Potts Model Research Articles

Metamagnetic anomalies in the kinetic Ashkin–Teller model

This study investigates the metamagnetic anomaly in the Ashkin–Teller spin-1/2 model in a ferromagnetic state, utilizing dynamic mean-field theory on a cubic lattice from a stochastic theory. We analyze instantaneous magnetizations over time, dynamic order parameters, and magnetic susceptibilities concerning the constant bias field, as well as examining dynamic phase diagrams. Our observations reveal that dynamic phase transitions occur at specific points of the constant bias field, where instantaneous magnetization transitions from oscillating between negative and positive states to a strictly positive state. We also observe a narrow region in the dynamic phase diagram between phase transitions of different order parameters of the model, for some conditions among the couplings of these order parameters. These results contribute to an understanding of the dynamic behavior of the spin-1/2 AT model under ferromagnetic conditions, providing interesting insights into phase transitions and the metamagnetic anomaly.

Quantifying nonuniversal corner free-energy contributions in weakly anisotropic two-dimensional critical systems.

We derive an exact formula for the corner free-energy contribution of weakly anisotropic two-dimensional critical systems in the Ising universality class on rectangular domains, expressed in terms of quantities that specify the anisotropic fluctuations. The resulting expression agrees with numerical exact calculations that we perform for the anisotropic triangular Ising model and quantifies the nonuniversality of the corner term for anisotropic critical two-dimensional systems. Our generic formula is expected to apply also to other weakly-anisotropic critical two-dimensional systems that allow for a conformal field theory description in the isotropic limit. We consider the three-states and four-states Potts models as further specific examples.

CTMRG study of the critical behavior of an interacting-dimer model

The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.

Perfect sampling of $q$-spin systems on $\mathbb{Z}^{2}$ via weak spatial mixing

We present a perfect marginal sampler of the unique Gibbs measure of a spin system on \mathbb{Z}^{2} . The algorithm is an adaptation of a previous “lazy depth-first” approach by the authors, but relaxes the requirement of strong spatial mixing to weak. Our result is a step towards methods of efficient sampling using only weak spatial mixing. The work exploits a classical result in statistical physics relating weak spatial mixing on \mathbb{Z}^{2} to strong spatial mixing on squares. When the spin system exhibits weak spatial mixing, the run-time of our sampler is linear in the size of sample. Applications of note are the ferromagnetic Potts model at supercritical temperatures and the ferromagnetic Ising model with consistent non-zero external field at any non-zero temperature.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

Simulation of the Five-Component Potts Model on Triangular Lattice by the Monte Carlo Method in Pure and Diluted Modes

Simulation of the Five-Component Potts Model on Triangular Lattice by the Monte Carlo Method in Pure and Diluted Modes

Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model

Topological/perfectly-transmissive defects play a fundamental role in the analysis of the symmetries of two dimensional conformal field theories (CFTs). In the present work, spin chain regularizations for these defects are proposed and analyzed in the case of the three-state Potts CFT. In particular, lattice versions for all the primitive defects are presented, with the remaining defects obtained from the fusion of the primitive ones. The defects are obtained by introducing modified interactions around two given sites of an otherwise homogeneous spin chain with periodic boundary condition. The various primitive defects are topological on the lattice except for one, which is topological only in the scaling limit. The lattice models are analyzed using a combination of exact diagonalization and density matrix renormalization group techniques. Low-lying energy spectra for different defect Hamiltonians as well as entanglement entropy of blocks located symmetrically around the defects are computed. The latter provides a convenient way to compute the g-function which characterizes various defects. Finally, the eigenvalues of the line operators in the “crossed channel” and fusion of different defect lines are also analyzed. The results are all in agreement with expectations from conformal field theory.

Dynamical critical behavior of the two-dimensional three-state Potts model.

We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.

Simulation of Soluble and Bound VEGF-stimulated in vitro Capillary-like Network Formation on Deformed Substrate.

Capillary plexus cultivation is crucial in tissue engineering and regenerative medicine. Theoretical simulations have been conducted to supplement the expensive experimental works. However, the mechanisms connecting mechanical and chemical stimuli remained undefined, and the functions of the different VEGF forms in the culture environment were still unclear. In this paper, we developed a hybrid model for simulating short-term in vitro capillary incubations. We used the Cellular Potts model to predict individual cell migration, morphology change, and continuum mechanics to quantify biogel deformation and VEGF transport dynamics. By bridging the mechanical regulation and chemical stimulation in the model, the results showed good agreement between the predicted network topology and experiments, in which elongated cells connected, forming the network cords and round cells gathered, creating cobblestone-like aggregates. The results revealed that the capillary-like networks could develop in high integrity only when the mechanical and chemical couplings worked adequately, with the cell morphology and haptotaxis driven by the soluble and bound forms of VEGF, respectively, functioning simultaneously.

Modelling the effect of cell motility on mixing and invasion in epithelial monolayers

Collective cell invasion underlies several biological processes such as wound healing, embryonic development, and cancerous invasion. Here, we investigate the impact of cell motility on invasion in epithelial monolayers and its coupling to cellular mechanical properties, such as cell-cell adhesion and cortex contractility. We develop a two-dimensional computational model for cells with active motility based on the cellular Potts model, which predicts that the cellular invasion speed is mainly determined by active cell motility and is independent of the biological and mechanical properties of the cells. We also find that, in general, motile cells out-compete and invade non-motile cells, however, this can be reversed by differential cell proliferation. Stable coexistence of motile and static cell types is also possible for certain parameter regimes.

Critical behaviors of non-stabilizerness in quantum spin chains

Non-stabilizerness – commonly known as magic – measures the extent to which a quantum state deviates from stabilizer states and is a fundamental resource for achieving universal quantum computation. In this work, we investigate the behavior of non-stabilizerness around criticality in quantum spin chains. To quantify non-stabilizerness, we employ a monotone called mana, based on the negativity of the discrete Wigner function. This measure captures non-stabilizerness for both pure and mixed states. We introduce Rényi generalizations of mana, which are also measures of non-stabilizerness for pure states, and utilize it to compute mana in large quantum systems. We consider the three-state Potts model and its non-integrable extension and we provide strong evidence that the mutual mana exhibits universal logarithmic scaling with distance in conformal field theory, as is the case for entanglement.

Cycling and spiral-wave modes in an active cyclic Potts model.

We studied the nonequilibrium dynamics of a cycling three-state Potts model using simulations and theory. This model can be tuned from thermal-equilibrium to far-from-equilibrium conditions. At low cycling energy, the homogeneous dominant state cycles via nucleation and growth, while spiral waves are formed at high energy. For large systems, a discontinuous transition occurs from these cyclic homogeneous phases to spiral waves, while the opposite transition is absent. Conversely, these two modes can coexist for small systems. The waves can be reproduced by a continuum theory, and the transition can be understood from the competition between nucleation and growth.

Investigation of Partition Function Transformation for the Potts Model into a Dichromatic Knot Polynomial 74

Investigation of Partition Function Transformation for the Potts Model into a Dichromatic Knot Polynomial 74

Phase Transitions in the Four-Component Potts Model on a Triangular Lattice

Phase Transitions in the Four-Component Potts Model on a Triangular Lattice

APW: An ensemble model for efficient wheat spike counting in unmanned aerial vehicle images

The accurate estimation of wheat ear is of great significance to gain high yield for ensuring food security. The use of computer vision for wheat ear counting to improve the estimation efficiency and accuracy has been explored by many scholars. However, existing single-computer vision recognition methods have insufficient robustness and recognition of adherent wheat ears. This study mainly focused on to solve the problem of adherent wheat ears, and proposed a combined algorithm called “APW” that ensemble with alternating direction methods of multipliers (ADMM), Potts algorithm and Watershed to achieve fast recognition and counting of wheat ears. The images of wheat ears were collected under three different scenarios including growth stages, planting densities, and flight heights by using unmanned aerial vehicle (UAV), and the Potts model was used to recognise the images. The masked images of the wheat ear were converted into a greyscale image, and were layered using a greyscale gradient. Finally, the watershed algorithm was used to segment adherent wheat ears in the layered image, and the centroids were labelled and counted. The results shows that the best accuracy was obtained under low planting density scenario, with an R2 of 0.89 and a root mean square error (RMSE) of 3.72, suggesting that APW can handle scenarios with large background differences and low target adherence. These findings provides a new approach for efficient counting of wheat ears in UAV acquired images.

Machine learning in biological physics: From biomolecular prediction to design

Machine learning has been proposed as an alternative to theoretical modeling when dealing with complex problems in biological physics. However, in this perspective, we argue that a more successful approach is a proper combination of these two methodologies. We discuss how ideas coming from physical modeling neuronal processing led to early formulations of computational neural networks, e.g., Hopfield networks. We then show how modern learning approaches like Potts models, Boltzmann machines, and the transformer architecture are related to each other, specifically, through a shared energy representation. We summarize recent efforts to establish these connections and provide examples on how each of these formulations integrating physical modeling and machine learning have been successful in tackling recent problems in biomolecular structure, dynamics, function, evolution, and design. Instances include protein structure prediction; improvement in computational complexity and accuracy of molecular dynamics simulations; better inference of the effects of mutations in proteins leading to improved evolutionary modeling and finally how machine learning is revolutionizing protein engineering and design. Going beyond naturally existing protein sequences, a connection to protein design is discussed where synthetic sequences are able to fold to naturally occurring motifs driven by a model rooted in physical principles. We show that this model is “learnable” and propose its future use in the generation of unique sequences that can fold into a target structure.

Poissonian Cellular Potts Models Reveal Nonequilibrium Kinetics of Cell Sorting

Cellular Potts models are broadly applied across developmental biology and cancer research. We overcome limitations of the traditional approach, which reinterprets a modified Metropolis sampling as dynamics, by introducing a physical timescale through Poissonian kinetics and by applying principles of stochastic thermodynamics to separate thermal and relaxation effects from athermal noise and nonconservative forces. Our method accurately describes cell-sorting dynamics in mouse-embryo development and identifies the distinct contributions of nonequilibrium processes, e.g., cell growth and active fluctuations. Published by the American Physical Society 2024

On the tractability of sampling from the Potts model at low temperatures via random-cluster dynamics

On the tractability of sampling from the Potts model at low temperatures via random-cluster dynamics

Weakening and disappearance of the jaming behavior in systems of self-propelled particles

In this study, we utilize the Eigen Microstate Approach (EMA) to investigate the active four-state Potts model, with a focus on the impact of volume repulsion. Our findings reveal that the introduction of long-range interactions significantly weakens or completely eliminates the jamming state, as indicated by changes in the probability curve of the second-largest eigen microstate. The emergence of sub-band structures in the system correlates with the appearance of a two-peak structure in the probability curve. We construct a phase diagram to explore the complex interplay between long-range interactions and emergent system behaviors. Additionally, we conjecture that the active four-state Potts model with long-range interactions undergoes a transition higher than second order.

Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree

Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree