Planar Potts Model Research Articles

Structure of Gibbs measure for planar FK-percolation and Potts models

We prove that all Gibbs measures of the $q$-state Potts model on $\mathbb{Z}^2$ are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is at $T=T_{c}(q)$ when $q>4$. In this case the structure of Gibbs measures is the most complex in the sense that there exist $q+1$ distinct extremal measures. Most of the work is devoted to the FK-percolation model on $\mathbb{Z}^{2}$ with $q\geq 1$, where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation. Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop $O(n)$ model in the range of parameters for which its spin representation is positively associated.

Emergence of Concentration Effects in the Variational Analysis of theN‐Clock Model

AbstractWe investigate the relationship between theN‐clock model (also known as planar Potts model or‐model) and theXYmodel (at zero temperature) through a Γ‐convergence analysis of a suitable rescaling of the energy as both the number of particles andNdiverge. We prove the existence of rates of divergence ofNfor which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of theN‐clock model in this regime features an energy that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex‐vortex interaction energy. © 2022 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC.

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

Optical Potts machine through networks of three-photon down-conversion oscillators

AbstractIn recent years, there has been a growing interest in optical simulation of lattice spin models for applications in unconventional computing. Here, we propose optical implementation of a three-state Potts spin model by using networks of coupled parametric oscillators with phase tristability. We first show that the cubic nonlinear process of spontaneous three-photon down-conversion is accompanied by a tristability in the phase of the subharmonic signal between three states with 2π/3 phase contrast. The phase of such a parametric oscillator behaves like a three-state spin system. Next, we show that a network of dissipatively coupled three-photon down-conversion oscillators emulates the three-state planar Potts model. We discuss potential applications of the proposed system for all-optical optimization of combinatorial problems such as graph 3-COL and MAX 3-CUT.

Simulating Ising and n-State Planar Potts Models and External Fields with Nonequilibrium Condensates.

Classical spin models with discrete or continuous degrees of freedom arise in many studies of complex physical systems. A wide class of hard real-life optimization problems can be formulated as a minimization of a spin Hamiltonian. Here we show how to simulate the discrete Ising and n-state planar Potts models with or without external fields using the physical gain-dissipative platforms with continuous phases, such as lasers and various nonequilibrium Bose-Einstein condensates. The underlying operational principle originates from a combination of resonant and nonresonant pumping. Our results lay grounds for the physical simulations of a broad range of Hamiltonians with complex interactions that can vary in time and space and with combined symmetries.

Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with $${1 \le q \le 4}$$ 1 ≤ q ≤ 4

This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. 2. A new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions, - the uniqueness of infinite-volume measures, - the sub-exponential decay of the two-point function for free boundary conditions, - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions. The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $1\le q \le 4$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

Two-Dimensional Disorder in Black Phosphorus and Monochalcogenide Monolayers

Ridged, orthorhombic two-dimensional atomic crystals with a bulk Pnma structure such as black phosphorus and monochalcogenide monolayers are an exciting and novel material platform for a host of applications. Key to their crystallinity, monolayers of these materials have a 4-fold degenerate structural ground state, and a single energy scale EC (representing the elastic energy required to switch the longer lattice vector along the x- or y-direction) determines how disordered these monolayers are at finite temperature. Disorder arises when nearest neighboring atoms become gently reassigned as the system is thermally excited beyond a critical temperature Tc that is proportional to EC/kB. EC is tunable by chemical composition and it leads to a classification of these materials into two categories: (i) Those for which EC ≥ kBTm, and (ii) those having kBTm > EC ≥ 0, where Tm is a given material's melting temperature. Black phosphorus and SiS monolayers belong to category (i): these materials do not display an intermediate order-disorder transition and melt directly. All other monochalcogenide monolayers with EC > 0 belonging to class (ii) will undergo a two-dimensional transition prior to melting. EC/kB is slightly larger than room temperature for GeS and GeSe, and smaller than 300 K for SnS and SnSe monolayers, so that these materials transition near room temperature. The onset of this generic atomistic phenomena is captured by a planar Potts model up to the order-disorder transition. The order-disorder phase transition in two dimensions described here is at the origin of the Cmcm phase being discussed within the context of bulk layered SnSe.

The phase transitions of the planar random-cluster and Potts models with $$q \ge 1$$ are sharp

We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling. Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions. Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.

Series Analysis of a Kosterlitz-Thouless Transition: The 6-State Planar Potts Model

A new implementation of the Finite Lattice Method of series expansion is used to derive low temperature series for the 6-state planar Potts model. This is expected to have Kosterlitz-Thouless transitions between the ordered low-temperature phase and a massless phase, and between the massless phase and a high-temperature disordered phase. In an exploratory study, we have analysed series for the order parameter to order x59. These series have proved particularly difficult to analyse and when seeking to both locate the transition and confirm the form of the Kosterlitz-Thouless transition, considerable ambiguity is found. If however the form of the Kosterlitz-Thouless transition is assumed, then the series analysis consistently indicates that the lower transition occurs at xL=0.4886(10). A more conservative analysis, which made no assumption about the nature of the transition, led to the estimate xL=0.485(8).

Effect of uniaxial stress on the electromechanical properties in ferroelectric thin films under combined loadings

The electromechanical properties of ferroelectric thin films under an alternating electric field and a static uniaxial compressive stress are investigated using the modified planar four-state Potts model. To implement the electromechanical properties and the coupling of the electrical and mechanical response, the mechanical energy density as well as the energy due to anisotropic switching between a-domain and c-domain are incorporated in the Hamiltonian. Besides, there are two contributions to the strain at each cell: eigenstrain and elastic strain. Our simulation results show that the longitudinal strain-electric field butterfly loop shifts downward along strain axis and that for the transverse strain shifts upward as the stress magnitude is increased. Moreover, the polarization-electric field hysteresis loop becomes a double-loop under a large compressive stress. The piezoelectric coefficient increases with the stress magnitude and reaches a maximum value at a critical stress level. It then gradually decreases to a small value at large stress magnitudes. Our results qualitatively agree with experimental ones.

Scaling and stress dependence of sub-coercive field dynamic hysteresis in0.6Pb(Zr1/2Ti1/2)O3–0.4Pb(Zn1/3Nb2/3)O3 ceramic

The scaling behavior of sub-coercive field dynamic ferroelectric hysteresisunder the influence of stress was investigated in rhombohedral0.6Pb(Zr0.5Ti0.5)O3–0.4Pb(Zn1/3Nb2/3)O3 (0.6PZT–0.4PZN) bulk ceramic. The scaling relation of hysteresis area against frequency f, field amplitude E0, and stress σ for the minor loops takes the form of , indicating the difference in the energy dissipation between the stressedand stress-free conditions. While the scaling obtained is very similarto that of soft and hard PZT ceramics, slightly faster responses tof andE0 indicate the ease of polarization orientation in this ceramic with a simpler domain structurecompared to commercial PZT ceramics. However, the difference in mechanical properties ofthese materials could contribute to a variation in the response to stress. While theE0 exponent obtained in this study agreed well with that derived from the MonteCarlo simulation based on the Q-state planar Potts model, the difference in thef exponent obtained experimentally and theoretically was attributed to the depolarizingeffects presented in the bulk ceramics.

Crystal vs. glass formation in lattice models with many coexisting ordered phases

We present here new evidence that after a quench the planar Potts model on the square lattice relaxes towards a glassy state if the number of states q is larger than four. By extrapolating the finite size data we compute the average energy of these states for the infinite system with periodic boundary conditions, and find that it is comparable with that previously found using fixed boundary conditions. We also report preliminary results on the behaviour of these states in the presence of thermal fluctuations.

Glassy behaviour in short-range lattice models without quenched disorder

We investigate the quenching process in lattice systems with short range interaction and several crystalline states as ground states. We consider in particular the following systems on square lattice: - hard particle (exclusion) model; - q states planar Potts model. The system is initially in a homogeneous disordered phase and relaxes toward a new equilibrium state as soon as the temperature is rapidly lowered. The time evolution can be described numerically by a stochastic process such as the Metropolis algorithm. The number of pure, equivalent, ground states is q for the Potts model and r for the hard particle model, and it is known that for r or q larger or equal to d+1, the final equilibrium state may be polycrystalline, i.e. not made of a uniform phase. We find that in addition n_g and q_g exist such that for r > r_g, or q > q_g the system evolves toward a glassy state, i.e. a state in which the ratio of the interaction energy among the different crystalline phases to the total energy of the system never vanishes; moreover we find indications that r_g=q_g. We infer that q=q_g (and r=r_g) corresponds to the crossing from second order to discontinuous transition in the phase diagram of the system.

Glassy behaviour in short-range lattice models without quenched disorder

We investigate the quenching process in lattice systems with short range interaction and several crystalline states as ground states. We consider in particular the following systems on square lattice: - hard particle (exclusion) model; - q states planar Potts model. The system is initially in a homogeneous disordered phase and relaxes toward a new equilibrium state as soon as the temperature is rapidly lowered. The time evolution can be described numerically by a stochastic process such as the Metropolis algorithm. The number of pure, equivalent, ground states is q for the Potts model and r for the hard particle model, and it is known that for r or q larger or equal to d+1, the final equilibrium state may be polycrystalline, i.e. not made of a uniform phase. We find that in addition n_g and q_g exist such that for r > r_g, or q > q_g the system evolves toward a glassy state, i.e. a state in which the ratio of the interaction energy among the different crystalline phases to the total energy of the system never vanishes; moreover we find indications that r_g=q_g. We infer that q=q_g (and r=r_g) corresponds to the crossing from second order to discontinuous transition in the phase diagram of the system.

Scaling behavior of dynamic hysteresis in multi-domain spin systems

The dynamic hysteresis in two-dimensional multi-domain spin systems is simulated using Monte-Carlo method based on Q-state planar Potts model. The pattern and scaling property of the hysteresis as a function of the oscillating time-varying external field are studied and compared qualitatively with the experimental data on PZT ferroelectric films and theoretical predictions. While the scaling of the hysteresis area with frequency f and amplitude E 0 of the external field at a form of f 1/3 E 0 2/3 over the low- f range is well-established, the high- f range scaling relations as derived from the simulation, experiment and theoretical model differ from one and another.

Some possible effects of periodic pinning in high Tcsuperconductors

We are assuming that the pinning points could be created in the center of each unit cell of the planar superconducting lattice. The existence of periodic pinning suggests a particular dynamics of vortices in the mixed state of 2D and 3D high Tcsuperconductors. Such a moving mechanism may be studied using a planar Potts model. We used such a model to perform some numerical simulations focusing on the melting of the Abrikosov lattice of vortices in a system of Josephson arrays of superconducting grains. A Monte Carlo analysis on the resulting hamiltonian in a field given by a starting filling factor off=1/2 exhibits two phase transitions, in both 2D and 3D. The 2D model gives the critical temperatures[formula]and[formula]. The~3D one gives[formula]and[formula]. The numerical estimation of the Challa parameters for both transitions indicates their continuous nature in both 2D and 3D versions of this model. The results suggest that, below Tc1, the superconductor is in the Meissner phase, between Tc1and Tmit is in the mixed state and above Tmit is in the melted flux state, i.e. technological normal state.

Interfacial adsorption in planar potts models

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).

High-field Polynomial Expansions for the Six-state Planar Potts Model

High-field polynomial expansions (through order 9) are derived for the six-state planar Potts model and analysed. Suggestive evidence is found for an intermediate phase in which the exponent 0 varies continuously as a function of temperature. This behaviour is consistent with recent results predicting that the model exhibits two transitions separating a 'topologically ordered' phase analogous to that found in the planar rotor model.

Phase boundary for planar site-bond percolation problems from a generalised star-triangle transformation

On the basis of a generalised star-triangle transformation and of existing conjectures concerning the critical point of some planar Potts models, the phase boundary is determined for the combined site-bond percolation problem on a number of lattices, including the triangular and the honeycomb lattice. As a special case, the critical probability for the honeycomb site problem is found to be pc-1/ square root 2.

Critical point of planar Potts models

The critical point of the Potts model is conjectured for the following planar lattices: (i) generalised (chequerboard) square lattice; (ii) Kagome lattice; (iii) triangular lattice with two- and three-site interactions. As a result, the critical probability for bond percolation on the Kagome lattice is determined to be pc=0.524430.