Computational Physics - Department of Physics

13.3 Ising Model and Phase Transitions in Magnetic Systems 427

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7

CV

Inverse TemperatureβJ

Fig. 13.3Heat capacity per spin (CV/(N− 1 )kBas function of inverse temperatureβfor the one-dimensional
Ising model.

function (with periodic boundary conditions)

ZN= ∑

s 1 =± 1

... ∑

sN=± 1

exp(β

N

∑

j= 1

(Jsjsj+ 1 +

B

2

(sj+sj+ 1 )),

which yields a new transfer matrix with matrix elementst 11 =eβ(J+B),t 1 − 1 =e−βJ,t− 11 =eβJ
andt− 1 − 1 =eβ(J−B)with eigenvalues

λ 1 =eβJcosh(βJ)+ (e^2 βJsinh^2 (βB)+e−^2 βJ)^1 /^2 ,

and
λ 1 =eβJcosh(βJ)−(e^2 βJsinh^2 (βB)+e−^2 βJ)^1 /^2.

The partition function is given byZN=λ 1 N+λ 2 Nand in the thermodynamic limit we obtain the
following free energy

F=−NkBT ln

(

eβJcosh(βJ)+ (e^2 βJsinh^2 (βB)+e−^2 βJ)^1 /^2

)

.

It is now useful to compute the expectation value of the magnetisation per spin

〈M/N〉=

1

NZ

M

∑

i

Mie−βEi=−

1

N

∂F

∂B

,

resulting in

〈M/N〉=

( sinh(βB)

sinh^2 (βB)+e−^2 βJ)^1 /^2

).

We see that forB= 0 the magnetisation is zero. This means that for a one-dimensional Ising
model we cannot have a spontaneous magnetization. For the two-dimensional model however,
see the discussion below, the Ising model exhibits both a spontaneous magnetisation and a
specific heat and susceptibility which are discontinuous oreven diverge. However, except for
the simplest case such as 2 × 2 lattice of spins, with the following partition function