13.6 Selected Results for the Ising Model 443
" 100 × 100 "
" 80 × 80 "
" 40 × 40 "
" 10 × 10 "
kBT
E/
J
1.6 1.8 2 2.2 2.4 2.6
0
-0.5
-1
-1.5
-2
-2.5
Fig. 13.5Average energy per spin as function of the lattice size for the two-dimensional Ising model.
We note that with increasing lattice size we approach a steeper line and the transition from
a smaller magnetisation to a larger one becomes sharper. This is a possible sign of a phase
transition, where we move from a state where all spins (or most of them) align in a specific
direction (high degree of order) to a phase where both spin directions are equally probable
(high degree of disorder) and result in zero net magnetisation. The ordered phase at low
temperatures is called a ferromagnetic phase while the disordered phase is called the param-
agnetic phase, with zero net magnetisation. Since we are plotting the absolute value, our net
magnetisation will always be above zero since we are taking the average of a number which
is never negative.
" 100 × 100 "
" 80 × 80 "
" 40 × 40 "
" 10 × 10 "
kBT
〈|M
|〉
1.6 1.8 2 2.2 2.4 2.6 2.8 3
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Fig. 13.6Absolute value of the average magnetization per spin as function of the lattice size for the two-
dimensional Ising model.