We consider the exact correlation length calculations for the two-dimensional Potts model at the transition point βt by Klümper, Schadschneider and Zittartz, and by Buffenoir and Wallon. We argue that the correlation length calculated by the latter authors is the correlation length in the disordered phase and then combine their result with duality and the assumption of complete wetting to give an explicit formula for the order-disorder interface tension σod of this model. The result is used to clarify a controversy stemming from different numerical simulations of σod.
We consider the exact correlation length calculations for the two-dimensional Potts model at the transition point βt by Klümper, Schadschneider and Zittartz, and by Buffenoir and Wallon. We argue that the correlation length calculated by the latter authors is the correlation length in the disordered phase and then combine their result with duality and the assumption of complete wetting to give an explicit formula for the order-disorder interface tension σod of this model. The result is used to clarify a controversy stemming from different numerical simulations of σod.
We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
We report numerical simulations of two-dimensional q-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function Gdiam(x), which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter x, ${G}^{\mathrm{diam}}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}x/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with q=10, 15, and 20 on large square lattices of size $300\ifmmode\times\else\texttimes\fi{}300$, $120\ifmmode\times\else\texttimes\fi{}120$, and $80\ifmmode\times\else\texttimes\fi{}80$, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ${\ensuremath{\xi}}_{d}({\ensuremath{\beta}}_{t})$ in the disordered phase at the first-order transition point ${\ensuremath{\beta}}_{t}$ with an accuracy of about 1--2 for all considered values of q. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison.
The surface and bulk properties of the two-dimensional Q>4 state Potts model in the vicinity of the first-order bulk transition point have been studied by exact calculations and by density-matrix renormalization-group techniques. For the surface transition the complete analytical solution of the problem is presented in the $\stackrel{\ensuremath{\rightarrow}}{Q}\ensuremath{\infty}$ limit, including the critical and tricritical exponents, magnetization profiles, and scaling functions. We have shown in an accurate numerical study that the universality class of the surface transition is independent of the value of Q>4. For the bulk transition we have numerically calculated the latent heat and the magnetization discontinuity and we have shown that the correlation lengths in the ordered and disordered phases are identical at the transition point.
We calculate the large-q expansion of the second moment correlation length at the first order phase transition point of the q-state Potts model in two dimensions both in the ordered and disordered phases to order 21 in 1/q. They coincide with each other to the third term of the series but differ a little in higher orders. Numerically the ratio of the second moment correlation length in the two phases is not far from unity in all region of q>4. The ratio of the second moment correlation length to the standard correlation length in the disordered phase is far from unity, which suggests that the second largest and smaller eigenvalues of the transfer matrix form a continuum spectrum not only in the large-q region but also in all the region of q>4.
We study the interface tension of the 4-state Potts model in three dimensions using the Wang- Landau algorithm. The interface tension is given by the ratio of the partition function with a twisted boundary condition in one direction and periodic boundary conditions in all other directions over the partition function with periodic boundary conditions in all directions. With the Wang-Landau algorithm we can explicitly calculate both partition functions and obtain the result for all temperatures. We find solid numerical evidence for perfect wetting. Our algorithm is tested by calculating thermodynamic quantities at the phase transition point.