434 13 Monte Carlo Methods in Statistical Physics
and if we assume thatmis real we have two solutions
m= 0 ,or
m^2 =
b
2 c(
− 1 ±
√
1 − 4 ac/b^2)
.
This relation can be used to describe both first and second-order phase transitions. Here
we consider the second case. We assume thatb> 0 and leta≪ 1 since we want to study a
perturbation aroundm= 0. We reach the critical point whena= 0 , that is
m^2 =
b
2 c(
− 1 ±
√
1 − 4 ac/b^2)
≈−a/b.We define the temperature dependent function
a(T) =α(T−TC),withα> 0 andTCbeing the critical temperature where the magnetization vanishes. Ifais
negative we have two solutions
m=±√
−a/b=±√
α(TC−T)
b ,meaning thatmevolves continuously to the critical temperature whereF= 0 forT≤TC
We can now compute the entropy as follows
S=−(
∂F
∂T
)
.
ForT≥TCwe havem= 0 and
S=−(
∂F 0
∂T
)
,
and forT≤TC
S=−(
∂F 0
∂T
)
−α^2 (TC−T)/ 2 b,and we see that there is a smooth crossover atTC.
In theories of critical phenomena one has that
CV∼∣∣
∣∣ 1 −T
TC
∣∣
∣∣
−α
,and Onsager’s result is a special case of this power law behavior. The limiting form of the
function
limα→ 0
1
α
(Y−α− 1 ) =−log(Y),meaning that the closed-form result is a special case of the power law singularity withα= 0.
13.5 The Metropolis Algorithm and the Two-dimensional Ising Model
We switch now back to the Ising model in two dimensions and explore how to write a pro-
gram that will allow us to compute various thermodynamical quantities. The algorithm of