428 13 Monte Carlo Methods in Statistical Physics
Z= 2 e−^8 Jβ+ 2 e^8 Jβ+ 12 ,and resulting mean energy
〈E〉=−J
Z
(
16 e^8 Jβ− 16 e−^8 Jβ)
,
it is a highly non-trivial task to find the closed-form expression forZNin the thermodynamic
limit. The closed-form expression for the Ising model in twodimensions was obtained via a
mathematical tour de force in 1944 by the Norwegian chemist Lars Onsager [73]. The exact
partition function forNspins in two dimensions and with zero magnetic fieldBis given by
ZN=[
2 cosh(βJ)eI]N
,
with
I=1
2 π∫π
0dφln[
1
2
(
1 + ( 1 −κ^2 sin^2 φ)^1 /^2)]
,
and
κ= 2 sinh( 2 βJ)/cosh^2 ( 2 βJ).
The resulting energy is given by
〈E〉=−Jcoth( 2 βJ)[
1 +^2
π
( 2 tanh^2 ( 2 βJ)− 1 )K 1 (q)]
,
withq= 2 sinh( 2 βJ)/cosh^2 ( 2 βJ)and the complete elliptic integral of the first kind
K 1 (q) =∫π/ 2
0dφ
√
1 −q^2 sin^2 φ.
Differentiating once more with respect to temperature we obtain the specific heat given by
CV=^4 kB
π
(βJcoth( 2 βJ))^2{
K 1 (q)−K 2 (q)−( 1 −tanh^2 ( 2 βJ))[π
2
+ ( 2 tanh^2 ( 2 βJ)− 1 )K 1 (q)]}
,
(13.1)
where
K 2 (q) =∫π/ 2
0dφ√
1 −q^2 sin^2 φ, (13.2)is the complete elliptic integral of the second kind. Near the critical temperatureTCthe
specific heat behaves as
CV≈−2
π(
2 J
kBTC) 2
ln∣∣
∣∣ 1 −T
TC
∣∣
∣∣+const. (13.3)In theories of critical phenomena one ca show that for temperaturesTbelow a critical
temperatureTC, the heat capacity scales as [75]
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→ 01
α
(Y−α− 1 ) =−lnY,
can be used to infer that the closed-form result is a special case of the power law singularity
withα= 0.
Similar relations applies to other expectation values. An example is the the spontaneous
magnetisation per spin. This quantity is also highly non-trivial to compute. Here we simply