Statistical Mechanics - Physics at Oregon State University

198 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
Differentiating both sides and using h = 0 gives
β
χ(h = 0, T ) =
cosh 2 {qJχ(h = 0, T ) + 1} (9.44)
βJqm
If T is larger than Tc we have m = 0 and we find that near Tc
χ(0, T ) ≈
1
kB(T − Tc)
(9.45)
If, on the other hand, T is less than Tc, the magnetization m is non-zero. Near Tc
the value of m is small, however, and we can expand the square of the hyperbolic
cosine
cosh 2 βqJm ≈ 1 + (βqJ) 2 α 2 (Tc − T ) (9.46)
where we used that near the critical temperature m ≈ α √ Tc − T . The value of
α follows from the results in the previous chapter and we find α2 = 3 . From
Tc
9.44 we get
Since βqJ = Tc−T
T
cosh 2 βqJm − βqJ χ(h = 0, T ) = β (9.47)
1 ≈ Tc (Tc − T ) near Tc we find
1
χ(0, T ) ≈
2kB(Tc − T )
Hence near the critical temperature we find in general that
χ(0, T ) = A±|T − Tc| −γ
(9.48)
(9.49)
where the value of the critical exponent γ = 1, just like we found in mean field
theory in chapter four.
The calculation in the Bethe approximation is harder. We have from the
previous chapter:
m = cosh β(J + h′ + h) − cosh β(J − h ′ − h)
cosh β(J + h ′ + h) + cosh β(J − h ′ − h)
with self-consistency equation 8.129:
cosh(β(J + h + h ′ ))
cosh(β(J − h − h ′ 2
= e q−1
)) βh′
The first of these two equations gives
β −1
∂m
∂h
T
=
(9.50)
(9.51)
[(sinh β(J + h ′ + h) + sinh β(J − h ′ − h))(cosh β(J + h ′ + h) + cosh β(J − h ′ − h))−