Statistical Mechanics - Physics at Oregon State University

9.5. EXACT SOLUTION FOR THE ISING CHAIN. 209
This formula shows clearly that there is no phase transition in the one-dimensional
Ising chain. In the limit h → 0 the magnetization m is zero as long as T = 0.
Next we ask the question how large a field do we need to get a large magnetization.
The condition for a value of m close to one is
sinh 2 (βh) ≫ e −4βJ
(9.95)
Note that this is never satisfied in the limit h → 0, because in that limit the right
hand side goes to zero very rapidly. If we define a field ho by sinh 2 (βho) = e −4βJ ,
then this field represents the field at which the chain becomes magnetic. In the
limit T → 0 the exponent will be very small, and therefore the hyperbolic sine
is very small. As a result we find
T → 0 ho ≈ kBT e −2βJ
(9.96)
Although the Ising chain is non-magnetic at low temperatures, we only need
a very small magnetic field to change its state into a magnetic one. It goes
to zero exponentially fast. We see the susceptibility at zero field is very large
near T = 0, which is the situation close to a phase transition. We can evaluate
χ(h, T, N) =
∂m
∂h and find
χ(h, T, N) =
T,N
At h = 0 this is equal to
eβJ
β cosh(βh) e2βJ sinh 2 (βh) + e−2βJ −
eβJ sinh(βh) 1
2 [e2βJ sinh 2 (βh) + e−2βJ 1 − ] 2 e2βJ 2 sinh(βh) cosh(βh)β
e2βJ sinh 2 (βh) + e−2βJ (9.97)
χ(h = 0, T, N) = eβJ β √ e −2βJ
e −2βJ
= βe 2βJ
(9.98)
which indeed diverges for T → 0.
The solution for m is singular at T = 0, and the effects of this singularity
are felt at small temperatures. It is not surprising that one might interpret the
data as showing a phase transition. For example, the following figure shows m
versus T and h with 4J = 1:
This figure seems to indicate that at low temperatures there is an abrupt
change from negative to positive magnetization as a function of h, with a critical
temperature of about 0.5, which is indeed equal to 2J. But now look at it with
h plotted on a logarithmic scale. Now we see that we always go through an area
with zero magnetization.
For a real phase transition we have a positive critical temperature, below
which we see in the m(T, h) plots an abrupt change. The key word is below the
critical temperature. In the one dimensional Ising chain this point is pushed
to T = 0, and we cannot get below anymore. We still see the effects of the
singularity at T = 0, though. The ordered state at h = 0 might live long
enough that for all practical purposes the system behaves like a system with a