Showing papers on "Potts model published in 1983"
TL;DR: In this article, an Ising and aq-state Potts model on a diamond hierarchical lattice is considered and the distribution of zeros of the partition function in the complex plane of temperatures for several choices of q.
Abstract: We consider an Ising and aq-state Potts model on a diamond hierarchical lattice. We give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq. These zeros are just the Julia set corresponding to the renormalization group transformation.
179 citations
165 citations
TL;DR: In this paper, a one-dimensional quantum model with three states per site is considered and its ground state shows several commensurate-incommensurate transitions analogous to ones previously studied in two-dimensional statistical mechanics.
148 citations
TL;DR: The extended scaling relations for the leading and next-to-leading magnetic critical exponents of the two-dimensional Potts model are derived from the spin-spin correlation function in the Coulomb-gas representation as discussed by the authors.
Abstract: The extended scaling relations for the leading and next-to-leading magnetic critical exponents of the two-dimensional Potts model are derived from the spin-spin correlation function in the Coulomb-gas representation.
116 citations
TL;DR: In this paper, the simulation results of simulations sur ordinateur de la cinetique d'un modele de Potts a etat Q ferromagnetique are presented.
Abstract: Resultats de simulations sur ordinateur de la cinetique d'un modele de Potts a etat Q ferromagnetique
91 citations
TL;DR: In this article, the authors studied the phase structure of the q = 3, d = 3 Potts model with external fields and showed that the confinement/deconfinement phase transition is first order, but highly sensitive to external fields.
80 citations
TL;DR: In this paper, a disordered and frustrated Potts model is described which exhibits a number of curious features, such as symmetry breaking effects, spin glass transition, and infinite-range model is always ferromagnetic.
Abstract: A disordered and frustrated Potts model is described which exhibits a number of curious features. For the m-state model the authors find a spin glass transition which within replica symmetric theory is continuous for m 6. Replica symmetry breaking effects are much stronger than for conventional systems and the conventional Parisi ansatz fails for m>or=4. At low temperatures an infinite-range model is always finally ferromagnetic.
72 citations
TL;DR: In this article, the q-state ferromagnetic Potts model (FPM) and antiferromagnetic potts model were solved on Bethe lattices for all values of the external magnetic field and temperature, and exact expressions of all thermodynamic functions of interest in the FPM and APM were calculated.
Abstract: The q-state ferromagnetic Potts model (FPM) and antiferromagnetic Potts model (APM) are solved on Bethe lattices for all values of the external magnetic field and temperature. The exact expressions of all thermodynamic functions of interest in the FPM and APM are calculated. The authors find the complete phase diagrams for both systems. In the FPM there are first-order phase transitions at the critical point for every q>2. In the APM they find second-order phase transitions along a critical line for every q>or=2.
70 citations
70 citations
TL;DR: The approach to equilibrium of ordered superlattice structures on surfaces is analyzed in this paper, where both Monte Carlo methods and analytic models of interacting domain walls are used to study the kinetics of quenched, two-dimensional systems with $Q$ degenerate equilibrium states.
Abstract: The approach to equilibrium of ordered superlattice structures on surfaces is analyzed. Both Monte Carlo methods as well as analytic models of interacting domain walls are used to study the kinetics of quenched, two-dimensional systems with $Q$ degenerate equilibrium states. In particular, the growth of long-range order in the magnetic analogs ($Q$-component Potts models) of the commensurate superlattice structures found on surfaces is analyzed in a study of the kinetics of typical domain geometries that are found when systems are quenched from high (disordered state) to low temperatures. Calculations and simulations of the time and temperature dependence of the domain sizes for the model system of isolated domains indicate that for two dimensions roughening fluctuations of the domain walls strongly influence the grain growth kinetics. Strip-shaped domains approach equilibrium almost entirely through these fluctuations, while circular domains shrink due to deterministic curvature, but with a rate that is strongly temperature dependent even for temperatures outside the critical region. Pinning effects in systems with $Q\ensuremath{\ge}3$ lead to extremely slow kinetics for particular domain geometries. Low-temperature quenches on a triangular lattice indicate that these pinned geometries are rarely nucleated, while quenches on a square lattice equilibrate much more slowly due to pinning. The application of the theory to superlattice grain growth on surfaces is discussed in the following paper which presents results for the kinetics of quenched systems in terms of the simple domain geometries examined here.
69 citations
TL;DR: In this article, the melting transition of the two-dimensional, three-state, asymmetric or chiral clock model is examined, and the chiral symmetry-breaking operator is relevant at the symmetric (or pure Potts) critical point with a crossover exponent of o ≈ 0.2.
Abstract: The melting transition of the two-dimensional, three-state, asymmetric or chiral clock model is examined. Evidence from scaling arguments and analysis of perturbation series is presented, indicating that the chiral symmetry-breaking operator is relevant at the symmetric (or pure Potts) critical point with a crossover exponent of o ≈ 0.2. The remainder of the commensurate-disordered phase boundary therefore appears to be in a new universality class, distinct from the pure three-state Potts transition. An interfacial wetting transition that plays an important role in the crossover between the two types of critical behavior is discussed. The location and exponents of this wetting transition are obtained both in a low-temperature limit using generating function techniques and in a systematic low-temperature expansion of the transfer matrix.
TL;DR: In this article, the authors present some analytical results for the Potts model, using the symmetry group generated by the inverse relation and other symmetries of this model, and find the critical manifolds and study the relationship between this group and the Lee-Yang singularities in the complex plane.
Abstract: The authors present some analytical results for the Potts model, using the symmetry group (acting on its parameters) generated by the inverse relation and other symmetries of this model. In particular, they find the critical manifolds and study the relationship between this group and the Lee-Yang singularities in the complex plane. For those cases where the symmetry group is finite, they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials.
TL;DR: In this paper, the authors generalized the Ising model of a spin-glass to the Potts model, which possesses a frustration and a guage symmetry similar to those of the random Ising models.
Abstract: The Ising model of a spin-glass is generalized to the Potts model. Our model possesses a frustration and a guage symmetry similar to those of the random Ising model. The gauge symmetry enables us to obtain exact results for the internal energy, specific heat, and correlation functions. The infinite-range version of this model is discussed at the same level of accuracy as the Sherrington-Kirkpatrick solution of the Ising spin-glass and the phase diagram is obtained. The model is solved exactly on the Bethe lattice and a nontrivial phase diagram is presented.
TL;DR: In this article, the s-state Potts model with a gaussian distribution of pair couplings is found to possess, in the infinite-range limit, a replica symmetric (RS) solution that exhibits a spin glass transition which appears to be second order for s 2.
Abstract: The s-state Potts model with a gaussian distribution of pair couplings is found to possess, in the infinite-range limit, a replica symmetric (RS) solution that exhibits a spin glass transition which appears to be second order for s 2, the stability of the PM phase breaks down above the transition temperature indicated by the RS solution suggesting a first-order transition which, however, cannot be located by the RS solution.
TL;DR: For a Potts spin glass model with Sherrington-Kirkpatrick exchange disorder, the full phase structure in (J0, J, h, T) is indicated as mentioned in this paper, which is analogous to those of a Heisenberg spin glass but there are important differences of transition orders and regions of stability.
Abstract: For a Potts spin glass model with Sherrington-Kirkpatrick exchange disorder the full phase structure in (J0, J, h, T) is indicated. The phases predicted are analogous to those of a Heisenberg spin glass but there are important differences of transition orders and regions of stability.
TL;DR: In this article, the authors studied the adaption of non-boundary states at interfaces in two-dimensional Potts models using Monte Carlo techniques, and showed that the results for the exponents ω anda are consistent with a scaling argument that yields ω=ν-β anda=1-(β/ν).
Abstract: The net adsorption,W, of non-boundary states at interfaces in two-dimensional Potts models is studied using Monte Carlo techniques. For the three-and four-state Potts model the net adsorption is found to diverge when the critical point is approached asW∼t−ω, wheret=(Tc-T)/T. Finite-size effects have been also studied; for anN x N lattice the net adsorption at the bulk critical temperature behaves asWc∼Na. The numerical results for the exponents ω anda are consistent with a scaling argument that yields ω=ν-β anda=1-(β/ν).
TL;DR: The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays three phase transitions in different universality classes as the magnetic field is increased as discussed by the authors.
Abstract: The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first...
TL;DR: In this article, the Bethe lattice was used to solve the percolation problem on the s-state Potts model with different z-spin interactions on the two sublattices and the critical concentration was found to be pc(a)=0.21 whereas the correlation length exponent nu a seems to converge towards the accepted percolations value nu p=1.333.
Abstract: The antipercolation problem is solved on the Bethe lattice. The critical exponents are identical to the percolation exponents when the coordination number z is greater than a critical value zc=3, for which the problem has new exponents satisfying extended universality and below which there is no transition. For alternate lattices the problem may be transformed into a percolation problem with different occupation probabilities on the two sublattices. This allows a connection with an s-state Potts model with different z-spin interactions on the two sublattices. In two dimensions there is no transition on the alternate square and honeycomb lattices whereas a transition exists on the triangular lattice. Using the phenomenological renormalisation group method, the critical concentration is found to be pc(a)=0.21 whereas the correlation length exponent nu a seems to converge towards the accepted percolation value nu p=1.333.
TL;DR: In this paper, the phase diagram of the Potts antiferromagnet in a field is studied by exact mappings and Monte Carlo methods, and it is shown that systems with macroscopic ground-state degeneracy may exhibit conventional-type critical ordering.
Abstract: The phase diagram of the $q=3$ state Potts antiferromagnet in a field is studied by exact mappings and Monte Carlo methods, and it is shown that systems with macroscopic ground-state degeneracy may exhibit conventional-type critical ordering. Our Monte Carlo data for the square-lattice case indicate a finite-temperature ordering in a range of field values where the entropy of the ground state is extensive. In a particular limit this phase transition can be shown to be of the Ising type by mapping the Potts model onto a colored, nearest-neighbor exclusion lattice gas. We also discuss the phase diagram of the body-centered-cubic Ising antiferromagnet in a field in order to show that renormalization-group arguments alone are not sufficient to restrict the diversity of possible low-temperature phases occurring in systems with a macroscopically degenerate ground state.
TL;DR: In this article, the (1+1)-dimensional Hamiltonian Potts model was studied for q>or=4 using finite lattice extrapolation techniques and the ground state energy and its first derivative gave information about the free energy and the latent heat.
Abstract: The (1+1)-dimensional Hamiltonian Potts model is studied for q>or=4 using finite lattice extrapolation techniques. The ground-state energy and its first derivative give information about the free energy and the latent heat of the classical two-dimensional Potts model, while the gap in the excitation energy corresponds to the inverse of the correlation length. It is shown that the finite latent heat for q>4, when the transition is of first order, comes from the crossing of levels; nevertheless there are no excitations for which the gap would vanish in the thermodynamic limit i.e. the correlation length is also finite.
TL;DR: In this paper, the anisotropic triangular nearest-neighbor Ising model with antiferromagnetic interactions is studied, and two commensurate ordered phases are found.
Abstract: The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied The phase diagram, as a function of temperature and field, has a complex structure Two commensurate ordered phases are found The transition to one is Ising-type The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model These two transition regimes are found to be separated by a Lifshitz point The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods
TL;DR: In this article, a decorated Ising model with classical vector spins on a square lattice is investigated in detail, and three successive phase-transition temperatures are obtained and four states, namely, paramagnetic, antiferromagnetic, again paramagnetic and ferromagnetic states are realized as the temperature is decreased.
Abstract: A decorated Ising model with classical vector spins on a square lattice is investigated in detail. The partition function is reduced to the one of the Ising model with effective exchange integrals. Three successive phase-transition temperatures are obtained and four states, namely, paramagnetic, antiferromagnetic, again paramagnetic and ferromagnetic states are realized as the temperature is decreased. For systems on other two- and three-dimensional loose-packed lattices, the situation is the same as the system on the square lattice.
TL;DR: In this paper, the break-collapse method for the q-state Potts model is adapted for resistor networks, which greatly simplifies the calculation of the conductance of an arbitrary two-terminal d-dimensional array of conductances.
Abstract: The break-collapse method recently introduced for the q-state Potts model is adapted for resistor networks. This method greatly simplifies the calculation of the conductance of an arbitrary two-terminal d-dimensional array of conductances, obviating the use of either Kirchhoff's laws or the star-triangle transformation. In addition, a real-space renormalisation group based on a new type of averaging gives excellent results for the conductivity of the random-resistor network on the square lattice.
TL;DR: In this article, the convexity of the free energy is studied for several lattice models in situations in which a parameter which is normally a positive integer takes on noninteger real values, such as the numbern of components in then-vector model, the number of states in the Potts model, and the dimensionality of the lattice.
Abstract: The convexity of the free energy is studied for several lattice models in situations in which a parameter which is normally a positive integer takes on noninteger real values. Examples include the numbern of components in then-vector model, the number of states in the Potts model, and the dimensionality of the lattice. In a typical case there is a critical value of the parameter such that convexity is preserved when the parameter exceeds the critical value, but can be violated for appropriate Hamiltonians whenever the parameter is less than the critical value, but not a positive integer. In several cases the critical value of the parameter increases with the size of the system, thus raising questions about the significance of a continuous variation of the parameter in the thermodynamic limit.
TL;DR: In this paper, a large q expansion of the partition function of the checkerboard q-state Potts model is given up to sixth order in q-12, where the latent heat and correlation functions are discussed from the point of view of their functional dependence in the parameters.
Abstract: A large q expansion of the partition function of the checkerboard q-state Potts model is given up to sixth order in q-12/. This expansion is used to discuss the following two points. First, on the critical manifold, the large q expansion agrees up to this order, with the 'minimal' solution suggested by the group structure associated with this model. Secondly, the latent heat and some correlation functions are discussed from the point of view of their functional dependence in the parameters.
TL;DR: In this article, the critical temperature of the q-state ferromagnetic Potts model on a hypercubic lattice in d dimensions is expressed as an explicit function of both d and q.
Abstract: The critical temperature of the q-state ferromagnetic Potts model on a hypercubic lattice in d dimensions is expressed as an explicit function of both d and q. The author's formula agrees very well with the existing accurate results and numerous numerical studies for particular values of d and q, and it seems to be an excellent approximation for all d and q.
TL;DR: In this article, a Potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given, and four different critical behaviours are found, corresponding to random animal, collapse or theta point, percolation and compact cluster.
Abstract: A Potts model formulation of the statistics of branched polymers or lattice animals in a solvent is given. The Migdal-Kadanoff renormalisation group is employed to study the critical behaviour or fractal dimension of the branched polymer. Four different critical behaviours are found, corresponding to random animal, collapse or theta point, percolation and compact cluster. The theta point behaviour is described by a tricritical point while percolation corresponds to a higher-order critical point, where the effect of the solvent on the branched polymer is the same as the screening effect of the other clusters in percolation.
TL;DR: In this article, a two-dimensional Ising model on a hexagonal lattice is considered and the partition function is evaluated exactly by the method of Pfaffian, which is a special case of the model.
Abstract: The authors have considered a two-dimensional Ising model on a ruby lattice. The partition function is evaluated exactly by the method of Pfaffian. The Ising model on a hexagonal lattice is a special case of the model.
TL;DR: In this paper, a Monte Carlo renormalization-group scheme was used to investigate the transition from a floating phase to a liquid phase at a temperature much higher than that found by series-expansion methods.
Abstract: Layers of atoms or molecules adsorbed on surfaces may exhibit floating critical phases. To investigate this phenomenon we have developed a Monte Carlo renormalization-group scheme and performed calculations on the three-state chiral clock model. The results indicate a continuous transition from a floating phase to a liquid phase at a temperature much higher than that found by series-expansion methods.