Statistical Mechanics - Physics at Oregon State University

9.6. SPIN-CORRELATION FUNCTION FOR THE ISING CHAIN. 217
and we expect to observe things that are dependent on the sample size below
that temperature.
Finally, we can check the relation between the spin correlation function and
the susceptibility. We found before that χ = β Γi and this gives
β
Γi = β
tanh i β
(βJ) =
1 − tanh(βJ)
i
i
(9.138)
and for small temperatures this is about 1
2 βe2βJ . That is off by a factor of
two. Again, this is not surprising. We used the result obtained after taking
the thermodynamic limit to calculate the result. We need to take the whole
expression as a function of N, though, and evaluate
This is equal to
or
or
or
N−1
j=0
1
1 + tanh N ⎛
N−1
⎝
(βJ)
1
1 + tanh N (βJ)
1
1 + tanh N (βJ)
j=0
tanh j (βJ) + tanh N−j (βJ)
1 + tanh N (βJ)
tanh j (βJ) + tanh N N−1
(βJ)
(9.139)
tanh −j ⎞
(βJ) ⎠ (9.140)
j=0
1 − tanh N (βJ)
1 − tanh(βJ) + tanhN (βJ) 1 − tanh−N (βJ)
1 − tanh −1
(βJ)
1 − tanh N (βJ)
1 − tanh(βJ) + tanh(βJ)tanhN
(βJ) − 1
tanh(βJ) − 1
1 − tanh N (βJ)
1 + tanh N 1 + tanh(βJ)
(βJ) 1 − tanh(βJ)
(9.141)
(9.142)
(9.143)
and now the results are consistent. If we take the thermodynamic limit in the
last equation we get an extra factor in the denominator, which is equal to two
at low temperature, and which is needed to get the results we derived before.
Also, if we first take the limit to zero temperature the result becomes N, which
is correct because all atoms now act in the same way and the magnetization of
one atom as a function of h follows Curie’s law with χ = Nβ.
The observations in the last section might seem a bit esoteric, but they
are important. Too often approximated results are used to obtain erroneous
answers. One needs to know when approximations can be made.