Potts Spin Systems Research Articles

Topological aspects of the critical three-state Potts model

We explore the topological defects of the critical three-state Potts spin system on the torus, Klein bottle and cylinder. A complete characterization is obtained by breaking down the Fuchs–Runkel–Schweigert construction of 2D rational CFT to the lattice setting. This is done by applying the strange correlator prescription to the recently obtained tensor network descriptions of string-net ground states in terms of bimodule categories (Lootens et al 2021 SciPost Phys. 10 053). The symmetries are represented by matrix product operators (MPO), as well as intertwiners between the diagonal tetracritical Ising model and the non-diagonal three-state Potts model. Our categorical construction lifts the global transfer matrix symmetries and intertwiners, previously obtained by solving Yang–Baxter equations, to MPO symmetries and intertwiners that can be locally deformed, fused and split. This enables the extraction of conformal characters from partition functions and yields a comprehensive picture of all boundary conditions.

Loop-Cluster Coupling and Algorithm for Classical Statistical Models.

Potts spin systems play a fundamental role in statistical mechanics and quantum field theory and can be studied within the spin, the Fortuin-Kasteleyn (FK) bond or the q-flow (loop) representation. We introduce a Loop-Cluster (LC) joint model of bond-occupation variables interacting with q-flow variables and formulate an LC algorithm that is found to be in the same dynamical universality as the celebrated Swendsen-Wang algorithm. This leads to a theoretical unification for all the representations, and numerically, one can apply the most efficient algorithm in one representation and measure physical quantities in others. Moreover, by using the LC scheme, we construct a hierarchy of geometric objects that contain as special cases the q-flow clusters and the backbone of FK clusters, the exact values of whose fractal dimensions in two dimensions remain as an open question. Our work not only provides a unified framework and an efficient algorithm for the Potts model but also brings new insights into the rich geometric structures of the FK clusters.

A multispin algorithm for the Kob-Andersen stochastic dynamics on regular lattices

The aim of the paper is to propose an algorithm based on the Multispin Coding technique for the Kob-Andersen glassy dynamics. We first give motivations to speed up the numerical simulation in the context of spin glass models [M. Mezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987)], after defining the Markovian dynamics as in [W. Kob, H.C. Andersen, Phys. Rev. E 48, 4364 (1993)] as well as the related interesting observables, we extend it to the more general framework of random regular graphs, listing at the same time some known analytical results [C. Toninelli, G. Biroli, D.S. Fisher, J. Stat. Phys. 120, 167 (2005)]. The purpose of this work is a dual one, firstly, we describe how bitwise operators can be used to build up the algorithm by carefully exploiting the way data are stored on a computer. Since it was first introduced [M. Creutz, L. Jacobs, C. Rebbi, Phys. Rev. D 20, 1915 (1979), C. Rebbi, R.H. Swendsen, Phys. Rev. D 21, 4094 (1980)], this technique has been widely used to perform Monte Carlo simulations for Ising and Potts spin systems, however, it can be successfully adapted to more complex systems in which microscopic parameters may assume boolean values. Secondly, we introduce a random graph in which a characteristic para- meter allows to tune the possible transition point. A consistent part is devoted to listing the numerical results obtained by running numerical simulations.

Scaling Properties of Parallelized Multicanonical Simulations

We implemented a parallel version of the multicanonical algorithm and applied it to a variety of systems with phase transitions of first and second order. The parallelization relies on independent equilibrium simulations that only communicate when the multicanonical weight function is updated. That way, the Markov chains efficiently sample the temporary distributions allowing for good estimations of consecutive weight functions. The systems investigated range from the well known Ising and Potts spin systems to bead-spring polymers. We estimate the speedup with increasing number of parallel processes. Overall, the parallelization is shown to scale quite well. In the case of multicanonical simulations of the $q$-state Potts model ($q\ge6$) and multimagnetic simulations of the Ising model, the optimal performance is limited due to emerging barriers.

Setup of Frustration in Potts Spin Systems

Frustration in the Potts spin systems is proposed as an extension of the ordinary frustration in the Ising spin systems. We construct a variety of fully-frustrated p -state Potts models on the square lattice by applying the extended frustration inspired by the primitive form on a multi-decorated lattice; the universality class of those fully-frustrated p -state Potts models is the same as that of the ferromagnetic one-state Potts model, which is equivalent to the ordinary percolation system. It is shown that a typical case of the fully-frustrated p -state Potts models can be regarded as the p -mer packing problem at the macroscopically degenerated ground state.

Computer Physics Communications – List of Editors

Mesoscopic phase transitions and the critical behavior of nanostructured materials are summarized for three-dimensional systems (1) of simple particles starting from grains of fcc or hcp and two-dimensional spin-particle systems (2), and generalized q-state (q=3,4) Potts-spin systems (3), interacting magnetoelastically. The interactions are taken into account up to the third nearest neighbors of the Lennard–Jones potential. By generalizing Metropolis Monte Carlo and renormalization methods, the following results are discussed: (1) similarity and difference in nanostructured characteristics inherent in the initial fcc or hcp structure, (2) a lot of local, magnetostructural mesoscopic phase transitions, and (3) the adaptive, intelligent control of phase transition temperatures, in addition to (4) concepts of the bulk, “ordered and disordered phases, magnetic and/or elastic”, completely corrected in the mesoscopic phases, and (5) mesoscopic critical behaviors of N-component systems.

Mesoscopic phase transitions and their critical behavior of nanostructured materials

Abstract Mesoscopic phase transitions and the critical behavior of nanostructured materials are summarized for three-dimensional systems (1) of simple particles starting from grains of fcc or hcp and two-dimensional spin-particle systems (2), and generalized q -state ( q =3,4) Potts-spin systems (3), interacting magnetoelastically. The interactions are taken into account up to the third nearest neighbors of the Lennard–Jones potential. By generalizing Metropolis Monte Carlo and renormalization methods, the following results are discussed: (1) similarity and difference in nanostructured characteristics inherent in the initial fcc or hcp structure, (2) a lot of local, magnetostructural mesoscopic phase transitions, and (3) the adaptive, intelligent control of phase transition temperatures, in addition to (4) concepts of the bulk, “ordered and disordered phases, magnetic and/or elastic”, completely corrected in the mesoscopic phases, and (5) mesoscopic critical behaviors of N-component systems.

Thermodynamics and topology of disordered systems: Statistics of the random knot diagrams on finite lattices

The statistical properties of random lattice knots, the topology of which is determined by the algebraic topological Jones-Kauffman invariants, was studied by analytical and numerical methods. The Kauffman polynomial invariant of a random knot diagram was represented by a partition function of the Potts model with a random configuration of ferro-and antiferromagnetic bonds, which allowed the probability distribution of the random dense knots on a flat square lattice over topological classes to be studied. A topological class is characterized by the highest power of the Kauffman polynomial invariant and interpreted as the free energy of a q-component Potts spin system for q→∞. It is shown that the highest power of the Kauffman invariant correlates with the minimum energy of the corresponding Potts spin system. The probability of the lattice knot distribution over topological classes was studied by the method of transfer matrices, depending on the type of local junctions and the size of the flat knot diagram. The results obtained are compared to the probability distribution of the minimum energy of a Potts system with random ferro-and antiferromagnetic bonds.

A Duality and an Analytic Profile of Topological Entropy in Percolation Expressions of 2D Potts Spin Systems

By the Kasteleyn and Fortuin formulation, a Potts spin system can be expressed as a percolation system of spin clusters. The topological entropy can be defined from the number of spin cluster patterns as a function of the numbers of bonds and clusters. Using a self duality, the topological entropy is shown to have the maximum value at the transition temperature for the square lattice. It also shows a delicate singular structure around the transition point even at the unity degree of internal freedom, the one-state Potts model, as a unified generating function. The strongly-frustrated Ising system is understood as the one-state Potts model with the real spin clusters.

Explicit KF-Percolation Transitions in 2D Strongly-Frustrated Spin Systems

An Ising spin system can be expressed as a percolation system, formulated by Kasteleyn and Fortuin (KF), of the spin clusters coupled by interactions effective against the thermal agitation. For the 2D strongly-frustrated Ising spin system, the free energy does not show any criticality at the percolation transition temperature of the explicit singularity in the KF-percolation generating function. By extending it into a frustrated Potts spin system with a general degree of internal freedom, the criticality of KF-percolation expression becomes explicit in the ordinary free energy. The universality class belongs to the ferromagnetic Potts model with the half degree of its Potts spin freedom; this means that the effective spin freedom of the frustrated Ising spin system is unity.

Theory of self-organization of cortical maps: Mathematical framework

We have constructed a general theory describing cortical map formation in the primary sensory cortices by using three main hypotheses: (1) competition among synapses due to the limitation of a postsynaptic factor secretion, (2) synaptic stabilization which takes place by a presynaptic factor secretion, and (3) synaptic stabilization according to a local Hebbian rule. Based on these hypotheses, we have expressed a process of synaptic stabilization in terms of a nonlinear dynamical equation. From detailed consideration of this equation, we have found that the problem for cortical map formation is equivalent to that for the thermodynamics in the Potts spin system whose spin variables typically have complex internal states. This equivalence means that the developmental process of the central nervous system is the dynamical process near equilibrium. Furthermore, we can systematically deal with various cortical map formations within a unified framework of the thermodynamics. Therefore, we can interpret use-dependent self-organization of a cortical map as a kind of phase transition phenomena. Finally we can suggest that there exists a relationship among plasticity, effective temperature, presynaptic factor secretion and critical period.

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O( N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g 2-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly. The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P( N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P( N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a 1 N expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.