Investigation of the critical behavior of the three-dimensional weakly diluted Potts model

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v.     

Phase transition properties of three-dimensional systems described by diluted potts model

Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.     

Mean-field approximation for the potts model of a diluted magnet in the external field

The Potts model of a diluted magnet with an arbitrary number of states placed in the external field has been considered. Phase transitions of this model have been studied in the mean-field approximation, the dependence of the critical temperature on the external field and the density of magnetic atoms has been found, and the magnetic susceptibility has been calculated. An improved mean-field technique has been proposed, which provides more accurate account of the effects associated with nonmagnetic dilution. The influence of dilution on the first-order phase transition curve and the magnetization jump at the phase transition has been studied by this technique.     

Histogram data analysis for a three-dimensional diluted ferromagnetic 3- and 4-state potts models

On the basis of a histogram data analysis, phase transitions (PTs) in a three-dimensional diluted ferromagnetic 3- and 4-state Potts models are investigated. Systems with linear dimensions of L = 20–60 and spin concentrations of p = 1.00, 0.95, and 0.65 are studied. It is shown that the introduction of weak disorder (p ～ 0.95) into the system in the three-dimensional Potts model with q = 3 is sufficient to change a first-order phase transition to a second-order one, whereas, in the three-dimensional Potts model with q = 4, the change of a first-order phase transition to a second-order one occurs only in the presence of strong disorder (p ～ 0.65).     

T>0 thermal properties of the generalized three-state potts model

A generalized three-state Potts model is proposed, the decimation renormalization-group method (DRG) is applied to the hamiltonian version of this three-state generalized Potts model, and T>0 thermal properties have been calculated. We find a nontrivial unstable fixed line in the finite temperature case, when T approaches zero it agrees with the result by using the T = 0 block renormalization-group method (BRG). The possibility of extending both the reliability of range of the DRG method is also mentioned.     

Interface properties of the three-state potts model in two dimensions

Interface properties, in particular the interface free energy and the interface profile of the three-state Potts model in two dimensions are studied using Monte Carlo techniques and a generalized version of the method of Müller-Hartmann and Zittartz. The role of the third state in characterizing the interface between the two other states is elucidated.     

Renormalization group approach to the surface and defect critical behavior in the potts model

A simple real-space renormalization group method with two-terminal clusters is used to treat the critical behavior of Potts ferromagnet with free surface and defect plane on the same footing both for square and cubic lattices. For a square lattice, quite different critical behaviors are found for the cases of line defect and free surface. Whenq is larger than three, like the case ofa line type defect in ‘diamond’ hierarchical lattice, the order parameter on a defect line increases discontinuously at the bulk critical point if the defect interaction is sufficiently strong. This behavior, on the contrary, does not occur on the surface of a semi-infinite plane. For a cubic lattice, the phase diagram and renormalization group flow properties are obtained explicitly for bothq=1 (bond percolation) andq=2 (Ising model). In both cases, our calculations whow that the critical behavior on the surface of a semi-infinite system belongs to a different universality class from the critical behavior on the defect plane of a bulk system.     

Ground-state properties of the antiferromagnetic potts model in an external field

The ground-state properties of the antiferromagnetic q-state Potts model in an external field are studied. At the upper critical field this model may be mapped onto a hard-core lattice gas with activity z = q ?1. This allows us to get some exact results for the triangular lattice on which the corresponding hard hexagon problem has been recently solved by Baxter.     

Large-q asymptotics of the random-bond potts model

We numerically examine the large-q asymptotics of the q-state random bond Potts model. Special attention is paid to the parametrization of the critical line, which is determined by combining the loop representation of the transfer matrix with Zamolodchikov's c-theorem. Asymptotically the central charge seems to behave like c(q)=1 / 2 log(2)(q)+O(1). Very accurate values of the bulk magnetic exponent x(1) are then extracted by performing Monte Carlo simulations directly at the critical point. As q-->infinity, these seem to tend to a nontrivial limit, x(1)-->0.192+/-0.002.     

Critical exponents of the three-state potts model

The Fernandez-Pacheco duality invariant renormalization group is applied to the hamiltonian version of the two-dimensional three-state Potts model. The fixed point is located at exactly the self-dual critical point K¹ = 1. The thermal exponent is calculated to be y_T=1.1814. This value is in excellent agreement with the low temperature series expansion result of Zwanzig and Ranshaw (y_T = 1.174) and the strong coupling expansion result of Elitzur, Pearson and Shigemitsu (y_T = 1.190). It also seems to lend strong support to den Nijs' recent conjecture that the exact value should be y_T = 6/5.     

New critical frontiers for the potts and percolation models

We obtain the critical threshold for a host of Potts and percolation models on lattices having a structure which permits a duality consideration. The consideration generalizes the recently obtained thresholds of Scullard and Ziff for bond and site percolation on the martini and related lattices to the Potts model and to other lattices.     

Features of the phase transitions in the three-dimensional diluted potts model with number of spin states <Emphasis Type="Italic">q</Emphasis> = 3

The effect of frozen disorder, implemented in the form of nonmagnetic impurities, on phase transitions in the three-dimensional Potts model with number of states q = 3 has was investigated by the Monte Carlo method, using the Wolf single-cluster algorithm. Systems with linear sizes L = 20–44 were considered at spin concentrations p = 1.00–0.65. The method of fourth-order Binder cumulants was used to demonstrate that a first-order phase transition is observed for the pure model (p = 1.00) and a second-order phase transition occurs at concentrations p = 0.90, 0.80, 0.70, and 0.65.     

Analyticity properties of the weakly coupled double-well model

In this note we study lattice Φ⁴-models with Hamiltonian $$H = \tfrac{1}{2}(\varphi , - \Delta \varphi ) + \lambda \Sigma \left( {\varphi _i^2 - \frac{{m^2 }}{{8\lambda }}} \right)^2$$ and Gaussian boundary conditions. Using the polymer expansion we obtain analyticity of the pressure and the correlation functions in the infinite volume limit in a region $$\left\{ {\left. \lambda \right| \left| \lambda \right|< \varepsilon ,\left| {arg } \right.\left. \lambda \right|< \frac{\pi }{2} - \delta } \right\}$$ for every δ>0.     

Some AB-percolation problems in the antiferromagnetic potts model

The AB-correlated-site/random-bond percolation problem in a q-state antiferromagnetic Potts model on Bethe lattices is solved. We find the analytic expression of the AB-percolation characteristic functions in terms of the temperature, the external field and the active bond concentration p_B. The AB-threshold and the phase boundary of the system coincide at zero temperature and at most in two other points for every constant p_B > 1?σ. The properties of the Bethe lattice allow us to find the temperature dependent p_B which defines the AB-droplets, i.e. those special AB-clusters which diverge with thermal exponents along the phase boundary.     

Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex ^N y ^N+x ^N+y ^N+1/k ²=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.     

The dynamic critical exponent of the three-dimensional Ising model

We measure the dynamic exponent of the three-dimensional Ising model using a damage spreading Monte Carlo approach as described by MacIsaac and Jan. We simulate systems fromL=5 toL=60 at the critical temperature,T _c =4.5115. We report a dynamic exponent,z=2.35±0.05, a value much larger than the consensus value of 2.02, whereas if we assume logarithmic corrections, we find thatz=2.05±0.05.     

Electrophysical properties of the rayleigh three-dimensional model

A new approach to solving the problem of conductivity of composites with a regular structure is suggested that provides a fundamental possibility of analyzing media with arbitrarily shaped inclusions. The conductivity, its derivatives with respect to both arguments, and the two-parameter function entering into the expression for the effective Hall constant are considered using the Rayleigh three-dimensional model, a periodic system with spherical inclusions, as an example. The results of the numerical analysis of general equations obtained in this way are represented in graphic form.