p212p6 - Physics 212 Statistical mechanics II Fall 2006...

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Problem set 6 : due Tuesday, 12/5/06 NO CREDIT can be given if turned in after Thursday 12/7/06 1. Find the exponent α for random walks in two dimensions on the square lattice. In the notes we showed α = 5 / 2 in one dimension. This exponent gives the number of walks that return to the origin: q N = μ N N α- 3 . (1) 2. Calculate the correlation function of the operator cos( pθ ) in the Gaussian model in two dimensions, where the partition function is Z = Z Dθ ( r ) e- K 2 R ( ∇ θ ) 2 d 2 r . (2) Hint: Start from the fact shown in class (in Lecture 21 on the course webpage) that the correlations of θ are logarithmic since the Coulomb potential in 2D is logarithmic, with a coefficient in front of the log determined by K : ˜ G ( r ) ∼ Z a- 1 dk k =- log( r/a ) 2 πK + ..., (3) Next prove the identity that h e ipθ (0) e- ipθ ( r ) i = e p 2 ( G (0)- G ( r )) . (4) What is the scaling dimension [cos( pθ )] as a function of p , at the Gaussian fixed point? At what value K ( p ) does the operator R cos( pθ ) d 2 x become relevant at the Gaussian fixed point? Note that p = 1 corresponds to an ordinary magnetic field in the XY system.= 1 corresponds to an ordinary magnetic field in the XY system....
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This note was uploaded on 08/01/2008 for the course PHYSICS 212 taught by Professor Moore during the Fall '06 term at Berkeley.

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p212p6 - Physics 212 Statistical mechanics II Fall 2006...

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