Showing papers on "Potts model published in 1980"

Duality in field theory and statistical systems

Abstract: This paper presents a pedagogical review of duality (in the sense of Kramers and Wannier) and its application to a wide range of field theories and statistical systems. Most of the article discusses systems in arbitrary dimensions with discrete or continuous Abelian symmetry. Globally and locally symmetric interactions are treated on an equal footing. For convenience, most of the theories are formulated on a $d$-dimensional (Euclidean) lattice, although duality transformations in the continuum are briefly described. Among the familiar theories considered are the Ising model, the $x\ensuremath{-}y$ model, the vector Potts model, and the Wilson lattice gauge theory with a ${Z}_{N}$ or $U(1)$ symmetry, all in various dimensions. These theories are all members of a more general heirarchy of theories with interactions which are distinguished by their geometrical character. For all these Abelian theories it is shown that the duality transformation maps the high-temperature (or, for a field theory, large coupling constant) region of the theory into the low-temperature (small coupling constant) region of the dual theory, and vice versa. The interpretation of the dual variables as disorder parameters is discussed. The formulation of the theories in terms of their topological excitations is presented, and the role of these excitations in determining the phase structure of the theories is explained. Among the other topics discussed are duality for the Abelian Higgs model and related models, duality transformations applied to random systems (such as theories of a spin glass), duality transformations in the "lattice Hamiltonian" formalism, and a description of attempts to construct duality transformations for theories with a non-Abelian symmetry, both on the lattice and in the continuum.

Magnetic exponents of the two-dimensional q-state Potts model

Abstract: The results of a variational renormalisation-group calculation for the magnetic exponent yH of the two-dimensional q-state Potts model suggest a simple relationship between yH and the exactly known critical exponent yT8v of the eight-vertex model. The relation allows one to predict the critical and tricritical magnetic exponent delta of the q-state Potts model as a function of q.

Scaling Theory of the Potts Model Multicritical Point

Abstract: A theory of the scaling behavior near the ${q}_{c}=4$ state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for $q$ near ${q}_{c}$. For the pure Potts model with $q={q}_{c}$ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At ${T}_{c}$ we find $\mathrm{ln}(r)$ corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with $q={q}_{c}$ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as $T$ approaches the multicritical point from the first-order side. Results for $qg{q}_{c}$ and $ql{q}_{c}$ are also given.

Conjecture for the extended Potts model magnetic eigenvalue

Abstract: I exhibit a form for the magnetic eigenvalue of the extended Potts model which I believe to be exact. This result complements the discussion of the thermal eigenvalue given by Nienhuis, Berker, Riedel, and Schick.

Singularities and Scaling Functions at the Potts-Model Multicritical Point

Abstract: Differential renormalization equation for the $q$-state Potts model are proposed, and the critical behavior of the model near $q={q}_{c}$ discussed. The equations give rise to critical and tricritical fixed points which merge at $q={q}_{c}$ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of $q={q}_{c}$, in accordance with the exact result of Baxter, and, for $q={q}_{c}$, to a logarithm correction to the power-law behavior of the free energy as a function of $T\ensuremath{-}{T}_{c}$.

Finite size scaling and critical point exponents of the potts model

Abstract: The calculation of critical exponents by combining finite size scaling and the transfer matrix technique is proposed and applied to the two-dimensional q-state Potts model. The exact results for q = 2 are very accurately reproduced. For q = 3, our results suggest α = 13 and δ = 14. Convergence of our results for q ⩾ 4 is poor but it is suggested that α $12 and δ #62 14 for q = 4. A preliminary result for the dynamical exponent of the stochastic Ising model is reported.

Variational renormalisation-group approach to the q-state Potts model in two dimensions

Abstract: The two-dimensional q-state Potts model is investigated by means of a Kadanoff lower-bound renormalisation-group transformation that utilises a recent suggestion to identify disordered cells with vacancies. The topology of the phase diagram is obtained, including first- and second-order transitions for q>qc and q

Potts model in the many-component limit

Abstract: The mean-field theory of the q-component Potts model is shown to be exact in the limit q to infinity . This proves a conjecture by Mittag and Stephen, (1974).

On the triangular Potts model with two- and three-site interactions

Abstract: The equivalence of the triangular Potts model having two- and three-site interactions with a 20-vertex Kelland model is rederived using a graphical method. The conjectured critical point of this Potts model is shown to agree with the known results in two instances.

Crossover from percolation to random animals and compact clusters

Abstract: Introduces a field-like variable to develop a generating function for the percolation problem which, in the appropriate limits, also describes the statistics of random animals (dilute branched polymers) and compact clusters (collapsed branched polymers). The crossover from percolation to random animals and compact clusters is studied using a two-parameter position space renormalisation-group approach. The authors obtain the global flow diagram in the two-parameter space and calculate the critical properties. They find that the critical behaviour is described below the percolation threshold pc by the random-animal fixed point, and above pc by the compact-cluster fixed point. A Hamiltonian formulation using the Q-state Potts model is proposed. The crossover from percolation to random animals can be described by taking a specific limit of the field-like variable.

Conservation laws in the quantum version of $N$-positional Potts model

Abstract: A quantum analogue of theN-positional Potts model is constructed. The system is shown to possess an infinite set of involutory conservation laws in the phase transition point.

Critical dynamics of the Potts model

Abstract: The dynamical critical index $z$ is calculated for the two-dimensional Potts model on square lattice using a Migdal-type recursion method generalized to dynamics.

Critical behaviour of the dilute Ashkin-Teller-Potts model

Abstract: A real-space rescaling technique (decimation) is used to study the critical properties of the bond-dilute, S state Ashkin-Teller-Potts model on the square lattice. Fixed points and critical exponents are obtained and critical curves plotted for values of S from 1 to 4. For S=1, 2 the dilute model exhibits the same critical behaviour as the pure system. However, for S=3, 4, there is crossover to a new dilute fixed point. The transition remains second order but exhibits new values of the critical exponents. This behaviour is in accord with the Harris criterion (1974).

Exact results for a dilute Potts model

Abstract: Exact results are obtained for the annealed, dilute,q-component Potts model on the decorated square lattice. The phase diagram is found to consist of a high-temperature region, a low-temperature region, and a two-phase region in between which arises only forq>4: exact expressions for the phase boundary and the critical probability are derived. At the critical point the specific heat is generally finite and has a cusp; the slope of the cusp is finite forq=4 and infinite (vertical) forq=2 and 3.

Series analysis for the three-state Potts model

Abstract: Low-temperature series expansions for the zero-field partition function and order parameter of the three-state Potts model have been extended to u31. Estimates for alpha ' and beta show rather poor convergence but are consistent with the values alpha '=1/3, beta =1/9 which characterise the 'hard hexagons' lattice gas.

Phase diagram of the Z(5) model on a square lattice

Abstract: A general five-state model, which contains the five-state Potts model and a solid-on-solid model as special cases, is studied. The authors find that the high-temperature paramagnetic and the low-temperature ordered phases are separated either by a line of first-order transitions or by an intermediate phase with algebraic decay of correlations. The phase diagram is proposed on the basis of general considerations and Monte-Carlo simulations.

Fixed points of the hierarchical Potts model

Abstract: The renormalisation group equations for the q-state Potts model having the hierarchical interaction is investigated. A state fixed point is found to exist when q 1/3. Critical exponents associated with the new fixed point are also obtained to the first order in Delta sigma .

Quenched bond disorder in the Potts model: single-bond effective interaction approximation

Abstract: An effective-medium approximation is proposed for thermal phase transitions. For quenched bond disorder in the Potts model, the exact critical line is obtained when s=1; for s=2 (Ising), some exact results in the dilute limit and the correct behaviour for Tc(p) near pc are recovered. The critical concentrations are in good agreement with the known values, in both two and three dimensions. For s>4 (first-order transition) the phase diagram shows a two-phase region.

Critical exponents of the four-state Potts model

Abstract: The two-dimensional four-state Potts model at finite temperature can be transformed, via the transfer matrix, into a one-dimensional quantum mechanical model at zero temperature The duality invariant renormalisation group introduced by Fernandez-Pacheco (1979) is then employed to study the ground-state critical properties of this model The fixed point is located at exactly the self-dual critical point K*=1 The thermal exponent is calculated to be yT=13219 It is in excellent agreement with the recent series value of Ditzian and Kadanoff (1979) (yT=133) Although it is not inconsistent with den Nijs's (1979) conjectured exact value of 3/2, the difference is nevertheless substantial

Effects of symmetry-breaking perturbations on the three-state Potts model

Abstract: The effects of linear and quadratic symmetry-breaking perturbations on the continuous version of the three-state Potts model are analysed, using both Landau's theory of phase transitions and renormalisation-group techniques. Variation of the strength of the perturbations produces many different types of phase diagrams, featuring lambda lines, first-order lines, critical, tricritical and triple points and other complexities. Universal amplitude ratios characterising the multicritical points are calculated to first order in epsilon =4-d.

Universality in a generalised Potts model

Abstract: A three-state Potts model with pure three-site interactions is analysed using low-temperature series expansions. Universality predicts that the exponents alpha ' and beta should be the same as for the 'hard-hexagons' lattice gas: 1/3 and 1/9 respectively. The series estimates are consistent with these predictions, and the estimates of gamma ' are consistent with the scaling value of 14/9.

Multicritical points in structural phase transitions

Abstract: Recent theoretical (renormalization group) results on phase diagrams and multicritical points in systems undergoing structural phase transitions are reviewed. Particular attention is given to (a) the effects of anisotropic stress on cubic systems, e.g. perovskites undergoing antiferrodistortive phase transitions (bicritical, tricritical, Potts model, Lifshitz points, etc.), (b) phase diagrams of alloys, (c) the effects of electric fields on anisotropic antiferroelectrics.

Critical curve of the dilute random-bond Potts model

Abstract: The phase boundary of the bond diluted s-state Potts model on the square lattice is calculated for the case of quenched randomness Using a duality transformation and the replica method the author finds the phase boundary to be given by exp(-sJ/Tc)=(sxp/-1)/(s-1), where x=p-pc

A polymer model equivalent to the Ising model

Abstract: It is shown that the zero-field Ising model is equivalent to a polymer model in which a bond of a polymer chain can assume one gauche and one trans state. The Ising model is defined on a lattice graph G of degree d=4, whereas the polymer model is defined on a covering lattice digraph Dc of out-degree d*=2. As an example, a polymer model, defined on the Manhattan square lattice, is shown to be equivalent to the zero-field Ising model on the square lattice. The polymer model can be used to discuss the melting transition in polymers.

Antiferroelectric transition in β-alumina, a realization of the D = 2, s = 3 Potts model ?

Abstract: 2014 A model based on electric dipole-dipole interactions is proposed to explain the recently observed ordering transition in stoichiometric silver 03B2-alumina. The low temperature phase is a three-fold degenerate antiferroelectric array of dipoles associated with Ag+ and O-5 ions. When T ~ Tc with (T Tc)/Tc > 10-2, the correlation length 03BE increases in the D = 2 layers with a critical behaviour 03BE ~ (T 2014 Tc)-03BD which is supposed to belong to the D = 2, s = 3 Potts model universality class : we find 03BDexp = 0.85 ± 0.1 as 03BDth ~ 0.83. J. Physique LETTRES 41 (1980) L-115 L-117 I er MARS 1980, Classification Physics Abstracts 64.70K 66.30 77.80 It has recently been observed that stoichiometric [1] ] Ag+ #-alumina undergoes a structural ordering around 300 K, the low temperature structure being an hexagonal superlattice structure [2, 3] with a’ = a ~/3. The purpose of this paper is first to remark that the observed structure can be predicted in a simple picture which takes into account the existence of off-centred positions (o.c.p.) for the Ag+ and 0~’ ions. The model is basically that of an order-disorder antiferroelectric transition in two dimensions [4]. Antiferroelectric interactions between o.c.p. have been already claimed to be responsible for the anomalous static and dynamic behaviour of CU3VS4 at low temperature and the existence of off-centred positions seems to play an important role in a number of fast ion conductors [5, 6]. We show that the transition can be related to recent (*) Groupe de Recherches du Centre National de la Recherche Scientifique. (**) Laboratoire associé au Centre National de la Recherche Scientifique. theoretical and experimental studies of phase transition in 2-dimensions and that it constitutes the first physical realization of the D = 2, three states Potts model [7] where the critical exponent v for the correlation length ~ can be measured from scattering techniques. To clarify the following discussion one should stress that in fast ion conductors one can consider two kinds of order-disorder transitions : the first is the classical transition concerning the occupation of Beevers-Ross (BR) and the anti-Beevers-Ross (aBR) sites. We shall call it the BR-aBR transition and is usually considered as closely related to fast ion conduction (lattice gas models [8]). Here we shall consider an other type of transition in which what is considered is the ordering between the equivalent off-centre positions of the ions. We show in this letter that the order-disorder transition in ~-alumina when studied in the quasi-2D regime belongs to the same universality class as the D = 2 three states (s = 3) Potts model [7]. The only other realization [9] of this model is the transition Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01980004105011500 L-116 JOURNAL DE PHYSIQUE LETTRES from a liquid to a commensurate solid for adsorbed helium and Kr atoms on graphite [10, 11]. For 4He specific heat measurements have given a critical exponent [10, 12] a = 0.36 and oc = 0.28. Contrary to Landau theory which predicts a first-order transition, the strong fluctuations in 2D lead for the Potts model with s 4 to a second-order transition [13]. Recent conjectures [14] and real space renormalization group calculations [15] predict for the three states D = 2 Potts models 5/6andx=2-2y ~ 1/3. The fact that the ordering seems to take place between a liquid like phase at high temperature (probably highly conducting) and an antiferroelectric phase at low temperature suggests that both phenomena are :upt independent. The superstructure which exists below Tc = 300 K is shown in figure 1. It is due to an ordering of the silver and the oxygen ions between the otherwise equivalent o.c.p. It is reminiscent of the J3 x J3 superstructure found in commensurate solid phases of rare gas adsorbed on graphite [10] (see also ref. [15] for further references). Microdensitometer readings of the X-rays pattern show that approaching Tc from above there exists a region T > T * for which correlations are purely 2D. [(T* rj/r, a few 10-2]. When T > T* the system behaves as if it was 2-dimensional and only very close to Tc, (T T*) the behaviour crosses over to the truly 3-dimensional regime. If the two.dimensional regime is second order a critical behaviour is expected in the region T -~ T~ ; T > T * with ~ ^’ (T 7,)-B In our case, the interaction between the o.c.p. can be modeled by electric dipole-dipole interactions [5] Fig. 1. Low temperature structure of stoichiometric silver ~-alumina. The unit cell of size a J3 is represented by the dotted line. The full circles represent the classical BR sites. The open circles represent the classical 05 sites. The square represent the aBR sites. The Ag+ ions in BR sites and the 05 ions are found displaced towards an aBR site represented by an open square (site A). The other two degenerate fundamental states corresponds to displacement towards the B and C sites. where the dipoles di can take six different values (3 for 05 and 3 for Ag+). The ground state of a layer is obtained numerically through the following iterative process : we first consider a small cluster and find the lower energy configurations. We then put several clusters together and determine the lower energy configurations of this larger cluster from a combination of the lower states of the small clusters. The ground state structure is found to depend on the ratio of the electric dipole associated with the two species Ag+ and 0. The structure represented in figure 1 is obtained only if this ratio is smaller than 2.583. Experimentally we can estimate from [2] that d(Ag+?/d(o5 -) ~ 0.8/2 x 0.3 ~ 1.33 a value compatible with the limit of 2.583 obtained from the numerical calculation. The fundamental state is found three-fold degenerate. One of these states is shown in figure 1. It is obtained by letting the Ag+ and 0 5 ions move towards the aBR site named A in the figure. There are two other states of same energy named B and C in which the ions move respectively towards the sites B and C. In fact each of these states can be build by a tile pavement of only three types of rigid hexagons of dipoles. It is important to stress that the first excitations of this system corresponds to a domain structure of the three degenerate ground states and that excited states corresponding to the destruction of the basic hexagons by letting the Ag+ and 05 ions move independently are much higher in energy. They can be neglected near the transition. The ions positions in the ordered phase can then be expressed in terms of the two functions t/J 1 = exp(ik1 R) and t/J 2 = exp(lk2 R) where k = 7~(1, 0, 0) , k2 = 7~(1/2, ~/3/2, 0) as for the order parameter corresponding to the superlattice structure of He on graphite. The LandauGinzburg Hamiltonian takes the form : where { 91, ~p2 } is the real basis of the irreducible representation to which belongs the order parameter. Up to the third-order this system and the three states Potts [7] model have the same Landau-Ginzburg Hamiltonians. Strong universality assumption implies the same critical behaviour for both systems. The same assumption was made for helium adsorbed layers and seems to be confirmed by experiments [12, 13]. For #-alumina a preliminary analysis of the experimental correlation length (Fig. 2) gives vexp = 0.85 ± 0.1 while theory [14, 15] predicts v~ ~ 0.83. The uncertainty in this analysis comes from the difficulty to choose Tc and from the onset of the 2D-3D crossover. L-117 ANTIFERROELECTRIC TRANSITION IN ~-ALUMINA Fig. 2. Log. Log plot of the correlation length ~ as a function of T Tc. These data are the best obtained by Boilot et al. [17]. The data of reference [2] has been also analysed and give compatible results. The vertical error bars are estimated errors in the measurement of ~. In this figure Tc has been chosen to be equal to 300 K. Other choice of 7~ compatible with the experiment, results in the quoted error for the exponent v. For (T Tc)/Tc > 0.2 the effective exponent Veff = Log ~/Log AT crosses over to the mean field value 0.5 . The observed 3D low temperature superstructure is a superposition of layer superstructures. One layer is of type A. The next layer is of type B with a permutation between the oxygen and the silver ions. This keeps the value of the lattice spacing c. Preliminary calculations using dipolar interactions for two layers cannot decide between AA and AB for the lowest energy configuration. In fact the basic hexagonal tiles with which the fundamental states are constructed have no net dipolar moment. These hexagons have only quadrupolar interaction varying as r5. Moreover the difference between the AA and AB structure goes more like r6. Experimentally [3] the crossover to the 3D ordering occurs within a few degrees from 7~ in agreement with the weak interaction between planes. Because of the permutation of ions from one layer to the next the 3D structure can be only either AAA... or ABAB... We cannot exclude that the observed 3D arrangement comes from weak interactions through the spinel structure between the layers. The 3D ground state is six-fold degenerate and the order parameter has now four components. The Landau-Ginzburg Hamiltonian would have the same form as for two coupled s = 3 Potts models. In 3D a first-order transition is certainly expected. (The 3D, s > 2 Potts model is considered to lead to a first-order transition [16].) More experimental work including detailed measurements of ~ between 300 and 330 K and specific heat measurements are needed to verify our model for the transition. Measurements of the conductivity in the same temperature range should give information about the relation between the o.c.p. ordering and the BR-aBR ordering. Acknowledgments. The authors wish to thank J. P. Boilot and G. Collin for communicating their unpublished results and for infor