Potts Model Research Articles

The fuzzy Potts model in the plane: scaling limits and arm exponents

AbstractWe consider a critical Fortuin–Kasteleyn (FK) percolation with cluster weight $$q \in [1,4)$$ q ∈ [ 1 , 4 ) in the plane, and color its clusters in red (respectively blue) with probability $$r \in (0,1)$$ r ∈ ( 0 , 1 ) (respectively $$1-r$$ 1 - r ), independently of each other. We study the resulting fuzzy Potts model, which corresponds to the critical Ising model in the special case $$q=2$$ q = 2 and $$r=1/2$$ r = 1 / 2 . We show that under the assumption that the critical FK percolation converges to a conformally invariant scaling limit (which is known to hold for the FK-Ising model,i.e. $$q=2$$ q = 2 ), the obtained coloring converges to variants of Conformal Loop Ensembles constructed, described and studied by Miller, Sheffield and Werner. Based on discrete considerations, we also show that the arm exponents for this coloring in the discrete model are identical to the ones of the continuum model. Using the values of these arm exponents in the continuum, we determine the arm exponents for the fuzzy Potts model.

Robust Teleportation of a Surface Code and Cascade of Topological Quantum Phase Transitions

Teleportation is a facet where quantum measurements can act as a powerful resource in quantum physics, as local measurements allow us to steer quantum information in a nonlocal way. While this has long been established for a single Bell pair, the teleportation of a many-qubit entangled state using nonmaximally entangled resources presents a fundamentally different challenge. Here, we investigate a tangible protocol for teleporting a long-range entangled surface-code state using elementary Bell measurements and its stability in the presence of coherent errors that weaken the Bell entanglement. We relate the underlying threshold problem to the physics of anyon condensation under weak measurements and map it to a variant of the Ashkin-Teller model of statistical mechanics with Nishimori-type disorder, which gives rise to a cascade of phase transitions. Tuning the angle of the local Bell measurements, we find a continuously varying threshold. Notably, the threshold moves to infinity for the X+Z angle along the self-dual line—indicating that infinitesimally weak entanglement is sufficient in teleporting a self-dual topological surface code. Our teleportation protocol, which can be readily implemented in dynamically configurable Rydberg-atom arrays, thereby gives guidance for a practical demonstration of the power of quantum measurements. Published by the American Physical Society 2024

Small-amplitude synchronization in driven Potts models

We study driven q-state Potts models with thermodynamically consistent dynamics and global coupling. For a wide range of parameters, these models exhibit a dynamical phase transition from decoherent oscillations into a synchronized phase. Starting from a general microscopic dynamics for individual oscillators, we derive the normal form of the high-dimensional Hopf bifurcation that underlies the phase transition. The normal-form equations are exact in the thermodynamic limit and close to the bifurcation. Exploiting the symmetry of the model, we solve these equations and thus uncover the intricate stable synchronization patterns of driven Potts models, characterized by a rich phase diagram. Making use of thermodynamic consistency, we show that synchronization reduces dissipation in such a way that the most stable synchronized states dissipate the least entropy. Close to the phase transition, our findings condense into a linear dissipation-stability relation that connects entropy production with phase-space contraction, a stability measure. At finite system size, our findings suggest a minimum-dissipation principle for driven Potts models that holds arbitrarily far from equilibrium. Published by the American Physical Society 2024

Magnetic Properties in the Spin-3/2 Ashkin-Teller Model from Mean Field Theory

Magnetic Properties in the Spin-3/2 Ashkin-Teller Model from Mean Field Theory

Investigating the evolution of U-10Mo fuel foil microstructures during multi-stage hot rolling using coupled potts model-finite element method simulations

In this work, we study the microstructural evolution of U-10Mo foils over multiple stages of hot-rolling and reheating using our previously validated method coupling the Kinetic Monte Carlo Potts Model with the finite element method. Hot rolling and reheating refine the U-10Mo foil microstructure, but the relationships between the foil microstructure and recrystallization behavior over multiple successive reductions have complex relationships with the rolling schedule that have not yet been well quantified. Simulations were employed to forecast the impact of hot rolling reduction per pass, grain size variations, and uranium carbide (UC) distribution on the Johnson Mehl Avrami Kolmogorov (JMAK) recrystallization kinetics of the U-10Mo alloy, as well as it's post-rolling grain growth kinetics. Initial homogenized grain sizes varying from 100 µm to 1 mm and UC volume fractions ranging from 0 to 2 vol% were parametrically evaluated as a part of this study. While some of our simulation results support the findings of our previous analysis for conditions of single-pass rolling and annealing, our extended analysis shows that the grain size within the as-cast and homogenized foil can lead to significant changes in the in the distribution of strain within the final microstructure, which can slow grain coarsening over the final anneal. The magnitude of the hot rolling reduction per pass had a similarly strong impact on the distribution of strain within the foil microstructure and its subsequent grain growth behavior. The implications of these results on U-10Mo fuel foil fabrication procedures are discussed.

From in vitro to in silico: a pipeline for generating virtual tissue simulations from real image data.

3D cell culture models replicate tissue complexity and aim to study cellular interactions and responses in a more physiologically relevant environment compared to traditional 2D cultures. However, the spherical structure of these models makes it difficult to extract meaningful data, necessitating advanced techniques for proper analysis. In silico simulations enhance research by predicting cellular behaviors and therapeutic responses, providing a powerful tool to complement experimental approaches. Despite their potential, these simulations often require advanced computational skills and significant resources, which creates a barrier for many researchers. To address these challenges, we developed an accessible pipeline using open-source software to facilitate virtual tissue simulations. Our approach employs the Cellular Potts Model, a versatile framework for simulating cellular behaviors in tissues. The simulations are constructed from real world 3D image stacks of cancer spheroids, ensuring that the virtual models are rooted in experimental data. By introducing a new metric for parameter optimization, we enable the creation of realistic simulations without requiring extensive computational expertise. This pipeline benefits researchers wanting to incorporate computational biology into their methods, even if they do not possess extensive expertise in this area. By reducing the technical barriers associated with advanced computational modeling, our pipeline enables more researchers to utilize these powerful tools. Our approach aims to foster a broader use of in silico methods in disease research, contributing to a deeper understanding of disease biology and the refinement of therapeutic interventions.

Exploring transitions in finite-size Potts model: comparative analysis using Wang–Landau sampling and parallel tempering

Abstract This research provides a examination of transitions within the various-state Potts model in two-dimensional finite-size lattices. Leveraging the Wang–Landau sampling and parallel tempering, we systematically obtain the density of states, facilitating a comprehensive comparative analysis of the results. The determination of the third-order transitions location are achieved through a meticulous examination of the density of states using microcanonical inflection-point analysis. The remarkable alignment between canonical and microcanonical results for higher-order transition locations affirms the universality of these transitions. Our results further illustrate the universality of the robust and microcanonical inflection-point analysis of Wang–Landau sampling.

Competition between microstructural factors affecting growth of abnormally large grains in thin Cu foils

Grain boundary types and local boundary curvatures are generally considered to be important microstructural factors controlling grain boundary migration during grain growth. In this work, grain growth in thin copper foils is studied during annealing at a temperature of 1040 °C near the melting point by ex-situ experiments and Monte Carlo simulations. Few grains, stimulated by slight deformation at the sample edge, are observed to grow abnormally into the cube-oriented recrystallized microstructure with columnar grains spanning the foil thickness. The grain boundaries of these abnormally growing grains and the grain sizes in the adjacent polycrystalline recrystallized regions are analyzed. The experimental results suggest that spatial heterogeneities in the distribution of small recrystallized grains have a significant effect on the migrating boundaries. Potts model simulations confirm that grain boundary segments with small grains in front are more likely to migrate than segments facing coarser grains. The simulations also demonstrate the importance of grain morphology. Altogether, this work highlights the effects of a heterogeneous recrystallized microstructure on abnormal grain growth in thin foil samples.

Confining strings in three-dimensional gauge theories beyond the Nambu-Gotō approximation

We carry out a systematic study of the effective bosonic string describing confining flux tubes in SU(N) Yang-Mills theories in three spacetime dimensions. While their low-energy properties are known to be universal and are described well by the Nambu-Gotō action, a non-trivial dependence on the gauge group is encoded in a series of undetermined subleading corrections in an expansion around the limit of an arbitrarily long string. We quantify the first two of these corrections by means of high-precision Monte Carlo simulations of Polyakov-loop correlators in the lattice regularization. We compare the results of novel lattice simulations for theories with N = 3 and 6 color charges, and report an improved estimate for the N = 2 case, discussing the approach to the large-N limit. Our results are compatible with analytical bounds derived from the S-matrix bootstrap approach. In addition, we also present a new test of the Svetitsky-Yaffe conjecture for the SU(3) theory in three dimensions, finding that the lattice results for the Polyakov-loop correlation function are in excellent agreement with the predictions of the Svetitsky-Yaffe mapping, which are worked out quantitatively applying conformal perturbation theory to the three-state Potts model in two dimensions. The implications of these results are discussed.

Hyperuniformity in Ashkin–Teller model

We show that equilibrium systems inddimension that obey the inequalitydν>2,known as Harris criterion, exhibit suppressed energy fluctuation in their critical state. Ashkin-Teller model is an example ind = 2 where the correlation length exponentνvaries continuously with the inter-spin interaction strengthλand exceeds the valued2set by Harris criterion whenλis negative; there, the variance of the subsystem energy across a length scalelvaries asld-αwith hyperuniformity exponentα=2(1-ν-1).Point configurations constructed by assigning unity to the sites which has coarse-grained energy beyond a threshold value also exhibit suppressed number fluctuation and hyperuniformiyty with same exponentα.

Lattice Realization of Complex Conformal Field Theories: Two-Dimensional Potts Model with Q>4 States.

The two-dimensional Q-state Potts model with real couplings has a first-order transition for Q>4. We study a loop-model realization in which Q is a continuous parameter. This model allows for the collision of a critical and a tricritical fixed point at Q=4, which then emerge as complex conformally invariant theories at Q>4, or even complex Q, for suitable complex coupling constants. All critical exponents can be obtained as analytic continuation of known exact results for Q≤4. We verify this scenario in detail for Q=5 using transfer-matrix computations.

Power Consumption Comparison of GPU Linear Solvers for Cellular Potts Model Simulations

Power Consumption Comparison of GPU Linear Solvers for Cellular Potts Model Simulations

Statistical mechanics of stochastic quantum control: d-adic Rényi circuits.

The dynamics of quantum information in many-body systems with large onsite Hilbert space dimension admits an enlightening description in terms of effective statistical mechanics models. Motivated by this fact, we reveal a connection between three separate models: the classically chaotic d-adic Rényi map with stochastic control, a quantum analog of this map for qudits, and a Potts model on a random graph. The classical model and its quantum analog share a transition between chaotic and controlled phases, driven by a randomly applied control map that attempts to order the system. In the quantum model, the control map necessitates measurements that concurrently drive a phase transition in the entanglement content of the late-time steady state. To explore the interplay of the control and entanglement transitions, we derive an effective Potts model from the quantum model and use it to probe information-theoretic quantities that witness both transitions. The entanglement transition is found to be in the bond-percolation universality class, consistent with other measurement-induced phase transitions, while the control transition is governed by a classical random walk. These two phase transitions can be made to coincide by varying a parameter in the model, producing a picture consistent with behavior observed in previous small-size numerical studies of the quantum model.

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.

Entropy-induced phase transitions in a hidden Potts model.

A hidden state in which a spin does not interact with any other spin contributes to the entropy of an interacting spin system. We explore the q-state Potts model with extra r hidden states using the Ginzburg-Landau formalism in the mean-field limit. We analytically demonstrate that when 1<q≤2, the model exhibits a rich phase diagram comprising a variety of phase transitions such as continuous, discontinuous, two types of successive, and hybrid transitions; moreover, several characteristics such as critical point, critical endpoint, and tricritical point are identified. The critical line and critical endline merge in a singular form at the tricritical point. That supercritical behavior emerging at the tricritical point is, to the best our knowledge, novel. We microscopically investigate the origin of the discontinuous transition; it is induced by the competition between the interaction and entropy of the system in the Ising limit, whereas by the bistability of the hidden spin states in the percolation limit. Finally, we discuss the similarity and difference of the phase diagram generated in two types of Ising spins model.

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