Computational Physics - Department of Physics

13.3 Ising Model and Phase Transitions in Magnetic Systems 425

The partition function forNspins is given by

ZN= ∑

s 1 =± 1

... ∑

sN=± 1

exp(βJ

N− 1

∑

j= 1

sjsj+ 1 ),

and since the last spin occurs only once in the last sum in the exponential, we can single out
the last spin as follows
∑
sN=± 1

exp(βJsN− 1 sN) = 2 cosh(βJ).

The partition function consists then of a part from the last spin and one from the remaining
spins resulting in
ZN=ZN− 12 cosh(βJ).

We can repeat this process and obtain

ZN= ( 2 cosh(βJ))N−^2 Z 2 ,

withZ 2 given by
Z 2 = ∑
s 1 =± 1

∑

s 2 =± 1

exp(βJs 1 s 2 ) = 4 cosh(βJ),

resulting in
ZN= 2 ( 2 cosh(βJ))N−^1.

In the thermodynamical limit where we letN→∞, the way we treat the ends does not matter.
However, since our computations will always be carried out with a limited value ofN, we
need to consider other boundary conditions as well. Here we limit the attention to periodic
boundary conditions.
If we use periodic boundary conditions, the partition function is given by

ZN= ∑

s 1 =± 1

... ∑

sN=± 1

exp(βJ

N

∑

j= 1

sjsj+ 1 ),

where the sum in the exponential runs from 1 toNsince the energy is defined as

E=−J

N

∑

j= 1

sjsj+ 1.

We can then rewrite the partition function as

ZN= ∑

{si=± 1 }

N

∏

i= 1

exp(βJsisi+ 1 ),

where the first sum is meant to represent all lattice sites. Introducing the matrixTˆ(the so-
called transfer matrix)

Tˆ=

(

eβJ e−βJ

e−βJ eβJ

)

,

with matrix elementst 11 =eβJ,t 1 − 1 =e−βJ,t− 11 =eβJandt− 1 − 1 =eβJwe can rewrite the parti-
tion function as
ZN= ∑
{si=± 1 }

Tˆs 1 s 2 Tˆs 2 s 3 ...TˆsNs 1 =TrTˆN.

The 2 × 2 matrixTˆis easily diagonalized with eigenvaluesλ 1 = 2 cosh(βJ)andλ 2 = 2 sinh(βJ).
Similarly, the matrixTˆNhas eigenvaluesλ 1 Nandλ 2 Nand the trace ofTˆNis just the sum over