Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 247
The roots of (5.79) are
λ± = e βJ coshβH ± e −2βJ +e 2βJ sinh 2 βH 1/2 . (5.80)
It is easy to show that λ+ > λ− for all β and H, and consequently (λ−/λ+) N → 0 as N → ∞. In
the thermodynamic limit N → ∞ we obtain from (5.78) and (5.80)
1
lim
N→∞ N lnZN(T,H)
= lnλ+ +ln 1+
and the free energy per spin is given by
λ−
λ+
N = lnλ+, (5.81)
f(T,H) = 1
N F(T,H) = −kT ln e βJ coshβH + e 2βJ sinh 2 βH +e −2βJ 1/2 . (5.82)
We can use (5.82) and (5.31) and some algebraic manipulations to find the magnetization per
spin m at nonzero T and H:
m = − ∂f
∂H =
sinhβH
(sinh 2 βH +e −4βJ ) 1/2.
(5.83)
A system is paramagnetic if m = 0 only when H = 0, and is ferromagnetic if m = 0 when H = 0.
From (5.83) we see that m = 0 for H = 0 because sinhx ≈ x for small x. Thus for H = 0,
sinhβH = 0 and thus m = 0. The one-dimensional Ising model becomes a ferromagnet only at
T = 0 where e −4βJ → 0, and thus from (5.83) |m| → 1 at T = 0.
Problem 5.11. Isothermal susceptibility of the Ising chain
More insight into the properties of the Ising chain can be found by understanding the temperature
dependence of the isothermal susceptibility χ.
(a) Calculate χ using (5.83).
(b) What is the limiting behavior of χ in the limit T → 0 for H > 0?
(c) Show that the limiting behavior of the zero-field susceptibility in the limit T → 0 is χ ∼ e 2βJ .
(The zero-field susceptibility is found by calculating the susceptibility for H = 0 and then
taking the limit H → 0 before other limits such as T → 0 are taken.) Express the limiting
behavior in terms of the correlation length ξ. Why does χ diverge as T → 0?
Because the zero-field susceptibility diverges as T → 0, the fluctuations of the magnetization
also diverge in this limit. As we will see in Section 5.6, the divergence of the magnetization
fluctuations is one of the characteristics of the critical point of the Ising model. That is, the phase
transition from a paramagnet (m = 0 for H = 0) to a ferromagnet (m = 0 for H = 0) occurs at
zero temperature for the one-dimensional Ising model. We will see that the critical point occurs
at T > 0 for the Ising model in two and higher dimensions.