A Potts/Ising Correspondence on Thin Graphs

We note that it is possible to construct a bond vertex model that displays q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary topology, which we denote as ``thin'' random graphs in contrast to the fat graphs of the planar diagram expansion. Since the four vertex model in question also serves to describe the critical behaviour of the Ising model in field, the formulation reveals an isomorphism between the Potts and Ising models on thin random graphs. On planar graphs a similar correspondence is present only for q=1, the value associated with percolation.Comment: 6 pages, 5 figure

Global Bethe lattice consideration of the spin-1 Ising model

The spin-1 Ising model with bilinear and biquadratic exchange interactions and single-ion crystal field is solved on the Bethe lattice using exact recursion equations. The general procedure of critical properties investigation is discussed and full set of phase diagrams are constructed for both positive and negative biquadratic couplings. In latter case we observe all remarkable features of the model, uncluding doubly-reentrant behavior and ferrimagnetic phase. A comparison with the results of other approximation schemes is done.Comment: Latex, 11 pages, 13 ps figures available upon reques

Thin Animals

Lattice animals provide a discretized model for the theta transition displayed by branched polymers in solvent. Exact graph enumeration studies have given some indications that the phase diagram of such lattice animals may contain two collapsed phases as well as an extended phase. This has not been confirmed by studies using other means. We use the exact correspondence between the q --> 1 limit of an extended Potts model and lattice animals to investigate the phase diagram of lattice animals on phi-cubed random graphs of arbitrary topology (``thin'' random graphs). We find that only a two phase structure exists -- there is no sign of a second collapsed phase. The random graph model is solved in the thermodynamic limit by saddle point methods. We observe that the ratio of these saddle point equations give precisely the fixed points of the recursion relations that appear in the solution of the model on the Bethe lattice by Henkel and Seno. This explains the equality of non-universal quantities such as the critical lines for the Bethe lattice and random graph ensembles.Comment: Latex, 10 pages plus 6 ps/eps figure

Spin Glasses on Thin Graphs

In a recent paper we found strong evidence from simulations that the Isingantiferromagnet on ``thin'' random graphs - Feynman diagrams - displayed amean-field spin glass transition. The intrinsic interest of considering such random graphs is that they give mean field results without long range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice, or in previous replica calculations. We then investigate numerically spin glasses with a plus or minus J bond distribution for the Ising and Q=3,4,10,50 state Potts models, paying particular attention to the independence of the spin glass transition from the fraction of positive and negative bonds in the Ising case and the qualitative form of the overlap distribution in all the models. The parallels with infinite range spin glass models in both the analytical calculations and simulations are pointed out.Comment: 13 pages of LaTex and 11 postscript figures bundled together with uufiles. Discussion of first order transitions for three or more replicas included and similarity to Ising replica magnet pointed out. Some additional reference

Phase diagram of the three states Potts model with next nearest neighbor interactions on the Bethe lattice

We have found an exact phase diagram of the Potts model with next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the Lifshitz point i.e. $p=1/3$. We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by 2-periodic points of an one dimensional dynamical system lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting

New order parameters in the Potts model on a Cayley tree

For the $q-$state Potts model new order parameters projecting on a group of spins instead of a single spin are introduced. On a Cayley tree this allows the physical interpretation of the Potts model at noninteger values q of the number of states. The model can be solved recursively. This recursion exhibits chaotic behaviour changing qualitatively at critical values of $q_0$ . Using an additional order parameter belonging to a group of zero extrapolated size the additional ordering is related to a percolation problem. This percolation distinguishes different phases and explains the critical indices of percolation class occuring at the Peierls temperature.Comment: 16 pages TeX, 5 figures PostScrip

Mean Field Critical Behaviour for a Fully Frustrated Blume-Emmery-Griffiths Model

We present a mean field analysis of a fully frustrated Ising spin model on an Ising lattice gas. This is equivalent to a degenerate Blume-Emery-Griffiths model with frustration, which we analyze for different values of the quadrupolar interaction. This model might be useful in the study of structural glasses and related systems with disorder

Phase transitions for PP-adic Potts model on the Cayley tree of order three

In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter \rho\in\bq_p. Such a measure is called {\it generalized $p$-adic quasi Gibbs measure}. When $\rho$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $\rho=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|\rho|_p$. Namely, in the first regime, one takes $\rho=\exp_p(J)$ for some J\in\bq_p, in the second one $|\rho|_p<1$. In each regime, we first find conditions for the existence of generalized $p$-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when $|\r|_p,|q|_p\leq p^{-2}$ we prove the existence of a quasi phase transition. It turns out that if $|\r|_p<|q-1|_p^2<1$ and \sqrt{-3}\in\bq_p, then one finds the existence of the strong phase transition.Comment: 27 page

Study of Percolative Transitions with First-Order Characteristics in the Context of CMR Manganites

The unusual magneto-transport properties of manganites are widely believed to be caused by mixed-phase tendencies and concomitant percolative processes. However, dramatic deviations from "standard" percolation have been unveiled experimentally. Here, a semi-phenomenological description of Mn oxides is proposed based on coexisting clusters with smooth surfaces, as suggested by Monte Carlo simulations of realistic models for manganites, also briefly discussed here. The present approach produces fairly abrupt percolative transitions and even first-order discontinuities, in agreement with experiments. These transitions may describe the percolation that occurs after magnetic fields align the randomly oriented ferromagnetic clusters believed to exist above the Curie temperature in Mn oxides. In this respect, part of the manganite phenomenology could belong to a new class of percolative processes triggered by phase competition and correlations.Comment: 4 pages, 4 eps figure

On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three

In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two now quasi equivalent QMC for the given family of interaction operators $\{K_{}\}$.Comment: 34 pages, 1 figur