306 The Monte Carlo method
separating the metastable from the stable phase. This can also be checked: below
the critical temperature, the Ising model exhibits a first order transition triggered
by the magnetic field. Going from a state with positive magnetisation and positive
magnetic field to negative (but small) magnetic field, the magnetisation will not
turn over. Making the field strongly negative will eventually pull the system over
the free energy barrier.
CheckProduce a graph of the specific heat (measured in units ofJ/kB) as a function
ofτ=kBT/Jand compare it withFigure 10.1.
You could be tempted to calculate the magnetisation in order to compare this
with Figure 7.3. As we have just mentioned, cooling the system down from a high
temperature fails unless you cool very slowly through the critical region. However,
even if you start with the low temperature phase and heat the system up, you will
notice that the magnetisation vanishes for temperatures just below the transition
temperature. The point is that in order to let the magnetisation flip from a positive
value to a negative one, a domain wall separating the two phases must be built up.
Below the transition temperature, domain walls have a positive wall tension, that is,
they carry a free energy cost per unit length. Therefore, flipping the magnetisation
in theinfinitesystem requires an infinite amount of free energy so that this will
never happen. In the finite system however, if the wall tension is still finite (and the
temperature therefore still below the finite-size critical point), the energy barrier
for a magnetisation flip is finite as the domain walls are necessarily finite. Hence
the magnetisation will fluctuate around a positive or negative equilibrium value for
relatively long periods. But now and then it may switch sign, so that its long time
average vanishes. Figure 10.2 shows the typical behaviour of the magnetisation in
this phase.
A histogram of the probability of occurrence for various magnetisations shows
a double-peaked shape. These problems also show up in the determination of the
magnetic susceptibility which can be expressed in terms of the variance of the
magnetisation.
Several methods exist for avoiding this problem:
- A restriction to – say – positive magnetisation is built in. This option has the
disadvantage that the system is distorted and the consequences of this are not a
priori clear. Moreover the average magnetisation is always positive so it
becomes hard, if not impossible, to see where it would vanish without this
restriction.
- Making a plot of the magnetisation as a function of time and taking averages
only on the plateaux where the magnetisation is either positive or negative. This
method is, however, difficult to apply close to the transition, as the strong
fluctuations there will make it difficult to distinguish these plateaux clearly.
