Feynman Path Integral Formulation

8.8 Numerical Determination of the Scaling Exponents 293field-theoretic justification]. After taking derivatives with respect to the fields t and{u j }, the FSS scaling form for physical observables follows,O(L,t) =L x O/ν [ ˜ f O (Lt ν )+O(L −ω ) ] , (8.70)where x O is the scaling dimension of the operator O, and f˜O (x) an arbitrary function.As an example, consider the average curvature R. From Eq. (8.70), with t ∼k c − k and x O = 1 − 4ν, one has[ (R(k,L) =L −(4−1/ν) R˜(k c − k) L 1/ν) ]+ O(L −ω ) , (8.71)where ω > 0 is a correction-to-scaling exponent. If scaling involving k and L holdsaccording to Eq. (8.70), then all points should lie on the same universal curve.Fig. 8.8 shows a graph of the scaled curvature R(k) L 4−1/ν for different valuesof L = 4,8,16, versus the scaled coupling (k c − k)L 1/ν . If scaling involving k andL holds according to Eq. (8.71), with x O = 1 − 4ν the scaling dimension for thecurvature, then all points should lie on the same universal curve. The data is in goodagreement with such behavior, and provides a further test on the exponent, whichseems consistent within errors with ν = 1/3.Fig. 8.8 Finite size scaling behavior of the scaled curvature versus the scaled coupling. Here L = 4for the lattice with 4 4 sites (✷), L = 8 for a lattice with 8 4 sites (△), and L = 16 for the latticewith 16 4 sites (◦). Statistical errors are comparable to the size of the dots. The continuous linerepresents a best fit of the form a + bx c . Finite size scaling predicts that all points should lie on thesame universal curve. At k c = 0.0637 the scaling plot gives the value ν = 0.333.As a second example consider the curvature fluctuation χ R . From the generalEq. (8.70) one expects in this case, for t ∼ k c − k and x O = 2 − 4ν,