This observation leads us to revise our definition of a second-order phase
transition. It turns out that actual discontinuities in the heat capacity almost
never occur. Instead, second-order phase transitions are
characterized by a *local quasi-singularity* in the heat capacity.

Recall, from Eq. (374), that the typical amplitude of energy fluctuations is proportional
to the square-root of the heat capacity (*i.e.*,
). It
follows that the amplitude of energy fluctuations becomes *extremely large* in the vicinity
of a second-order phase transition.

Now, the main difference between our mean field and Monte-Carlo calculations is the existence of residual magnetization for in the latter case. Figures 119-123 show the magnetization pattern of a array of ferromagnetic atoms, in thermal equilibrium and in the absence of an external magnetic field, calculated at various temperatures. It can be seen that for the pattern is essentially random. However, for , small clumps appear in the pattern. For , the clumps are somewhat bigger. For , which is just above the critical temperature, the clumps are global in extent. Finally, for , which is a little below the critical temperature, there is almost complete alignment of the atomic spins.

The problem with the mean field model is that it assumes that all atoms are situated
in identical environments. Hence, if the exchange effect is not sufficiently
large to cause global alignment of the atomic spins then there is no alignment at all.
What actually happens when the temperature exceeds the critical temperature
is that global alignment disappears, but local
alignment (*i.e.*, clumping) remains. Clumps are only eliminated by thermal
fluctuations once the temperature is significantly greater than the
critical temperature. Atoms in the middle of the clumps
are situated in a different environment than atoms on the clump boundaries. Hence,
clumps cannot occur in the mean field model.