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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XI The next step is to understand correlation functions from mean-field theory, just on the disor- dered side of the transition. First we rewrite the correlation function in a simpler form, where Tr denotes the sum over spin configurations: h i i = Tr i e K h ij i i j Tr e K h ij i i j = Tr i e K h ij i i j Tr e K h ij i i j , (1) where the second equality is restricted to spin configurations with = 1. Here the step used to show the second equality is inserting 1 = , 1 + ,- 1 , and using up-down symmetry. This exact relation means that the correlation function can be viewed as the response of other spins to fixing one spin up. Now we use the mean-field approximation. The mean-field equation is now more complicated because we must allow m to depend on i : summing over nearest neighbors, m i = tanh( X j K ( | i- j | ) m j ) (2) where before we had just m = tanh zKm . We are generalizing slightly to the possibility of inter- actions beyond nearest neighbor, in order to make the generalization to the continuum limit more simple. For spins j far away from the original spin at site 0, we assume m j is nearly the equilibrium value (0, when we are on the disordered side of the transition) and linearize the above, in order to get a tractable problem. This calculation of correlation functions is a good example of a case for which it is more useful to work in the continuum (below) than on the lattice. Then m i = X j K ( | i- j | ) m j + (corrections when | j | small) . (3) This is a linear equation, and has a simple Fourier transform: m ( k ) = K ( k ) m ( k ) + C, (4) where in writing the constant C we have assumed that the corrections localized around the origin are nearly a delta-function on large scales. (Otherwise, with no source, the resulting solution would just be m = 0.) Finally, we need an assumption about K ( k ) = J ( k ). This is nearly localized, and hence nearly a -function. Assuming it is short-ranged, we can expand J ( k ) J (1- R 2 k 2 ) where R 2 = r r 2 J ( r ) / (2 r J ( r )) is a measure of the range of the interaction. (See Appendix below for some more information on this method.) Finally m ( k ) C 1- J (1- R 2 k 2 ) . (5) The last step is to Fourier-transform this back to real space. First rewrite it as G ( k ) C KR 2 k 2 + 1- K = CR- 2 k 2 + tR- 2 = CR- 2 k 2 + - 2 (6) 1 with = Rt- 1 / 2 , and t = ( T- T c...
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