Statistical Mechanics - Physics at Oregon State University

9.3. CRITICAL EXPONENTS. 197
9.3 Critical exponents.
Most of the interesting behavior of the phase transitions occurs near the critical
temperature. We have shown that the mean field results are equivalent to the
Landau theory discussed in the thermodynamics, and hence near Tc we have the
magnetization proportional to the square root of Tc − T , and hence the critical
exponent β is 1 in mean field theory. In the Bethe approximation we have to
2
find the value of h ′ for temperatures just below Tc. It turns out that in the
Bethe approximation the phase transition is also second order, and hence near
Tc the value of h ′ will be small. Hence we can expand the equation for h ′ in
terms of h ′ and we find
cosh β(J + h ′ )
cosh β(J − h ′ ) ≈ 1 + h′ 2β tanh βJ + (h ′ ) 2 2β 2 tanh 2 βJ + O(h ′ ) 3
e 2βh′
q−1 ′ 2β
≈ 1 + h
q − 1 + (h′ ) 2 2β2 (q − 1) 2 + O(h′ ) 3
(9.38)
(9.39)
The condition for a phase transition is the equality of the derivatives at
h ′ = 0. This gave our self-consistency condition. If this condition is satisfied, the
second derivatives are also the same! It is not hard to show that the third order
derivatives are really different. Therefore, when h ′ is small, it is determined by
h ′ 2β tanh βJ + (h ′ ) 2 2β 2 tanh 2 βJ + a(h ′ ) 3 ′ 2β
= h
q − 1 + (h′ ) 2 2β2 (q − 1) 2 + b(h′ ) 3
(9.40)
where a and b are different numbers. At the critical temperature h ′ = 0 is a
triple solution. This has to be the case, since at a temperature just below the
critical temperature the solutions are h ′ = 0 and h ′ = ±ɛ, where ɛ is a small
number. These three solutions merge together at the critical temperature. At
a temperature slightly below the critical temperature we can write
0 = h ′ (T − Tc)(c + dh ′ ) + (a − b)(h ′ ) 2
(9.41)
where c and d are constants. This shows that near Tc the effective field h ′
is proportional to √ Tc − T . The equations in the previous chapter can be
used to show that for small values of h the magnetization m is proportional
to h ′ , and hence we find that the critical exponent β is equal to 1
2 in the Bethe
approximation, just as it is in mean field theory.
The susceptibility χ is defined by
∂m
χ(h, T ) =
(9.42)
∂h T
and can be calculated easily in the mean field approximation from
m = tanh β(qJm + h) (9.43)