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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XVII This lecture contains some fairly sophisticated notions from field theory, but for the case of the “Gaussian model” (related to mean-field theory) the predictions can be understood in terms of power-counting : the change in a quantity under rescaling can be understood by considering what its units should be. Let us return to Landau theory. So far we understand the consequences of rescaling maps near a critical point, but have few ways to construct such maps. The best way is to work with continuum theories, as introduced in this lecture. We assume that near the critical point, the coarse-grained spin m ( r ) varies only slowly for the lowest-energy configurations. (Note that here m will have dimensionality L (2- d ) / 2 , so it is not a density.) Hence we guess that the “Gibbs free energy” takes the Landau form, ψ ( m ( r ) ,H ( r )) = 1 2 |∇ m ( r ) | 2- m ( r ) h ( r ) + 1 2 r m ( r ) 2 + s m ( r ) 3 + u m ( r ) 4 + .... (1) The meaning of this is that the partition function can effectively be written in terms of an integral over m alone: Z ( T,H ) ≈ C Z ( Dm ( r )) e- R d d r ψ ( m,H ) . (2) This differs slightly from our previous definition in that now β has been absorbed into ψ for convenience below: R d d r ψ should be a dimensionless number. Consider only the quadratic terms in Landau theory, which is called the “Gaussian model.” This theory is exactly solvable even beyond mean-field theory, since we know how to do Gaussian integrals. It is also the starting point for the “interacting” theory obtained by keeping the quartic term. That much harder theory is known as φ 4 theory, as the field m is often written as φ . It is believed that φ 4 theory is in the same universality class as the Ising model, and its critical exponents have been calculated in great detail in a number of different limits. In fact the modern way of calculating Ising model critical exponents, or O ( n ) model exponents in general, is to work with φ 4 theory rather than on a lattice. Our main result in what follows will be an understanding of when the Gaussian model is good enough, and when the full φ 4 theory is necessary. We now introduce a very powerful way of understanding which operators are relevant or ir- relevant in a continuum theory like Landau theory. In particular, we want to ask whether the neglected m 4 ,m 6 ,... terms are relevant or irrelevant at the Gaussian fixed point ( b = 0). This will also motivate the idea of scaling dimension of a field, which is quite important in modern applications of field theory. In this lecture we just present the basic idea without justification; the justification will come in the next lecture....
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