Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
9.5. EXACT SOLUTION FOR THE ISING CHAIN. 209<br />
This formula shows clearly th<strong>at</strong> there is no phase transition in the one-dimensional<br />
Ising chain. In the limit h → 0 the magnetiz<strong>at</strong>ion m is zero as long as T = 0.<br />
Next we ask the question how large a field do we need to get a large magnetiz<strong>at</strong>ion.<br />
The condition for a value of m close to one is<br />
sinh 2 (βh) ≫ e −4βJ<br />
(9.95)<br />
Note th<strong>at</strong> this is never s<strong>at</strong>isfied in the limit h → 0, because in th<strong>at</strong> limit the right<br />
hand side goes to zero very rapidly. If we define a field ho by sinh 2 (βho) = e −4βJ ,<br />
then this field represents the field <strong>at</strong> which the chain becomes magnetic. In the<br />
limit T → 0 the exponent will be very small, and therefore the hyperbolic sine<br />
is very small. As a result we find<br />
T → 0 ho ≈ kBT e −2βJ<br />
(9.96)<br />
Although the Ising chain is non-magnetic <strong>at</strong> low temper<strong>at</strong>ures, we only need<br />
a very small magnetic field to change its st<strong>at</strong>e into a magnetic one. It goes<br />
to zero exponentially fast. We see the susceptibility <strong>at</strong> zero field is very large<br />
near T = 0, which is the situ<strong>at</strong>ion close to a phase transition. We can evalu<strong>at</strong>e<br />
χ(h, T, N) = <br />
∂m<br />
∂h and find<br />
χ(h, T, N) =<br />
T,N<br />
At h = 0 this is equal to<br />
eβJ <br />
β cosh(βh) e2βJ sinh 2 (βh) + e−2βJ −<br />
eβJ sinh(βh) 1<br />
2 [e2βJ sinh 2 (βh) + e−2βJ 1 − ] 2 e2βJ 2 sinh(βh) cosh(βh)β<br />
e2βJ sinh 2 (βh) + e−2βJ (9.97)<br />
χ(h = 0, T, N) = eβJ β √ e −2βJ<br />
e −2βJ<br />
= βe 2βJ<br />
(9.98)<br />
which indeed diverges for T → 0.<br />
The solution for m is singular <strong>at</strong> T = 0, and the effects of this singularity<br />
are felt <strong>at</strong> small temper<strong>at</strong>ures. It is not surprising th<strong>at</strong> one might interpret the<br />
d<strong>at</strong>a as showing a phase transition. For example, the following figure shows m<br />
versus T and h with 4J = 1:<br />
This figure seems to indic<strong>at</strong>e th<strong>at</strong> <strong>at</strong> low temper<strong>at</strong>ures there is an abrupt<br />
change from neg<strong>at</strong>ive to positive magnetiz<strong>at</strong>ion as a function of h, with a critical<br />
temper<strong>at</strong>ure of about 0.5, which is indeed equal to 2J. But now look <strong>at</strong> it with<br />
h plotted on a logarithmic scale. Now we see th<strong>at</strong> we always go through an area<br />
with zero magnetiz<strong>at</strong>ion.<br />
For a real phase transition we have a positive critical temper<strong>at</strong>ure, below<br />
which we see in the m(T, h) plots an abrupt change. The key word is below the<br />
critical temper<strong>at</strong>ure. In the one dimensional Ising chain this point is pushed<br />
to T = 0, and we cannot get below anymore. We still see the effects of the<br />
singularity <strong>at</strong> T = 0, though. The ordered st<strong>at</strong>e <strong>at</strong> h = 0 might live long<br />
enough th<strong>at</strong> for all practical purposes the system behaves like a system with a