Universal amplitude ratios in the two-dimensional
q-state Potts model and percolation from quantum field theory
We consider the scaling limit of the two-dimensional
q-state Potts model for
qleq4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit
qto1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.
The amplitude ratio of the susceptibility (or second size moment) for two-dimensional percolation is calculated by two series methods and also by Monte Carlo simulation. The first series method is an approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value Gamma-/Gamma+ = 162.5+/-2 , a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.
Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.
The hypothesis of universality implies that for every scaling relation among critical exponents there exists a universal ratio among the corresponding critical amplitudes. If one writes $B{|t|}^{\ensuremath{\beta}}$, ${{A}_{F}}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{2\ensuremath{-}\ensuremath{\alpha}}$, ${C}^{\ifmmode\pm\else\textpm\fi{}}{|t|}^{\ensuremath{-}\ensuremath{\gamma}}$, and ${\ensuremath{\xi}}_{0}{|t|}^{\ensuremath{-}\ensuremath{\nu}}$ [where $t=\frac{({p}_{c}\ensuremath{-}p)}{{p}_{c}}$,
p being the concentration of nonzero bonds, and +(-) stands for
p<pc (
p>pc)] for the leading singular terms in the probability to belong to the infinite cluster, the mean number of clusters, the clusters' mean-square size, and the pair connectedness correlation length, then it is shown that the ratios $\frac{{{A}_{F}}^{+}}{{{A}_{F}}^{\ensuremath{-}}}$, $\frac{{C}^{+}}{{C}^{\ensuremath{-}}}$, ${{A}_{F}}^{+}{B}^{\ensuremath{-}2}{C}^{+}$, $\frac{{\ensuremath{\xi}}_{0}^{+}}{{\ensuremath{\xi}}_{0}^{\ensuremath{-}}}$, and ${{A}_{F}}^{+}{({\ensuremath{\xi}}_{0}^{+})}^{d}$ (
d is the dimensionality) are universal. Similar quantities are found for the behavior at
p=pc (as a function of a "ghost" field). All of these universal ratios are derived from a universal scaled equation of state, which is calculated to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$. The (extrapolated) results are compared with available information in dimensionalities
d=2,3,4,5, with reasonable agreements. The amplitude relations become exact at
d=6, when logarithmic corrections appear. Additional universal ratios are obtained for the confluent correction to scaling terms.
The universal amplitude ratio
tildeRxi for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here.