Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 243
Problem 5.8. What is the maximum value of tanhβJ? Show that for finite values of βJ, G(r)
given by (5.54) decays with increasing r.
∗ General calculation of G(r) in one dimension. To calculate G(r) in the absence of an
external magnetic field we assume free boundary conditions. It is useful to generalize the Ising
model and assume that the magnitude of each of the nearest neighbor interactions is arbitrary so
that the total energy E is given by
N−1
E = − Jisisi+1, (5.59)
i=1
where Ji is the interaction energy between spin i and spin i+1. At the end of the calculation we
will set Ji = J. We will find in Section 5.5.4 that m = 0 for T > 0 for the one-dimensional Ising
model. Hence, we can write G(r) = sksk+r. For the form (5.59) of the energy, sksk+r is given by
where
sksk+r = 1
ZN
s1=±1
N−1
ZN = 2
···
sN=±1
N−1
sksk+r exp
i=1
βJisisi+1
, (5.60)
2coshβJi. (5.61)
i=1
The right-hand side of (5.60) is the value of the product of two spins separated by a distance r in
a particular microstate times the probability of that microstate.
We now use a trick similar to that used in Section 3.5 and the Appendix to calculate various
sums and integrals. If we take the derivative of the exponential in (5.60) with respect to Jk, we
bring down a factor of βsksk+1. Hence, the spin-spin correlation function G(r = 1) = sksk+1 for
the Ising model with Ji = J can be expressed as
sksk+1 = 1
ZN
s1=±1
= 1 1
ZN β
= 1 1
ZN β
∂
∂Jk
= sinhβJ
coshβJ
···
sN=±1
N−1
sksk+1exp
···
i=1
N−1
exp
s1=±1 sN=±1 i=1
∂ZN(J1,··· ,JN−1)
∂Jk
Ji=J
βJisisi+1
βJisisi+1
(5.62a)
(5.62b)
(5.62c)
= tanhβJ, (5.62d)
where we have used the form (5.61) for ZN. To obtain G(r = 2), we use the fact that s2 k+1
write sksk+2 = sk(sk+1sk+1)sk+2 = (sksk+1)(sk+1sk+2). We write
G(r = 2) = 1
N−1
sksk+1sk+1sk+2 exp βJisisi+1
ZN
{sj}
i=1
= 1
ZN
= 1 to
(5.63a)
1
β2 ∂2ZN(J1,··· ,JN−1)
= [tanhβJ]
∂Jk∂Jk+1
2 . (5.63b)