Statistical Mechanics - Physics at Oregon State University

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222 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS. if the series converges. At this point we have established a general procedure relating the partition function of a system to a partition function containing the combined effects of the spins in a block of two. The coupling constants ˜ J and ˜ h changed values, however. They had to be renormalized in order for the expressions to be valid. Hence the new system is different from the old one, since it is not described by the same Hamiltonian. It is possible that in certain cases both ˜ J ′ = ˜ J and ˜h ′ = ˜ h. In that case the old and new system do represent the same physics, no matter how many times we apply the demagnification operation. Critical points therefore correspond to fixed points of the renormalization formulas. One has to keep in mind that a fixed point does not necessarily correspond to a critical point; there are more fixed points than critical points. The one-dimensional Ising model depends only on two coupling constants. These represent the interaction energy between the spins and the energy of a spin in an external field. Both constants are scaled with respect to kBT . It is in general always possible to scale the constants with respect to the temperature, since the partition function always combines β and H in a product. In a general model, one has a number of coupling constants, and a search for critical points corresponds to a search for fixed points in a many-dimensional space. The easiest example of such a search is again for the one-dimensional Ising model, this time without an external field. Hence ˜ h = 0 and 9.161 shows that ˜ h ′ = 0 too. In every step of the renormalization procedure the coupling constant for the external field remains zero. The renormalization equation 9.160 is now very simple: ˜J ′ ( ˜ J, 0) = 1 2 log cosh(2 ˜ J) (9.167) Since cosh(x) ex we see that log cosh(2 ˜ J) log e2 ˜ J = 2J˜ and hence 0 ˜ J ′ ˜ J. This also implies g( ˜ J ′ , 0) g( ˜ J, 0), and hence the series 9.166 for the free energy converges and is bounded by 1 k k 2 g( J, ˜ 0) = 2g( J, ˜ 0). Suppose we start with a coupling constant ˜ J = ˜ J (0) . Each iteration adds a prime to the value according to the equation 9.160 and after k iterations we have the value ˜ J = ˜ J (k) . If we keep going, we arrive at limk→∞ ˜ J (k) = ˜ J (∞) . Because of the limiting conditions we have 0 ˜ J (∞) ˜ J. Since the function g is also decreasing, we can get a lower bound on the free energy by using the infinite value, and we find 2Ng( ˜ J (∞) , 0) −βG 2Ng( ˜ J (0) , 0) (9.168) It is clear that the value of ˜ J (∞) depends on the initial value of ˜ J and hence on the temperature T . The possible values of ˜ J (∞) can be found by taking the limit in 9.160 on both sides. That leads to ˜J (∞) = 1 2 log cosh(2 ˜ J (∞) ) (9.169)
9.7. RENORMALIZATION GROUP THEORY. 223 This equation has solutions ˜ J (∞) = 0 and ˜ J (∞) = ∞. These solutions are called the fixed points of the equation. If we start with one of these values, the scaling transformations will not change the values. Hence they represent physical situations that are length independent. In the case ˜ J (∞) = 0 we find that the free energy is given by −βG = 2N 1 4 log(4) = log(2N ) (9.170) The right hand side is equal to the entropy of a completely disordered chain divided by the Boltzmann constant. Therefore we have G = −T S, as expected for large temperatures or zero coupling constant. In the case ˜ J (∞) = ∞ we find that the free energy is given by −βG = 2N 1 4 log(2e2 ˜ J (∞) ) ⇒ G ≈ −NkBT ˜ J = −NJ (9.171) The right hand side is equal to the energy of a completely ordered chain, which happens if the temperature is zero of the coupling constant is infinity. If we start with an arbitrary value of ˜ J the next value will be smaller and so on. So we will end at the fixed point zero. The only exception is when we start exactly at infinity, then we stay at infinity. Hence we can say the following. If we start at a fixed point, we remain at that point. If we make a small deviation from the fixed point for our starting value, we always end up at zero. Therefore, ˜J (∞) = 0 is a stable fixed point and ˜ J (∞) = ∞ is an unstable fixed point. The stable fixed point ˜ J = 0 corresponds to T = ∞. At an infinite temperature any system is completely disordered and the states of the individual spins are completely random and uncorrelated. If we average random numbers the results will remain random and a system at an infinite temperature will look the same for any magnification. This fixed point is therefore a trivial fixed point and is expected to occur for any system. The same is true for the second fixed point in our simple model, which corresponds to T = 0. At zero temperature all spins are ordered and again the system looks the same under any magnification. It is again a fixed point which will always show up. It does not correspond to a critical point since it does not divide the temperature range in two parts, there are no negative temperatures. Close to zero temperature the effects of this fixed point are noticeable, however, and show up in a divergence of the correlation length to infinity at T = 0. Note that at infinite temperature the correlation length becomes zero. What we do in renormalization group theory is to replace one spin by the effective value of a block of spins. That changes the length scales of our problem, and in general changes our observations. That is not true, however, at the critical point. In that case the correlation length is infinity and repeated transformations will give the same results. Therefore , the critical point will show up as a fixed point in the scaling equations. The other case where changing length scales does not affect the physics is when the correlation length is zero. Hence we always have a fixed point corresponding to zero correlation length or infinite temperature.
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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Statistical Mechanics - Physics at Oregon State University

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