Phase Transition Of Potts Model Research Articles

Study of phase transition of Potts model with Domain Adversarial Neural Network

A transfer learning method, Domain Adversarial Neural Network (DANN), is introduced to study the phase transition of two-dimensional q-state Potts model. With the DANN, we only need to choose a few labeled configurations automatically as input data, then the critical points can be obtained after training the algorithm. By an additional iterative process, the critical points can be captured to comparable accuracy to Monte Carlo simulations as we demonstrate it for q=3,4,5,7 and 10. The type of phase transition (first or second-order) is also determined at the same time. Meanwhile, for the second-order phase transition at q=3, we can calculate the critical exponent ν by data collapse. Furthermore, compared to the traditional supervised learning, we found the DANN to be more accurate with lower cost.

Information flow in first-order potts model phase transition

Phase transitions abound in nature and society, and, from species extinction to stock market collapse, their prediction is of widespread importance. In earlier work we showed that Global Transfer Entropy, a general measure of information flow, was found to peaks away from the transition on the disordered side for the Ising model, a canonical second order transition. Here we show that (a) global transfer entropy also peaks on the disordered side of the transition of finite first order transitions, such as ecology dynamics on coral reefs, which have latent heat and no correlation length divergence, and (b) analysis of information flow across state boundaries unifies both transition orders. We obtain the first information-theoretic result for the high-order Potts model and the first demonstration of early warning of a first order transition. The unexpected earlier finding that global transfer entropy peaks on the disordered side of a transition is also found for finite first order systems, albeit not in the thermodynamic limit. By noting that the interface length of clusters in each phase is the dominant region of information flow, we unify the information theoretic behaviour of first and second order transitions.

A comprehensive neural networks study of the phase transitions of Potts model

Using the techniques of neural networks (NN), we study the three-dimensional (3D) five-state ferromagnetic Potts model on the cubic lattice as well as the two-dimensional (2D) three-state antiferromagnetic Potts model on the square lattice. Unlike the conventional approach, here we follow the idea employed by Li et al (2018 Ann. Phys., NY 391 312–331). Specifically, instead of numerically generating numerous objects for the training, the whole or part of the theoretical ground state configurations of the studied models are considered as the training sets. Remarkably, our investigation of these two models provides convincing evidence for the effectiveness of the method of preparing training sets used in this study. In particular, the results of the 3D model obtained here imply that the NN approach is as efficient as the traditional method since the signal of a first order phase transition, namely tunneling between two channels, determined by the NN method is as strong as that calculated with the Monte Carlo technique. Furthermore, the outcomes associated with the considered 2D system indicate even little partial information of the ground states can lead to conclusive results regarding the studied phase transition. The achievements reached in our investigation demonstrate that the performance of NN, using certain amount of the theoretical ground state configurations as the training sets, is impressive.

Renormalization-group theory for cooling first-order phase transitions in Potts models.

We develop a dynamic field-theoretic renormalization-group (RG) theory for cooling first-order phase transitions in the Potts model. It is suggested that the well-known imaginary fixed points of the q-state Potts model for q>10/3 in the RG theory are the origin of the dynamic scaling found recently from numerical simulations, apart from logarithmic corrections. This indicates that the real and imaginary fixed points of the Potts model are both physical and control the scalings of the continuous and discontinuous phase transitions, respectively, of the model. Our one-loop results for the scaling exponents are already not far away from the numerical results. Further, the scaling exponents depend on q only slightly, consistent with the numerical results. Therefore, the theory is believed to provide a natural explanation of the dynamic scaling including the scaling exponents and their scaling laws for various observables in the cooling first-order phase transition of the Potts model.

A motivic approach to phase transitions in Potts models

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion–contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.

Partial Order and Finite-Temperature Phase Transitions in Potts Models on Irregular Lattices

We evaluate the thermodynamic properties of the 4-state antiferromagnetic Potts model on the Union-Jack lattice using tensor-based numerical methods. We present strong evidence for a previously unknown, "entropy-driven," finite-temperature phase transition to a partially ordered state. From the thermodynamics of Potts models on the diced and centered diced lattices, we propose that finite-temperature transitions and partially ordered states are ubiquitous on irregular lattices.

Percolation transition and distribution of connected components in generalized random network ensembles

In this work, we study the percolation transition and large deviation properties of generalized canonical network ensembles. This new type of random networks might have a very rich complex structure, including high heterogeneous degree sequences, non-trivial community structure or specific spatial dependence of the link probability for networks embedded in a metric space. We find the cluster distribution of the networks in these ensembles by mapping the problem to a fully connected Potts model with heterogeneous couplings. We show that the nature of the Potts model phase transition, linked to the birth of a giant component, has a crossover from second to first order when the number of critical colors qc = 2 in all the networks under study. These results shed light on the properties of dynamical processes defined on these network ensembles.

Dynamics of the 2d Potts model phase transition

The dynamics of 2d Potts models, which are temperature driven through the phase transition using updating procedures in the Glauber universality class, is investigated. We present calculations of the hysteresis for the (internal) energy and for Fortuin-Kasteleyn clusters. The shape of the hysteresis is used to define finite volume estimators of physical observables, which can be used to study the approach to the infinite volume limit. We compare with equilibrium configurations and the preliminary indications are that the dynamics leads to considerable alterations of the statistical properties of the configurations studied.

Phase transition of Potts model on a frustrated 3D lattice

We report here on Monte Carlo simulations of the q-state Potts model on a fully frustrated cubic geometry. Most investigations so far were restricted to the case of Ising, XY, or Heisenberg spins. Nothing is known about the fully frustrated Potts model and on the disorder effects in such systems. The effects of frustration and weak disorder on the q=3 and q=4 fully frustrated Potts system are studied in the present work.

Numerical study of phase transitions in Potts models.

A model is defined that interpolates between integer dimension and integer q-state Potts models. Phase transitions in these models are studied, and in particular the critical value ${\mathit{q}}_{\mathit{c}}$, where the phase transition changes from second to first order, is determined for several dimensions. For (d,${\mathit{q}}_{\mathit{c}}$) the values (2.0,4.05), (2.5,2.68), (3.0,2.21), (3.5,2.15), and (4.0,2.07) are obtained.

Gaussian ensemble: An alternate Monte Carlo scheme

The recently introduced Gaussian ensemble involves a sample (of size N) thermally connected to a finite heat bath (of size N') with specific properties. Treating N' as a parameter, we use a leading-order analysis of the \ensuremath{\beta} (inverse temperature) -versus-E (energy of sample) curves to show how static properties of finite samples become ensemble dependent. Inflection points in \ensuremath{\beta}(E) at phase transitions, however, appear as nontrivial fixed points with respect to N' and are defined as the transition temperature of the sample. By developing a fluctuation relation for the heat capacity C we show that, for small N', states with Cl0 are accessible at first-order transitions resulting in van der Waals loops in \ensuremath{\beta}(E). Monte Carlo studies of phase transitions in Potts models on two- and three-dimensional lattices confirm the finite-N' and finite-N effects. We find that the method significantly reduces computer time (sometimes by a factor of 100) compared with canonical-ensemble simulations and is effective in diagnosing the order of phase transition. Specific-heat data at second-order transitions reveal a new phenomenon; the peak in C sharpens as N' becomes smaller, leading us to speculate on sharp transitions in finite samples.

Gaussian ensemble as an interpolating ensemble.

We consider a noncanonical ensemble which involves a sample thermally connected to a finite heat bath with specific properties. Treating the size of the heat bath as a parameter, we show that static properties of finite samples are ensemble dependent. Monte Carlo simulations of phase transitions in Potts models confirm the analysis and demonstrate significant reductions in computer time (sometimes by a factor of 100) compared to canonical-ensemble simulations. For finite samples, second-order transitions are sharpened and are clearly distinguished from first-order transitions as the heat bath becomes smaller.

First- and second-order phase transitions in Potts models: Competing mechanisms (invited)

Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.

First- and Second-Order Phase Transitions in Potts Models: Renormalization-Group Solution

A generalization of the Niemeijer-van Leeuwen renormalization-group transformation treating disordered cells of spins as vacancies is introduced and applied to the two-dimensional $q$-state Potts model. Finite-lattice approximations yield the changeover from continuous to first-order phase transitions predicted by Baxter but not observed in previous renormalization-group calculations. Exact results are conjectured for the tricritical exponents of dilute Potts models.

Prescribing a System of Random Variables by Conditional Distributions

Prescribing a System of Random Variables by Conditional Distributions

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity

The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity