Statistical Mechanics - Physics at Oregon State University

210 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
1.0
0.5
0.0
−0.5
−1.0
0.0
0.25
0.5
T −0.5
0.75
1.0
−1.0
0.0
h
Figure 9.6: Magnetization of the one dimensional Ising chain.
phase transition. This requires switching times much longer than the time of
the experiment.
Phase transitions are related to singular behavior of the free energy. The
previous example shows that the singularity can be at zero temperature. If one
extends the temperature range to negative values, the singular point will be
at negative values for systems without phase transitions. It is interesting to
study how the position of the singularity depends on parameters in the model
hamiltonian, so one can predict when phase transitions occur.
9.6 Spin-correlation function for the Ising chain.
The spin correlation function Γi is an important quantity used to study the
effects of the singular behavior at T = 0. Note that semantically a critical
temperature can never be zero, since it is not possible to go below the critical
temperature in that case. In this section we calculate the spin-correlation function
for the Ising chain without an external field, that is for h = 0. We use
periodic boundary conditions and assume that the temperature is non-zero.
The spin correlation function is related to the pair distribution function gi,
which is defined by
0.5
1.0
gi = 〈S0Si〉 = 〈σ0σi〉 (9.99)
This function contains the information that we need to discuss how values of
the spin on one site relate to values of the spin on another site. For quantities
that are not correlated we have < AB >=< A >< B > and hence if the values
on the different sites are not correlated we have gi = 〈σ0σi〉 = 〈σ0〉〈σi〉 = m 2 .
The spin correlation function measures the correlation between fluctuations
from average between different sites. We have