444 13 Monte Carlo Methods in Statistical Physics
The reason we choose to plot the average absolute value instead of the net magnetisation
is that slightly belowTC, the net magnetisation may oscillate between negative and positive
values since the system, as function of the number of Monte Carlo cycles is likely to have its
spins pointing up or down. This means that after a given number of cycles, the net spin may
be slightly positive but could then occasionaly jump to a negative value and stay there for
a given number of Monte Carlo cycles. Above the phase transition the net magnetisation is
always zero.
The fact that the system exhibits a spontaneous magnetization (no external field applied)
belowTCleads to the definition of the magnetisation as an order parameter. The order param-
eter is a quantity which is zero on one side of a critical temperature and non-zero on the other
side. Since the magnetisation is a continuous quantity atTC, with the closed-form results
[
1 −
( 1 −tanh^2 (βJ))^4
16 tanh^4 (βJ)
] 1 / 8
,
forT<TCand 0 forT>TC, our transition is defined as a continuous one or as a second order
phase transition. From Ehrenftest’s definition of a phase transition we have that a second
order or continuous phase transition exhibits second derivatives of Helmholtz’ free energy
(the potential in this case) with respect to e.g., temperature that are discontinuous or diverge
atTC. The specific heat for the two-dimensional Ising model exhibits a power-law behavior
aroundTCwith a logarithmic divergence. In Fig. 13.7 we show the corresponding specific
heat.
" 100 × 100 "
" 80 × 80 "
" 40 × 40 "
" 10 × 10 "
kBT
CV
/Jk
B
1.6 1.8 2 2.2 2.4 2.6
4
3.5
3
2.5
2
1.5
1
0.5
0
Fig. 13.7Heat capacity per spin as function of the lattice size for thetwo-dimensional Ising model.
We see from this figure that as the size of the lattice is increased, the specific heat develops
a sharper and shaper peak centered around the critical temperature. A similar behavior is
seen for the susceptibility as well, with an even sharper peak, as can be seen from Fig. 13.8.