We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.
We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.
We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.
We investigate the two-dimensional q=3 and 4 Potts models with a variable interaction range by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges as expressed by the number z of equivalent neighbors. For not-too-large z, the transitions fit well in the universality classes of the short-range Potts models. However, at longer ranges, the transitions become discontinuous. For q=3 we locate a tricritical point separating the continuous and discontinuous transitions near z=80, and a critical fixed point between z=8 and 12. For q=4 the transition becomes discontinuous for z>16. The scaling behavior of the q=4 model with z=16 approximates that of the q=4 merged critical-tricritical fixed point predicted by the renormalization scenario.
The fractal dimensions of the largest clusters of like spins connected by nearest-neighbor bonds in the critical two-dimensional Q-state Potts model have been determined via Monte Carlo simulations to be df=1.946(3), 1.922(5), and 1.909(8) for Q=2, 3, and 4, respectively. These dimensions are close to the magnetic tricritical exponents of the (Q-1)-state dilute Potts model (d-x=187/96=1.947..., 77/40=1.925, and 40/21=1.904... for Q-1=1, 2, and 3, respectively). The critical Q-state model and the tricritical (Q-1)-state model are related to a two-component Potts lattice gas; this model generates percolative susceptibilities of critical Potts clusters, and suggests a renormalization-group argument for df=d-x.
We study bond percolation in two dimensions between random site variables having one out of q colors, using transfer-matrix and Monte Carlo techniques. We determine the percolation threshold as a function of the Potts temperature T in the disordered Potts range ${T}_{c}\ensuremath{\leqslant}T<\ensuremath{\infty}$ for several q-state Potts Hamiltonians. For high T, these transitions fit, irrespective of q, in the universality class of the ordinary percolation transitions. However, for $T\ensuremath{\downarrow}{T}_{c}$, q-dependent crossover phenomena appear. The topology of the phase diagram changes in a qualitative sense at q=2. For q<2 the Potts critical state appears to enhance percolation, for q>2 it appears to suppress it. Remarkably, for q=2 the percolation line coincides with the only flow line extending to T>Tc from the critical fixed point associated with Potts clusters.
The invaded cluster approach is extended to the two-dimensional Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of the vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of the "geometrical disorder cluster." The location of the tricritical point and the concentration of vacancies for q=1,2,3 are found to be in good agreement with the best known results. Scaling properties of the percolating scaling cluster and related critical exponents are also presented.
The spin-1 Ising model on the square lattice with nearest-neighbor ferromagnetic exchange interactions [both bilinear (J) and biquadratic (K) and crystal-field interaction ($\ensuremath{\Delta}$) is studied via a renormalization-group transformation in position space. The phase diagram in J, K, $\ensuremath{\Delta}$ space is found to have one surface of critical phase transitions and two surfaces of first-order phase transitions. These surfaces are variously bounded by an ordinary tricritical line, an isolated critical line, and a line of critical end points. These three lines joint at a special tricritical point corresponding to the transition of the three-state Potts model. The over-all phase diagram is qualitatively similar to that obtained with the mean-field approximation, except in the vicinity of the Potts transition where a four-phase coexistence line in mean-field theory shrinks into a special tricritical point in renormalization-group theory. Symmetry considerations guide the construction of our truncated renormalization-group transformation. The global connectivity and local exponents of the thirteen separate fixed points underlying this quite complicated structure are determined. Local analysis with respect to magnetic field (H) and another odd interaction (L) is performed. A one-adjusted-parameter version of our transformation yields remarkably quantitative results, predicting the Potts transition temperature, for example, within 0.3% of the exact value.