Logarithmic corrections to finite-size scaling in the four-state Potts model

The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz equations for equivalent eigenstates of a modifiedXXZ chain. Scaled gaps are found to behave for large chain lengthL asx+dL+0[(lnL)^–1], wherex is the anomalous dimension of the associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudesd are in agreement with predictions of conformai invariance. The implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x andd are found to be extremely difficult even with data available from large lattices,L500.     

Corrections to scaling in the q-state Potts model within Gaussian closure approximation

Corrections to scaling in the q-state Potts model due to departures of the initial condition from scaling morphology are studied at zero temperature in phase-ordering kinetics within Gaussian closure approximation. When the corrections to scaling are included, the equal time correlation function has the form G(r,t) = g₀(r/L)+L^-ωg₁(r/L) + ..., where L is the coarsening length scale. Both the correction-to-scaling exponent ω and the correction-to-scaling function g₁(x) are calculated for different values of q. The correction-to-scaling exponent ω is found to be nontrivial and depends on q. The corrections to scaling are found to be large (relative to scaling function g₀(x) itself) at large scaling variable x.     

Discontinuity of surface tensions in the q-state Potts model

For the ferromagnetic scalar q-state Potts model on a d-dimensional cubic lattice we prove the following results: (1) We derive a correlation inequality and then we prove that the surface tension between two ordered phases exists in dimension d ⩾ 2 whenever q ⩾ 2 and it is discontinuous at the transition point whenever q is large enough. (2) At the limit q↗ ∞ the surface tension between an ordered phase and the disordered one vanishes everywhere except at the transition point.     

The Kadanoff lowerbound renormalization transformation for the q-state Potts model

The Kadanoff lowerbound renormalization transformation when applied to the 2-dimensional q-state Potts model, is found to show a bifurcation phenomenon at q = 4, that might be considered as signalling the onset to the first-order transition. At the value of the free parameter where the bifurcation is found, the specific heat exponent takes almost the value predicted by weak universality $\text{α(4) =}\text{2}\text{3}$, while the cross-over exponent in the Potts-lattice gas direction becomes marginal. The cross-over exponent in the cubic direction is found already to be irrelevant for q > 3.3. Further a duality relation for a class of models obtained by a partial breaking of the Potts symmetry in the hamiltonian, including the cubic model is derived.     

Phase transitions of the q-state Potts model on multiply-laced Sierpinski gaskets

We have investigated stress effects on yellow exciton states in a Cu₂O thin film sandwiched by MgO plates by measuring wavelength modulated (WM) absorption spectra and their thermal variations. In the WM absorption spectra, dispersive spectral structures owing to 2P~4P states in the yellow excitons are clearly resolved in the thin film sample. A stress due to the lattice mismatch between Cu₂O and MgO provides a large red shift of the band gap in the green excitonic system. However, in the yellow excitonic transitions, it is found that a red-shift of the 2P excitonic state is much smaller than that in the green excitonic system. This result suggests that a shallow potential minimum for the yellow excitons is built up in the Cu₂O thin film sandwiched by MgO plates.     

Finite-size scaling for first-order transitions: Potts model

The finite-size scaling algorithm based on bulk and surface renormalization of de Oliveira is tesed onq-state Potts models in dimensionsD=2 and 3. Our Monte Carlo data clearly distinguish between first- and second-order phase transitions. Continuous-q analytic calculations performed for small lattices show a clear tendency of the magnetic exponentY=D-/v to reach a plateau for increasing values ofq, which is consistent with the first-order transition valueY=D. Monte Carlo data confirm this trend.     

Violation of finite-size scaling in three dimensions

We reexamine the range of validity of finite-size scaling in the lattice model and the field theory below four dimensions. We show that general renormalization-group arguments based on the renormalizability of the theory do not rule out the possibility of a violation of finite-size scaling due to a finite lattice constant and a finite cutoff. For a confined geometry of linear size L with periodic boundary conditions we analyze the approach towards bulk critical behavior as at fixed for where is the bulk correlation length. We show that for this analysis ordinary renormalized perturbation theory is sufficient. On the basis of one-loop results and of exact results in the spherical limit we find that finite-size scaling is violated for both the lattice model and the field theory in the region . The non-scaling effects in the field theory and in the lattice model differ significantly from each other. Received 5 February 1999     

On finite-size scaling of the order parameter in percolation

Finite-size scaling of the order parameter O in the disordered phase is presented for percolation in 2d and 3d and compared to the behavior of the magnetization of the Ising model. We find that the order parameter O grows logarithmically with the system size L for p < p_c and thus the analogy with the Ising model is not complete. Furthermore we find evidence for scaling behavior of the type O ～ L^?y/v (p_c ? p)^β?y where the exponent y is not dv/2 as in the Is ing case.     

Decreasing of the finite-size effects through the twisted boundary conditions II

In the large-N limit it is shown that a model with twisted boundary conditions becomes equivalent to the U(N) invariant theory which has a volume N² times larger than the theory with periodic boundary conditions. Even for finite N, it is confirmed that the finite-size effects in the models with twisted boundary conditions rather decrease, compared with the ones with periodic boundary conditions, by performing a Monte Carlo simulation for the two-dimensional SU(3) chiral models.     

The four-dimensional site-diluted Ising model: A finite-size scaling study

Using finite-size scaling techniques, we study the critical properties of the site-diluted Ising model in four dimensions. We carry out a high-statistics Monte Carlo simulation for several values of the dilution. The results support the perturbative scenario: there is only the Ising fixed point with large logarithmic scaling corrections. We obtain, using the Perturbative Renormalization Group, functional forms for the scaling of several observables that are in agreement with the numerical data.     

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 q ⩽ 4. 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 q → 1 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.     

On finite-size scaling in the presence of dangerous irrelevant variables

A scaling hypothesis on finite-size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long-range interaction and general geometryL ^d–d× ^d . A characteristic length which obeys a universal finite-size scaling relation is defined. The general conjectures are based on exact results for the mean spherical model with inverse power law interaction.