Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 253
1.0
m
0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0
kT/J
2.5 3.0
Figure 5.10: The temperature dependence of the spontaneous magnetization m(T) of the twodimensional
Ising model.
than discontinuously. Because the transition occurs only at T = Tc and H = 0, the transition
occurs at a critical point.
So far we have introduced the critical exponents α, β, and γ to describe the behavior of the
specific heat, magnetization, and susceptibility near the critical point. We now introduce three
more critical exponents: η, ν, and δ (see Table 5.1). The notation χ ∼ |ǫ| −γ means that χ
has a singular contribution proportional to |ǫ| −γ . The definitions of the critical exponents given
in Table 5.1 implicitly assume that the singularities are the same whether the critical point is
approached from above or below Tc. The exception is m, which is zero for T > Tc.
The critical exponent δ characterizes the dependence of m on the magnetic field at T = Tc:
|m| ∼ |H| 1/15 ∼ |H| 1/δ
We see that δ = 15 for the two-dimensional Ising model.
T c
(T = Tc). (5.99)
The behavior of the spin-spin correlation function G(r) for T near Tc and large r is given by
G(r) ∼
1
r d−2+ηe−r/ξ
(r ≫ 1 and |ǫ| ≪ 1), (5.100)
where d is the spatial dimension and η is another critical exponent. The correlation length ξ
diverges as
ξ ∼ |ǫ| −ν . (5.101)
The exact result for the critical exponent ν for the two-dimensional (d = 2) Ising model is ν = 1.
At T = Tc, G(r) decays as a power law for large r:
G(r) =
1
r d−2+η (T = Tc, r ≫ 1). (5.102)
For the two-dimensional Ising model η = 1/4. The values of the various critical exponents for the
Ising model in two and three dimensions are summarized in Table 5.1.