1549380323-Statistical Mechanics Theory and Molecular Simulation

638 Critical phenomena

temperature where local ordering is expected and the middle spin is likely to cause
neighboring spins to align with it. The transformation functionT(σ′;σ 1 ,σ 2 ,σ 3 ) for
this example can expressed mathematically simply as

T(σ′;σ 1 ,σ 2 ,σ 3 ) =δσ′σ 2. (16.9.10)

The new spin lattice is shown below the original lattice in Fig. 16.14.
The transformation function in eqn. (16.9.10) is now used to compute the new
Hamiltonian Θ′ 0 according to

e−Θ

′ 0 ({σ′})

=

∑

σ 1

∑

σ 2

∑

σ 3

···

∑

σN

(

δσ′ 1 σ 2 δσ′ 2 σ 5 ···

)

eKσ^1 σ^2 eKσ^2 σ^3 eKσ^3 σ^4 eKσ^4 σ^5 ···

=

∑

σ 1

∑

σ 3

∑

σ 4

∑

σ 6

···eKσ^1 σ

′ 1

eKσ

1 ′σ^3

eKσ^3 σ^4 eKσ^4 σ

′ 2

···. (16.9.11)

Eqn. (16.9.11) encodes the information we need to determine the new coupling param-
eterK′. We will use the rule that when the sums overσ 3 andσ 4 are performed, the
new interaction betweenσ′ 1 andσ′ 2 gives the contribution exp(K′σ′ 1 σ 2 ′) to the partition
function. If this rule is satisfied, then the functional form of the Hamiltonian will be
preserved. The sum overσ 3 andσ 4 that must then be performed in eqn. (16.9.11) is
∑

σ 3

∑

σ 4

exp[Kσ′ 1 σ 3 ] exp[Kσ 3 σ 4 ] exp[Kσ 4 σ′ 2 ].

Note that the spin productσ 3 σ 4 has two possible values,σ 3 σ 4 =±1, which allows us
to employ a convenient identity:

e±θ= coshθ±sinhθ= coshθ[1±tanhθ]. (16.9.12)

Eqn. (16.9.12) allows us to express exp(Kσ 3 σ 4 ) as

eKσ^3 σ^4 = coshK[1 +σ 3 σ 4 tanhK].

If we definex= tanhK, the product of the three exponentials becomes

eKσ

′ 1 σ 3

eKσ^3 σ^4 eKσ^4 σ

2 ′

= cosh^3 K(1 +σ′ 1 σ 3 x)(1 +σ 3 σ 4 x)(1 +σ 4 σ′ 2 x)

= cosh^3 K(1 +σ′ 1 σ 3 x+σ 3 σ 4 x+σ 4 σ 2 ′x

+σ 1 ′σ 32 σ 4 x^2 +σ 1 ′σ 3 σ 4 σ′ 2 x^2 +σ 3 σ^24 σ′ 2 x^2

+σ 1 ′σ 32 σ^24 σ 2 ′x^3 ). (16.9.13)

When summed overσ 3 andσ 4 , most terms in eqn. (16.9.13) cancel, yielding
∑

σ 3

∑

σ 4

eKσ

′ 1 σ 3

eKσ^3 σ^4 eKσ^4 σ

′ 2

= 2cosh^3 K

[

1 +σ′ 1 σ′ 2 x^3

]

≡coshK′[1 +σ′ 1 σ 2 ′x′].

(16.9.14)
In eqn. (16.9.14), we have expressed the interaction in its original form but with a
new coupling constantK′. In order for the interaction term [1 +σ 1 ′σ′ 2 x′] to match the
original interaction [1 +σ′ 1 σ′ 2 x^3 ], we requirex′=x^3 or