Feynman Path Integral Formulation

292 8 Numerical Studiesthe scaling properties is the renormalization group behavior of the free energy F =− 1 V logZ F(t,{u j })=F reg (t,{u j })+b −d F sing (b y tt,{b y ju j }) , (8.64)where F sing is the singular, non-analytic part of the free energy, and F reg is the regularpart. b is the block size in the RG transformation, while y t and y j ( j ≥ 2) arethe relevant eigenvalues of the RG transformation, and t the reduced temperaturevariable that gives the distance from criticality. One denotes here by y t > 0therelevanteigenvalue, while the remaining eigenvalues y j ≤ 0 are associated with eithermarginal or irrelevant operators. Usually the leading critical exponent yt−1 is calledν, while the next subleading exponent y 2 is denoted −ω. For more details on themethod we have to refer to the comprehensive review in (Cardy, 1988).The correlation length ξ determines the asymptotic decay of correlations, in thesense that one expects for example for the two-point function at large distances< O(x)O(y) > ∼|x−y|≫ξ e−|x−y|/ξ . (8.65)The scaling equation for the correlation length itselfξ (t) =b ξ (b y tt) , (8.66)implies for b = t −1/y tthat ξ ∼ t −ν with a correlation length exponentν = 1/y t . (8.67)Derivatives of the free energy F with respect to t then determine, after setting thescale factor b = t −1/y t, the scaling properties of physical observables, including correctionsto scaling. Thus for example, the second derivative of the free energy withrespect to t yields the specific heat exponent α = 2 − d/y t = 2 − dν,∂ 2∂t 2 F(t,{u j}) ∼ t −(2−dν) . (8.68)In the gravitational case one identifies the scaling field t with k c − k, where k =1/16πG involves the bare Newton’s constant. The appearance of singularities inphysical averages, obtained from appropriate derivatives of F, is rooted in the factthat close to the critical point at t = 0 the correlation length diverges.The above results can be extended to the case of a finite lattice of volume V andlinear dimension L = V 1/d . The volume-dependent free energy is then written asF(t,{u j },L −1 )=F reg (t,{u j })+b −d F sing (b y tt,{b y ju j },b/L) . (8.69)For b = L (a lattice consisting of only one point) one obtains the Finite Size Scaling(FSS) form of the free energy [see for example (Brezin and Zinn-Justin, 1985) for a