13.9 Exercises 455
- Decrease the temperature step in this region and perform calculations for larger lattices
as well.
Forq= 6 andq= 10 we have a first order phase transition, the energy shows a discontinuity
at the critical temperature.
To compute the magnetisation in this case can lead to some preliminary conceptual prob-
lems. For theq= 2 case we can always assign the values of− 1 and+ 1 to the spins. We would
then get the same magnetisation as we had with the two-dimensional Ising model. However,
we could also assign the value of 0 and 1 to the spins. A simulation could then start with all
spins equal 0 at low temperatures. This is then the ordered state. Increasing the tempera-
ture and crossing the region where we have the phase transition, both spins value should be
equally possible. This means half of the spins take the value0 and the other half take the
value 1, yielding a final magnetisation per spin of 1 / 2. The important point is that we see
the change in magnetisation when we cross the critical temperature. For higherqvalues, for
exampleq= 3 we could choose something similar to the Ising model. The spins could take
the values− 1 , 0 , 1. We would again start with an ordered state and let temperature increase.
AboveTCall values are equally possible resulting again in a magnetisation equal zero. For
the values 0 , 1 , 2 the situation would be different. AboveTC, one third has value 0, another
third takes the value 1 and the last third is 2, resulting in a net magnetisation per spin equal
0 × 1 / 3 + 1 × 1 / 3 + 2 × 1 / 3 = 1.