Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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9.7. RENORMALIZATION GROUP THEORY. 223<br />
This equ<strong>at</strong>ion has solutions ˜ J (∞) = 0 and ˜ J (∞) = ∞. These solutions are<br />
called the fixed points of the equ<strong>at</strong>ion. If we start with one of these values,<br />
the scaling transform<strong>at</strong>ions will not change the values. Hence they represent<br />
physical situ<strong>at</strong>ions th<strong>at</strong> are length independent.<br />
In the case ˜ J (∞) = 0 we find th<strong>at</strong> the free energy is given by<br />
−βG = 2N 1<br />
4 log(4) = log(2N ) (9.170)<br />
The right hand side is equal to the entropy of a completely disordered chain<br />
divided by the Boltzmann constant. Therefore we have G = −T S, as expected<br />
for large temper<strong>at</strong>ures or zero coupling constant.<br />
In the case ˜ J (∞) = ∞ we find th<strong>at</strong> the free energy is given by<br />
−βG = 2N 1<br />
4 log(2e2 ˜ J (∞)<br />
) ⇒ G ≈ −NkBT ˜ J = −NJ (9.171)<br />
The right hand side is equal to the energy of a completely ordered chain, which<br />
happens if the temper<strong>at</strong>ure is zero of the coupling constant is infinity.<br />
If we start with an arbitrary value of ˜ J the next value will be smaller and so<br />
on. So we will end <strong>at</strong> the fixed point zero. The only exception is when we start<br />
exactly <strong>at</strong> infinity, then we stay <strong>at</strong> infinity. Hence we can say the following. If<br />
we start <strong>at</strong> a fixed point, we remain <strong>at</strong> th<strong>at</strong> point. If we make a small devi<strong>at</strong>ion<br />
from the fixed point for our starting value, we always end up <strong>at</strong> zero. Therefore,<br />
˜J (∞) = 0 is a stable fixed point and ˜ J (∞) = ∞ is an unstable fixed point.<br />
The stable fixed point ˜ J = 0 corresponds to T = ∞. At an infinite temper<strong>at</strong>ure<br />
any system is completely disordered and the st<strong>at</strong>es of the individual spins<br />
are completely random and uncorrel<strong>at</strong>ed. If we average random numbers the<br />
results will remain random and a system <strong>at</strong> an infinite temper<strong>at</strong>ure will look the<br />
same for any magnific<strong>at</strong>ion. This fixed point is therefore a trivial fixed point<br />
and is expected to occur for any system. The same is true for the second fixed<br />
point in our simple model, which corresponds to T = 0. At zero temper<strong>at</strong>ure all<br />
spins are ordered and again the system looks the same under any magnific<strong>at</strong>ion.<br />
It is again a fixed point which will always show up. It does not correspond to a<br />
critical point since it does not divide the temper<strong>at</strong>ure range in two parts, there<br />
are no neg<strong>at</strong>ive temper<strong>at</strong>ures. Close to zero temper<strong>at</strong>ure the effects of this fixed<br />
point are noticeable, however, and show up in a divergence of the correl<strong>at</strong>ion<br />
length to infinity <strong>at</strong> T = 0. Note th<strong>at</strong> <strong>at</strong> infinite temper<strong>at</strong>ure the correl<strong>at</strong>ion<br />
length becomes zero.<br />
Wh<strong>at</strong> we do in renormaliz<strong>at</strong>ion group theory is to replace one spin by the<br />
effective value of a block of spins. Th<strong>at</strong> changes the length scales of our problem,<br />
and in general changes our observ<strong>at</strong>ions. Th<strong>at</strong> is not true, however, <strong>at</strong> the<br />
critical point. In th<strong>at</strong> case the correl<strong>at</strong>ion length is infinity and repe<strong>at</strong>ed transform<strong>at</strong>ions<br />
will give the same results. Therefore , the critical point will show up<br />
as a fixed point in the scaling equ<strong>at</strong>ions. The other case where changing length<br />
scales does not affect the physics is when the correl<strong>at</strong>ion length is zero. Hence<br />
we always have a fixed point corresponding to zero correl<strong>at</strong>ion length or infinite<br />
temper<strong>at</strong>ure.