Computational Physics - Department of Physics

13.3 Ising Model and Phase Transitions in Magnetic Systems 421

and the entropy is given by

S=kBlnΞ+kBT

(

∂lnΞ

∂T

)

V,μ

,

while the mean number of particles is

〈N〉=kBT

(

∂lnΞ

∂ μ

)

V,T

The pressure is determined as

p=kBT

(

∂lnΞ

∂V

)

μ,T
In the pressure canonical ensemble we employ with Gibbs’ free energy as the potential. It
is related to Helmholtz’ free energy viaG=F+pV. The partition function is∆(N,p,T), with
temperature, pressure and the number of particles as variables. The pressure and volume
term can be replaced by other external potentials, such as anexternal magnetic field (or a
gravitational field) which performs work on the system. Gibbs’ free energy reads

G=−kBT ln∆,

and the entropy is given by

S=kBln∆+kBT

(

∂ln∆

∂T

)

p,N
We can compute the volume as

V=−kBT

(

∂ln∆

∂p

)

N,T

,

and finally the chemical potential

μ=−kBT

(

∂ln∆

∂N

)

p,T

In this chapter we work with the canonical ensemble only.

13.3 Ising Model and Phase Transitions in Magnetic Systems

13.3.1Theoretical Background

The model we will employ in our studies of phase transitions at finite temperature for mag-
netic systems is the so-called Ising model. In its simplest form the energy is expressed as

E=−J

N

∑

<kl>

sksl−B

N

∑

k

sk,

withsk=± 1 ,Nis the total number of spins,Jis a coupling constant expressing the strength
of the interaction between neighboring spins andBis an external magnetic field interacting
with the magnetic moment set up by the spins. The symbolindicates that we sum over
nearest neighbors only. Notice that forJ> 0 it is energetically favorable for neighboring spins
to be aligned. This feature leads to, at low enough temperatures, a cooperative phenomenon
called spontaneous magnetization. That is, through interactions between nearest neighbors,
a given magnetic moment can influence the alignment of spins that are separated from the