Statistical Mechanics - Physics at Oregon State University

8.5. CRITICAL TEMPERATURE IN DIFFERENT DIMENSIONS. 179
We can now distinguish two cases. If tlife ≫ tintro we find at each moment
many Bloch walls in the system. This means that the system looks like it is not
ordered. On the other hand, if tlife ≪ tintro most of the time there are no Bloch
walls in the system, and the chain is fluctuating back and forth between two
completely ordered states, one with all spins up and one with all spins down.
In both cases we have for the time average of a spin
τ
1
〈σi〉 = lim σi(t)dt (8.102)
τ→∞ τ 0
which is zero in both case. Note that we always need τ ≫ tlife, tintro.
What are values we expect for realistic systems. If we assume that attempts
are caused by phonons,the attempt frequencies for both creation and hopping
are of order 10 12 s −1 . This means that
tintro
tlife
≈ A−1 e 2Jβ
A −1 N 2
(8.103)
and with N = 10 8 as typical for a 1 cm chain and J = 1 meV as a typical spin
energy, we have tintro ≈ tlife for T ≈ 1 K. But this temperature depends on the
size of the system. At high temperatures the shorter time is the introduction
time. We always see many domain walls. This is the normal state of the system.
If we go to lower temperatures this picture should remain the same, but due
to the finite length of the chain we see a switch in behavior. Below a certain
temperature the chain at a given time is always uniformly magnetized, but
over time it switches. The temperature at which this change of behavior takes
place depends on the length of the chain. This is an example of a finite size
effect. For large systems finite size effects take place at very low temperatures,
see example numbers above. For small samples this can be at much higher
temperatures!
In the previous discussion we have three limits to worry about. In order
for the averages to be the same, 〈σ〉time = 〈σ〉ensemble, we need τ → ∞. We
are interested in low temperatures, or the limit T → 0. Finally, we have the
thermodynamic limit N → ∞. Since we need τ ≫ tlife = thopN 2 we see
that we need to take the limit τ → ∞ before the thermodynamic limit. Also,
because τ ≫ tintro = A −1 e 2Jβ we need to take the limit τ → ∞ before the limit
T → 0. Form our discussion it is clear that limτ→∞〈σ〉τ = 0. Hence we have
limT →0〈σ〉∞ = 0 and this remains zero in the TD limit. The one dimensional
Ising chain is never ordered!
Calculations at T = 0 do not make sense. At T = 0 we have tintro = ∞ and
we always have τ < tintro. That violates the ergodic theorem. So, even though
people do calculations at T = 0, one has to argue carefully if it makes sense.
That is what we did for the system of Fermions, for example.
After taking the limit τ → ∞, what does the system look like? If we now
take T → 0 we find a system that is most of the time completely ordered, but
once in a while flips. The net magnetization is zero. Therefore, the real T = 0
ground state of a finite system is a system that is ordered most of the time,