Showing papers on "Potts model published in 2016"

Quantum gravity and inventory accumulation

Abstract: We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on $\mathbb{Z}^{2}$. In more interesting versions, a $p$ fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on $p$. We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each $q>0$, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the $q$-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on $q$. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at $p=1/2$, $q=4$.

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

Abstract: We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(\pi^2/\sqrt{q-4})$ as $q$ tends to 4.

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Abstract: Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite $\Delta$-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree $\Delta$. To this end, we first present a detailed picture for the phase diagram for the infinite $\Delta$-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and order...

A conformal bootstrap approach to critical percolation in two dimensions

Abstract: We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

Structural phase transition in perovskite metal-formate frameworks: a Potts-type model with dipolar interactions.

Abstract: We propose a combined experimental and numerical study to describe an order–disorder structural phase transition in perovskite-based [(CH3)2NH2][M(HCOO)3] (M = Zn2+, Mn2+, Fe2+, Co2+ and Ni2+) dense metal–organic frameworks (MOFs). The three-fold degenerate orientation of the molecular (CH3)2NH2+ (DMA+) cation implies a selection of the statistical three-state model of the Potts type. It is constructed on a simple cubic lattice where each lattice point can be occupied by a DMA+ cation in one of the available states. In our model the main interaction is the nearest-neighbor Potts-type interaction, which effectively accounts for the H-bonding between DMA+ cations and M(HCOO)3− cages. The model is modified by accounting for the dipolar interactions which are evaluated for the real monoclinic lattice using density functional theory. We employ the Monte Carlo method to numerically study the model. The calculations are supplemented with the experimental measurements of electric polarization. The obtained results indicate that the three-state Potts model correctly describes the phase transition order in these MOFs, while dipolar interactions are necessary to obtain better agreement with the experimental polarization. We show that in our model with substantial dipolar interactions the ground state changes from uniform to the layers with alternating polarization directions.

Metastability of Non-reversible, Mean-Field Potts Model with Three Spins

Abstract: We examine a non-reversible, mean-field Potts model with three spins on a set with $$N\uparrow \infty $$ points. Without an external field, there are three critical temperatures and five different metastable regimes. The analysis can be extended by a perturbative argument to the case of small external fields, and it can be carried out in the case where the external field is in the direction or in the opposite direction to one of the values of the spins. Numerical computations permit to identify other phenomena which are not present in the previous situations.

Mixing of the Glauber dynamics for the ferromagnetic Potts model.

Abstract: We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely-matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ-regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Phase transition and chaos: P-adic Potts model on a Cayley tree

Abstract: In our previous investigations, we have developed the renormalization group method to p-adic models on Cayley trees, this method is closely related to the investigation of dynamical system associated with a given model. In this paper, we are interested in the following question: how is the existence of the phase transition related to chaotic behavior of the associated dynamical system (this is one of the important question in physics)? To realize this question, we consider as a toy model the p-adic q-state Potts model on a Cayley tree, and show, in the phase transition regime, the associated dynamical system is chaotic, i.e. it is conjugate to the full shift. As an application of this result, we are able to show the existence of periodic (with any period) p-adic quasi Gibbs measures for the model. This allows us to know that how large is the class of p-adic quasi Gibbs measures. We point out that a similar kind of result is not known in the case of real numbers.

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

Abstract: We prove that random-cluster models with $$q \ge 1$$ 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.

Graphical Representations for Ising and Potts Models in General External Fields

Abstract: This work is concerned with the theory of graphical representation for the Ising and Potts models over general lattices with non-translation invariant external field. We explicitly describe in terms of the random-cluster representation the distribution function and, consequently, the expected value of a single spin for the Ising and q-state Potts models with general external fields. We also consider the Gibbs states for the Edwards–Sokal representation of the Potts model with non-translation invariant magnetic field and prove a version of the FKG inequality for the so called general random-cluster model (GRC model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the almost sure quasilocality of the Gibbs measures for the GRC model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs measures for the ferromagnetic Ising model with a positive power-law decay magnetic field with small enough power, as conjectured in Bissacot et al. (Commun Math Phys 337: 41–53, 2015).

Potts-model critical manifolds revisited

Abstract: We compute critical polynomials for the q-state Potts model on the Archimedean lattices, using a parallel implementation of the algorithm of Jacobsen (2014 J. Phys. A: Math. Theor 47 135001) that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6 × 6 unit cells, and the root in the temperature variable is determined numerically at q = 1 for bases of size 8 × 8. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real (q, v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker–Kadanoff phase for the lattices studied.

Interplay between sign problem and Z 3 symmetry in three-dimensional Potts models

Abstract: We construct four kinds of ${Z}_{3}$-symmetric three-dimensional (3D) Potts models, each with a different number of states at each site on a 3D lattice, by extending the 3D 3-state Potts model. Comparing the ordinary Potts model with the four ${Z}_{3}$-symmetric Potts models, we investigate how ${Z}_{3}$ symmetry affects the sign problem and see how the deconfinement transition line changes in the $\ensuremath{\mu}\text{\ensuremath{-}}\ensuremath{\kappa}$ plane as the number of states increases, where $\ensuremath{\mu}$ ($\ensuremath{\kappa}$) plays a role of chemical potential (temperature) in the models. We find that the sign problem is almost cured by imposing ${Z}_{3}$ symmetry. This mechanism may happen in ${Z}_{3}$-symmetric QCD-like theory. We also show that the deconfinement transition line has stronger $\ensuremath{\mu}$ dependence with respect to increasing the number of states.

Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree

Abstract: We consider the Potts model with competing interactions on the Cayley tree of order k with k ≥ 2. We describe the sets of periodic and weakly periodic ground states corresponding to normal subgroups of the group representation of the Cayley tree of index 4.

On periodic Gibbs measures of p -adic Potts model on a Cayley tree

Abstract: In the present paper, we study the existence of periodic p-adic quasi Gibbs measures of p-adic Potts model over the Cayley tree of order two. We first prove that the renormalized dynamical system associated with the model is conjugate to the symbolic shift. As a consequence of this result we obtain the existence of countably many periodic p-adic Gibbs measures for the model.

Extraction of Active Regions and Coronal Holes from EUV Images Using the Unsupervised Segmentation Method in the Bayesian Framework

Abstract: The solar corona is the origin of very dynamic events that are mostly produced in active regions (AR) and coronal holes (CH). The exact location of these large-scale features can be determined by applying image-processing approaches to extreme-ultraviolet (EUV) data. We here investigate the problem of segmentation of solar EUV images into ARs, CHs, and quiet-Sun (QS) images in a firm Bayesian way. On the basis of Bayes’ rule, we need to obtain both prior and likelihood models. To find the prior model of an image, we used a Potts model in non-local mode. To construct the likelihood model, we combined a mixture of a Markov–Gauss model and non-local means. After estimating labels and hyperparameters with the Gibbs estimator, cellular learning automata were employed to determine the label of each pixel. We applied the proposed method to a Solar Dynamics Observatory/Atmospheric Imaging Assembly (SDO/AIA) dataset recorded during 2011 and found that the mean value of the filling factor of ARs is 0.032 and 0.057 for CHs. The power-law exponents of the size distribution of ARs and CHs were obtained to be −1.597 and −1.508, respectively, with the maximum likelihood estimator method. When we compare the filling factors of our method with a manual selection approach and the SPoCA algorithm, they are highly compatible.

The free energy in a class of quantum spin systems and interchange processes

Abstract: We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S=12, the model is the Heisenberg ferromagnet, and for general spin S∈12N, it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Toth and by Penrose when S=12). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.

Symmetry breaking and the geometry of reduced density matrices

Abstract: The concept of symmetry breaking and the emergence of corresponding local order parameters constitute the pillars of modern day many body physics. We demonstrate that the existence of symmetry breaking is a consequence of the geometric structure of the convex set of reduced density matrices of all possible many body wavefunctions. The surfaces of these convex bodies exhibit non-analyticities, which signal the emergence of symmetry breaking and of an associated order parameter and also show different characteristics for different types of phase transitions. We illustrate this with three paradigmatic examples of many body systems exhibiting symmetry breaking: the quantum Ising model, the classical q-state Potts model in two-dimensions at finite temperature and the ideal Bose gas in three-dimensions at finite temperature. This state based viewpoint on phase transitions provides a unique novel tool for studying exotic many body phenomena in quantum and classical systems.

A conformal bootstrap approach to critical percolation in two dimensions

Abstract: We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

Q-colourings of the triangular lattice: Exact exponents and conformal field theory

Abstract: We revisit the problem of Q-colourings of the triangular lattice using a mapping onto an integrable spin-one model, which can be solved exactly using Bethe Ansatz techniques. In particular we focus on the low-energy excitations above the eigenlevel g2, which was shown by Baxter to dominate the transfer matrix spectrum in the Fortuin- Kasteleyn (chromatic polynomial) representation for Q0 Q 4, where Q0 = 3:819 671 . We argue that g2 and its scaling levels dene a conformally invariant theory, the so-called regime IV, which provides the actual description of the (analytically continued) colouring problem within a much wider range, namely Q 2 (2; 4). The corresponding conformal eld theory is identied and the exact critical exponents are derived. We discuss their implications for the phase diagram of the antiferromagnetic triangular-lattice Potts model at non-zero temperature. Finally, we relate our results to recent observations in the eld of spin-one anyonic chains.

On Weakly Periodic Gibbs Measures for the Potts Model with External Field on the Cayley Tree

Abstract: We study the Potts model with external field on a Cayley tree of order k ≥ 2. For the antiferromagnetic Potts model with external field and k ≥ 6 and q ≥ 3, it is shown that the weakly periodic Gibbs measure, which is not periodic, is not unique. For the Potts model with external field equal to zero, we also study weakly periodic Gibbs measures. It is shown that, under certain conditions, the number of these measures cannot be smaller than 2 q − 2.

Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models

Abstract: Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the $q$-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 $XXX$ and anisotropic $XYX$ models in an external magnetic field, and the antiferromagnetic spin-1 $XXZ$ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.

Emergent O( n ) symmetry in a series of three-dimensional Potts models

Abstract: Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-${T}_{c}$ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of $q$-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O($n$) symmetry with $n$=$q$-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O($n$=$q$-1) universality classes.

Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree

Abstract: We study translation-invariant Gibbs measures on a Cayley tree of order k = 3 for the ferromagnetic three-state Potts model. We obtain explicit formulas for translation-invariant Gibbs measures. We also consider periodic Gibbs measures on a Cayley tree of order k for the antiferromagnetic q-state Potts model. Moreover, we improve previously obtained results: we find the exact number of periodic Gibbs measures with the period two on a Cayley tree of order k ≥ 3 that are defined on some invariant sets.

Correlation inequalities for the Potts model

Abstract: Correlation inequalities are presented for ferromagnetic Potts models with external field, using the random cluster representation of Fortuin and Kasteleyn, together with the FKG inequality. These results extend and simplify earlier inequalities of Ganikhodjaev and Razak, and also of Schonmann, and include GKS-type inequalities when the spin-space is taken as the set of qth roots of unity.

Density-of-states

Abstract: Although Monte Carlo calculations using Importance Sampling have matured into the most widely employed method for determining first principle results in QCD, they spectacularly fail for theories with a sign problem or for which certain rare configurations play an important role Non-Markovian Random walks, based upon iterative refinements of the density-of-states, overcome such overlap problems I will review the Linear Logarithmic Relaxation (LLR) method and, in particular, focus onto ergodicity and exponential error suppression Applications include the high-state Potts model, SU(2) and SU(3) Yang-Mills theories as well as a quantum field theory with a strong sign problem: QCD at finite densities of heavy quarks