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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XIII The first part of this lecture explains further the behavior of correlation functions near a second- order critical point. The second part returns to the 1D Ising model in order to understand how rescaling transformations tell us something about this solvable model. Then we will discuss the approximate behavior of the Ising model in higher dimensions, where there is a phase transition at some nonzero temperature T c between an ordered state and a disordered one. The concept of “anomalous dimensions” is one of the more fundamental in the field of statistical mechanics and field theory. Simply put, it means that quantities near a critical point, where the correlation length is very large, can depend in a rather nontrivial way on both the correlation length ξ and the lattice spacing or short-distance cutoff a . The term “anomalous” refers to the breakdown of simple dimensional analysis. As an example, consider the correlation function of spins at the critical point of the Ising model in three dimensions. One might guess that the spin-spin correlator should be independent of microscopic details in some sense, but taking this literally will be shown to lead to mean- field theory, which is quantitatively wrong for this transition. Really a fundamental breakthrough in the understanding of field theory and quantum phase transitions was in understanding how anomalous dimensions, i.e., violations of the scaling predicted by mean field theory, occur for almost all interesting problems. We found in mean-field theory that the correlation fell off as r- ( d- 2) up to the correlation length, then fell off exponentially. This happened because the Fourier transform of the correlation, for ξ = ∞ , went as 1 /k 2 , which in d dimensions corresponds to the Coulomb potential. That is, the correlations satisfy the homogeneous Laplacian equation ∇ 2 m = 0 at criticality, except for a source at the origin. Hence G ( r ) falls off logarithmically in 2D and as 1 /r in 3D. The correlation function is related to the susceptibility: χ ∼ R d d rG ( r ), so χ ∼ ξ 2 in MFT. This accords with our naive expectation that χ should depend only on the diverging length scale ξ . However, this is wrong for the 3D Ising model. In fact, χ does have a dependence on the short length scale a : χ ∼ a η ξ 2- η . (1) This η is an example of an anomalous dimension: the mean-field guess is incorrect and there is in some sense a dependence on the short-distance physics, but this dependence can still be universal ( η is equal for all problems in the same universality class). Understanding how this occurs is one of the major contributions of the RG to statistical physics and field theory....
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