Statistical Mechanics - Physics at Oregon State University

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180 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE. but sometimes flips. A bar magnet will switch poles once in a while (but don’t wait for it to happen, folks). If we would take the thermodynamic limit first, the ground state of the system would always be disordered! This is the wrong picture of the ground state with zero magnetization. On the other hand, if we take a system at a small non-zero temperature, the thermodynamic limit then implies that the equilibrium state of the system is a disordered state. Now the spatial and temporal averages of variables are the same. But at T = 0 they are not, and the point T = 0 is a singularity for the one dimensional Ising chain. Realistic systems have finite values of N and non-zero values of T . Keep this in mind, and watch out for sample size dependent behavior near phase transitions. Near a phase transition the correlation length becomes larger than the system size, just as in our example of the one dimensional Ising chain. Near a phase transition we have to look carefully at the thermodynamic limit and scale out finite size effects, using appropriate scaling laws. The same argument can be applied in two dimensions. Assume that the spins are arranged on a square of L × L sites. Consider all Bloch wall states which divide the sample into two parts, each of which is connected and contains at least one surface spin. The average length of the Bloch wall LW is proportional to L, LW = cL. The way we defined the Bloch wall does not allow for fractal dimensions, because we have discrete building blocks, and finite steps along the wall. The number of possible walls depends on the number of neighbors of each site. In first approximation it will depend on L in the form 2Lb LW . In this case b is the average number of choices we have for a wall to continue at a given point. For example, on a square lattice we expect b to be between two and three. At each site the next point in the Bloch wall can either go right, straight, or left. This would mean three choices for each link in the wall. Since we have to avoid wall crossings, the actual value will be lower. The pre-factor 2L is due to the fact that we have L possible starting points at a given side, and we can start at two sides. The increase in energy due to the wall will be LW 2J, because each element of the wall breaks a bond. Hence we have ∆G = 2JLW − kBT log(2[2Lb LW ]) + kBT log(2) (8.104) where we included a factor two because we can go spin up to down or down to up. This gives ∆G = 2JLW − kBT log(b LW ) − kBT log(2[2L]) + kBT log(2) (8.105) In the limit L → ∞ the last two terms are not important and we see that ∆G = LW (2J − kBT log(b)) (8.106) Therefore the ordered state is stable against the thermal introduction of Bloch walls if kBT log(b) < 2J. This gives an estimate of
8.6. BETHE APPROXIMATION. 181 kBTc = 2J log(b) The mean field results for a square lattice is kBT mf c (8.107) = 4J; if b > 1.6 the Landau estimate of Tc is below the mean-field result. The exact result for a two-dimensional square Ising lattice is kBTc = 2.27J, or b ≈ 2.4. This is a very reasonable value for b considering our discussion on the relation between the parameter b and choices of continuing a Bloch wall. The largest value of b we can have is three, which would give kBTc = 1.8J, and this gives a lower limit on calculations for Tc. 8.6 Bethe approximation. The next question is: how can we improve the mean-field results. In the mean field approximation we do not have any information on the collective behavior of two neighboring spins. We need that information, and want to find ways to obtain that information. One possibility is via the density matrices. If we introduce some correlation in the approximate density matrix, we will obtain better results. Mathematically, this is a rigorous procedure, and can be performed easily on a computer. A simple model of introducing correlations is the Bethe approximation, which does not introduce density matrices explicitly. The basic idea of the Bethe approximation is the following. Consider a cluster of atoms. The interactions between spins in this cluster are taken into account exactly, but the interactions with the environment are treated in a mean field manner. The basic philosophy of the mean field approximation was the following. The energy at a given site depends on the effects of the neighboring sites through some averaged additional magnetic field: Emf (σ0) = −(h + h ′ )σ0 + f(h ′ ) (8.108) where h is the regular magnetic field, and h ′ is the additional field. The term f(h ′ ) has to be introduced to avoid double counting of bonds. At the central site all bonds to neighbors are counted, and if we multiply by N that means that we would count the effect of each bond twice. We determine h ′ by requiring translational symmetry, 〈σ0〉 = m, which leads to m = tanh(β(h + h ′ )). We need an additional condition to relate h ′ and m and make the obvious choice h ′ = qJm. How can we improve the mean field result? If we want to describe bonds with neighbors exactly, we need to consider a cluster of atoms. All sites in the cluster are treated exactly. There is still an outside of the cluster, and that outside will be represented by some effective field. In this way we do describe correlations between spins in the cluster, and obtain more information that we had before. But since the outside is still represented by an average field, we expect that some elements of the mean field theory might still survive. We do
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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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VI CONTENTS
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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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10 CHAPTER 1. FOUNDATION OF STATIST
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44 CHAPTER 3. VARIABLE NUMBER OF PA
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68 CHAPTER 4. STATISTICS OF INDEPEN
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90 CHAPTER 5. FERMIONS AND BOSONS
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92 CHAPTER 5. FERMIONS AND BOSONS t
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94 CHAPTER 5. FERMIONS AND BOSONS w
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120 CHAPTER 6. DENSITY MATRIX FORMA
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230 APPENDIX A. SOLUTIONS TO SELECT
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