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.     

Integrable aspects of the scaling q-state Potts models II: finite-size effects

We continue our discussion of the q-state Potts models for q⩽4, in the scaling regimes close to their critical and tricritical points. In a previous paper, the spectrum and full S-matrix of the models on an infinite line were elucidated; here, we consider finite-size behaviour. TBA equations are proposed for all cases related to φ₂₁ and φ₁₂ perturbations of unitary minimal models. These are subjected to a variety of checks in the ultraviolet and infrared limits, and compared with results from a recently-proposed non-linear integral equation. A non-linear integral equation is also used to study the flows from tricritical to critical models, over the full range of q. Our results should also be of relevance to the study of the off-critical dilute A models in regimes 1 and 2.     

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.     

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.     

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q=2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen-Shannon divergence, the relative entropy of coherence, and the l₁ norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.     

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.     

First-order phase transitions and integrable field theory. The dilute q-state Potts model

We consider the two-dimensional dilute q-state Potts model on its first-order phase transition surface for 0 < q ⩽ 4. After determining the exact scattering theory which describes the scaling limit, we compute the two-kink form factors of the dilution, thermal and spin operators. They provide an approximation for the correlation functions whose accuracy is illustrated by evaluating the central charge and the scaling dimensions along the tricritical line.     

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.     

Corrections to finite-size-scaling in two dimensional Potts models

High resolution Monte Carlo simulations are used to examine the finite size behavior of Q-state Potts models in two dimensions. For Q = 3 we find that at the critical point bulk properties are subject to much larger corrections to finite size scaling than were previously realized. For Q = 4 we find that corrections to finite size scaling are subtle and that the multiplicative logarithmic correction is insufficient to correct the dominant terms.     

Free energy of the chiral Potts model in the scaling region

We explicitly calculate the free energy of the general solvableN-state chiral Potts model in the scaling region, forT<T _c . We do this from both of the two available results for the free energy, and verify that they are mutually consistent. Ift=T _c –T, then we find that - _c /t has a Taylor expansion in powers oft ^2/N (together with higher-order non-scaling terms of ordert, ort logt).     

Corrections to scaling for period doubling

We have studied a multiple scaling which describes corrections to scaling. For the period doubling in one-dimensional dissipative maps, two-dimensional areapreserving maps, and four-dimensional symplectic maps, the multiple scaling is seen to be well-obeyed, and new scaling factors have been found. The multiple scaling is also seen to be a very powerful tool for searching for scaling behavior.     

Corrections to scaling for branched polymers

We extend exact series enumerations for the mean square radius of gyration for lattice animals up to sizeN=19 on the square andN=12 on the simple cubic lattices. We have reanalysized the corrections to scaling exponents using the new data and found =0.5±0.05 and =0.45±0.1 in 2 and 3 dimensions respectively.     

Finite-size scaling of the three-state Potts model on a simple cubic lattice

Finite-size scaling is studied for the three-state Potts model on a simple cubic lattice. We show that the specific heat and the magnetic susceptibility scale accurately as the volume. The correlation length exhibits behaviors expected for a genuine first-order transition; the one extracted from the unsubtracted correlation function shows a characteristic finite-size behavior, whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point.     

Potts q-color field theory and scaling random cluster model

We study structural properties of the q-color Potts field theory which, for real values of q, describes the scaling limit of the random cluster model. We show that the number of independent n-point Potts spin correlators coincides with that of independent n-point cluster connectivities and is given by generalized Bell numbers. Only a subset of these spin correlators enters the determination of the Potts magnetic properties for q integer. The structure of the operator product expansion of the spin fields for generic q is also identified. For the two-dimensional case, we analyze the duality relation between spin and kink field correlators, both for the bulk and boundary cases, obtaining in particular a sum rule for the kink-kink elastic scattering amplitudes.     

Non-Gaussian scaling limits. Hierarchical model approximation

We extend a previous analysis of non-Gaussian fixed points of the block spin renormalization group in hierarchical models to a long distance analysis of the correlation functions.On leave from Department of Mathematical Methods of Physics, University, Warsaw.