1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 651

b. Consider trying to write one of the terms in brackets in the aboveexpres-
sion as
eK(σ^1 +σ^2 +σ^3 +σ^4 )+ e−K(σ^1 +σ^2 +σ^3 +σ^4 )=g(K)eK

′(σ 1 σ 2 +σ 2 σ 3 +σ 3 σ 4 +σ 4 σ 1 )

,

where the new coupling constantK′and the functiong(K) are to be de-

termined by requiring that this equation be satisfied by the four nonequiv-

alent choices of the spins:

σ 1 =σ 2 =σ 3 =σ 4 =± 1

σ 1 =σ 2 =σ 3 =−σ 4 =± 1

σ 1 =σ 2 =−σ 3 =−σ 4 =± 1

σ 1 =−σ 2 =σ 3 =−σ 4 =± 1.

Show thatg(K) andK′cannot be determined in this way.

c. Consider instead introducing several new coupling constants,K 1 ,K 2 , and

K 3 , and writing

eK(σ^1 +σ^2 +σ^3 +σ^4 )+ e−K(σ^1 +σ^2 +σ^3 +σ^4 )

=g(K)e(1/2)K^1 (σ^1 σ^2 +σ^2 σ^3 +σ^3 σ^4 +σ^4 σ^1 )+K^2 (σ^1 σ^3 +σ^2 σ^4 )+K^3 σ^1 σ^2 σ^3 σ^4.

By inserting the four nonequivalent choices of the spin variables from part

b, find expressions forK 1 ,K 2 ,K 3 , andg(K) in terms ofK. Interpret the

resulting partition function.

d. Note that the result of part c does not lead to an exact RG procedure.
Show that ifK 2 andK 3 are neglected, an RG equation of the form

K 1 =

1

4

ln cosh(4K)

results. Does this lead to a critical point?

e. In order to improve the results, it is proposed to neglect onlyK 3 and treat

theK 2 term approximately. Let Σ 1 and Σ 2 be the spin sums multiplying

K 1 andK 2 , respectively, in the partition function expression. Consider

the approximation

K 1 Σ 1 +K 2 Σ 2 ≈K′(K 1 ,K 2 )Σ′i,jσiσj,

where the sum is taken only over nearest neighbors. What is the partition

function that results from this approximation?

f. Define the free energy per spin as

a(K) =

1

N

lnQ(K).

Show that the free energy satisfies an RG equation of the form

a(K) =

1

2

lng(K) +

1

2

a(K′).