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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XVIII The last lecture had a number of new ideas, so lets quickly review the main ones. The definition of the scaling dimension of a local operator or field like ( x ) at a critical point was through its correlation function: [ ] = if h ( x ) (0) i = 1 x 2 . (1) For the Gaussian model, scaling dimensions are the same as engineering dimensions. In general, the engineering dimension of a field is minus the length dimension it would have to have to make the action (the quantity appearing in the exponent of Z = R e- S ) have zero length dimension. So for the Gaussian model, [ ] = ( d- 2) / 2 in any dimension. Let us now give a derivation of the rescaling equations in the Gaussian model that we conjectured last time, t = b 2 t u = b 4- d u v = b 6- 2 d v h = b d/ 2+1 h, (2) by imitating the steps that we followed to carry out rescaling transformations on the lattice. Our starting point is the partition function Z = Z ( D ) e- R d d x [ 1 2 ( ) 2 + a- 2 t 2 + a d- 4 u 4 + a- d/ 2- 1 h ] (3) We want to construct a transformation that will leave the Gaussian model invariant if t = u = v = h = 0, as that is the fixed point we are attempting to describe. First step : sum over some variables in order to generate an effective action for the remaining variables. The idea of the momentum-shell renormalization group is: starting from a theory with an ultraviolet (short-distance) cutoff a- 1 , integrate out the momentum components with ( ba )- 1 | k | < a- 1 , for some rescaling parameter b > 1. Writing the integration measure in Fourier space, we have Z = Z | k | <a- 1 d ( k ) e- R d d x = Z | k | < ( ba )- 1 d ( k ) Z ( ba )- 1 < | k | <a- 1 d ( k ) e- R d d x ! (4) If has only the two quadratic terms in it, then it has a very nice property: its integral is diagonal in momentum components, so that the integration over higher momentum components does not create any new nontrivial dependences on lower momentum components. (This separation of momentum scales is, we shall see, specific to the Gaussian model.) Assume that we are close to the Gaussian critical point and carry out the integrals over higher momentum components, which gives some overall factor. The part of the action containing nonintegrated momentum components is the same as before, except that the momentum components do not range quite as high as before. Second step : The new problem is not directly comparable to the original problem because the cutoff is different. We 1 would like to have a rescaling map that lives in the space of problems with the same cutoff. We can achieve this by rescaling length units so that the momentum cutoff is restored to a- 1 : looking at the Gaussian part in real space after the integration, we have Z d d x = Z ( ) 2 / 2 + a- 2 t 2 d d x. (5) We want to rescale length and the field so that the spatial cutoff is restored to a from ba...
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