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phys212ln12 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture XII The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is fixed up, m ( k ) C 1 - βJ (1 - R 2 k 2 ) . (1) Its Fourier transform is m ( r ), which we argued above to be equal to G ( r ), the correlation function between two spins separated by r . First rewrite it as G ( k ) C KR 2 k 2 + 1 - K = CR - 2 k 2 + tR - 2 = CR - 2 k 2 + ξ - 2 (2) with ξ = Rt - 1 / 2 , and t = ( T - T c ) /T . The result ξ = Rt - 1 / 2 defines the critical exponent ν = 1 2 for the growth of the correlation length near criticality. The Fourier transform described above, which in general dimensionality can be written in terms of Bessel functions, gives that the correlation function on long length scales falls off as (now forgetting about units; additional prefactors of C and the lattice spacing R must be added to get the units right) G ( r ) e - r/ξ /r ( d - 1) / 2 (3) starting from G ( r ) CR - 2 d d k exp( ikx ) k 2 + ξ - 2 = CR - 2 ξ 2 - d d d k exp( ikx/ξ ) k 2 + 1 . (4) Actually G ( r ) has two different asymptotic limits in MFT: for large r G ( r ) e - r/ξ r ( d - 1) / 2 if r ξ 1 r d - 2 if r ξ . (5) (Huang describes this Fourier transform in one problem but has a wrong power in the first limit.) Note that the first limit in d = 1 matches our result for the correlation function of the one- dimensional Ising chain, in the limit of large separation. With this information about correlations, we are able to justify our earlier claim about the validity of mean-field theory above four dimensions. The approach will be to estimate the neglected part of the Hamiltonian and compare it to the part we found in the mean-field calculation. This procedure is carried out for lattices in the book of Cardy; instead of working with the lattice mean field theory, we now will introduce a phenomenological approach to phase transitions in the continuum known as “Landau theory.” The continuum approach to phase transitions is a rare example where phenomenology can give exactly correct answers even for certain numerical quantities, such as critical exponents. For now, think of this phenomenology as motivated by a desire to describe continuum systems such as the liquid-gas transition in water, which was claimed in Lecture I to share some properties, including critical exponents, with the (lattice) Ising model. 1
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We start off by following Landau and conjecturing that the free energy density near a phase transition can be expanded in powers of the “order parameter” describing the transition. For the Ising transition, w choose the magnetization to be the order parameter: the requirement is that the symmetry that is broken at the transition should transform the order parameter from one ordered state to another. In terms of the dimensionless magnetization, we can write this expansion as F ( m ) = - m H kT + 1 2 r 0 m 2 + s 0 m 3 + u 0 m 4 + . . . . (6) Remark : We will later on want to consider spatially varying configurations and also fluctua- tions: let me quickly introduce a different way of looking at the Landau free energy. We assume
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