Two-dimensional Lattice Model Research Articles

Modelling the heart as a communication system.

Electrical communication between cardiomyocytes can be perturbed during arrhythmia, but these perturbations are not captured by conventional electrocardiographic metrics. We developed a theoretical framework to quantify electrical communication using information theory metrics in two-dimensional cell lattice models of cardiac excitation propagation. The time series generated by each cell was coarse-grained to 1 when excited or 0 when resting. The Shannon entropy for each cell was calculated from the time series during four clinically important heart rhythms: normal heartbeat, anatomical reentry, spiral reentry and multiple reentry. We also used mutual information to perform spatial profiling of communication during these cardiac arrhythmias. We found that information sharing between cells was spatially heterogeneous. In addition, cardiac arrhythmia significantly impacted information sharing within the heart. Entropy localized the path of the drifting core of spiral reentry, which could be an optimal target of therapeutic ablation. We conclude that information theory metrics can quantitatively assess electrical communication among cardiomyocytes. The traditional concept of the heart as a functional syncytium sharing electrical information cannot predict altered entropy and information sharing during complex arrhythmia. Information theory metrics may find clinical application in the identification of rhythm-specific treatments which are currently unmet by traditional electrocardiographic techniques.

The role of cellular coupling in the spontaneous generation of electrical activity in uterine tissue.

The spontaneous emergence of contraction-inducing electrical activity in the uterus at the beginning of labor remains poorly understood, partly due to the seemingly contradictory observation that isolated uterine cells are not spontaneously active. It is known, however, that the expression of gap junctions increases dramatically in the approach to parturition, by more than one order of magnitude, which results in a significant increase in inter-cellular electrical coupling. In this paper, we build upon previous studies of the activity of electrically excitable smooth muscle cells (myocytes) and investigate the mechanism through which the coupling of these cells to electrically passive cells results in the generation of spontaneous activity in the uterus. Using a recently developed, realistic model of uterine muscle cell dynamics, we investigate a system consisting of a myocyte coupled to passive cells. We then extend our analysis to a simple two-dimensional lattice model of the tissue, with each myocyte being coupled to its neighbors, as well as to a random number of passive cells. We observe that different dynamical regimes can be observed over a range of gap junction conductances: at low coupling strength, corresponding to values measured long before delivery, the activity is confined to cell clusters, while the activity for high coupling, compatible with values measured shortly before delivery, may spread across the entire tissue. Additionally, we find that the system supports the spontaneous generation of spiral wave activity. Our results are both qualitatively and quantitatively consistent with observations from in vitro experiments. In particular, we demonstrate that the increase in inter-cellular electrical coupling observed experimentally strongly facilitates the appearance of spontaneous action potentials that may eventually lead to parturition.

Heavy-Fermion Superconductivity and Localized-Itinerant Crossover in the Kondo Lattice

Superconductivity in the two-dimensional Kondo lattice model is derived by means of the dual-fermion approach. This framework takes full account of local correlations based on the dynamical mean-field theory and further includes influence of critical antiferromagnetic fluctuations. It is found that pairing with dx2 −y2 symmetry similar to the Hubbard model appears only in the weak-coupling regime. As the coupling is increased, we observe transition to p-wave spin-singlet superconductivity. The emergence of the unconventional p-wave pairing is ascribed to formation of large Fermi surface, which embodies itinerancy of f electrons.

Surprisal Metrics for Quantifying Perturbed Conformational Dynamics in Markov State Models.

Markov state models (MSMs), which model conformational dynamics as a network of transitions between metastable states, have been increasingly used to model the thermodynamics and kinetics of biomolecules. In considering perturbations to molecular dynamics induced by sequence mutations, chemical modifications, or changes in external conditions, it is important to assess how transition rates change, independent of changes in metastable state definitions. Here, we present a surprisal metric to quantify the difference in metastable state transitions for two closely related MSMs, taking into account the statistical uncertainty in observed transition counts. We show that the surprisal is a relative entropy metric closely related to the Jensen-Shannon divergence between two MSMs, which can be used to identify conformational states most affected by perturbations. As examples, we apply the surprisal metric to a two-dimensional lattice model of a protein hairpin with mutations to hydrophobic residues, all-atom simulations of the Fs peptide α-helix with a salt-bridge mutation, and a comparison of protein G β-hairpin with its trpzip4 variant. Moreover, we show that surprisal-based adaptive sampling is an efficient strategy to reduce the statistical uncertainty in the Jensen-Shannon divergence, which could be a useful strategy for molecular simulation-based ab initio design.

Surface-lattice model describes electrostatic interactions of ions and polycations with bacterial lipopolysaccharides: ion valence and polycation's excluded area.

The bacterial outer membrane (OM) is compositionally distinct and contains polyanionic lipopolysaccharide (LPS) in the outer layer as a main component. It has long been known that the cation-binding ability of LPS is one of the key determinants of OM permeability. Here we present a two-dimensional lattice model of the outer LPS layer, in which the lattice is decorated with bound ions or polycations; while small ions can occupy single binding sites, polycations, modeled as (charged) rods, compete for binding sites through their area exclusion, a consequence of their multisite binding. Our results suggest that in the parameter space of biological relevance, the effect of area exclusion is well-reflected in the competitive binding of Mg(2+) and polycations onto LPS; by reducing the apparent binding affinity of polycations, it enhances Mg(2+) binding. Despite simplifications, our results are generally consistent with the common view of Mg(2+) as OM-stabilizing and polycations as OM-perturbing agents. They will be useful for understanding how cationic antimicrobials can gain entry into the cytoplasmic membrane. We also outline a few strategies for extending our model toward a more realistic modeling of OM permeability.

Large-scale conductivity-tensor calculations for Hall effects in time-dependent wave-packet diffusion method

We present a computational methodology to evaluate the conductivity tensors of ``large-scale'' systems in a magnetic field based on the time-dependent wave-packet diffusion method. As demonstrations, we first apply the method to the two-dimensional square lattice model with static disorder and confirm appropriate magnetic-field dependence of conductivities from weak to strong magnetic-field regimes. Furthermore, we extend the method to apply to realistic systems and evaluate the influence of dynamical disorder on the Hall effects of organic semiconductors, taking microscopic molecular vibrations into account.

Dynamics of tunneling into nonequilibrium edge states

Time-dependent perturbations can drive a trivial two-dimensional band insulator into a quantum Hall-like phase, with protected nonequilibrium states bound to its edges. We propose an experiment to probe the existence of these topological edge states which consists of passing a tunneling current through a small two-dimensional sample out of equilibrium. The signature is a nonquantized metallic conductance near the edges of the sample and, in contrast, an excitation gap in the bulk. This proposal is demonstrated for the case of a two-dimensional lattice model of Dirac electrons with tunable mass in a strong electromagnetic field. In addition, we also study the tunneling conductance of the driven resonant level model and find a phenomenon similar to dynamic localization in which certain transport channels are suppressed.

Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems

We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath. We propose a phenomenological model for the resulting short-time dynamics that includes pair-creation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces. To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting. We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range between 13% to 25%. To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.

Coarse-grained molecular simulations of membrane adhesion domains.

We use a coarse-grained molecular model of supported lipid bilayers to study the formation of adhesion domains. We find that this process is a first order phase transition, triggered by a combination of pairwise short range attractive interactions between the adhesion bonds and many-body Casimir-like interactions, mediated by the membrane thermal undulations. The simulation results display an excellent agreement with the recently proposed Weil-Farago two-dimensional lattice model, in which the occupied and empty sites represent, respectively, the adhesion bonds and unbound segments of the membrane. A second phase transition, into a hexatic phase, is observed when the attraction between the adhesion bonds is further strengthened.

Phase transition in a two-dimensional triangular flow with consideration of optimal current difference effect

A modified two-dimensional triangular lattice model is presented by accounting the effect of optimal current difference on traffic dynamics and analyzed both theoretically and numerically. Based on the sensitivity and configurations of vehicles, two distinct types of jamming transitions occur: conventional jamming transition to the kink jam and jamming transition to the chaotic jam through kink jam. The chaotic region reduces with reaction coefficient and enhances when more number of vehicles move diagonally. It is shown that the incorporation of optimal current difference effect efficiently stabilizes the traffic flow and suppresses traffic jam for all possible configurations on triangular lattice.

Conflicting symmetries in topologically ordered surface states of three-dimensional bosonic symmetry protected topological phases

We study the Z2 topologically ordered surface state of three-dimensional bosonic SPT phases with the discrete symmetries G1 x G2. It has been argued that the topologically ordered surface state cannot be realized on a purely two-dimensional lattice model. We carefully examine the statement and show that the surface state should break G2 if the symmetry G1 is gauged. This manifests the conflict of the symmetry G1 and G2 on the surface of the three-dimensional SPT phase. Given that there is no such phenomena in the purely two-dimensional model, it signals that the symmetries are encoded anomalously on the surface of the three-dimensional SPT phases and that the surface state can never be realized on the purely two-dimensional models.

Periodic ordering of clusters and stripes in a two-dimensional lattice model. II. Results of Monte Carlo simulation.

The triangular lattice model with nearest-neighbor attraction and third-neighbor repulsion, introduced by Pȩkalski, Ciach, and Almarza [J. Chem. Phys. 140, 114701 (2014)] is studied by Monte Carlo simulation. Introduction of appropriate order parameters allowed us to construct a phase diagram, where different phases with patterns made of clusters, bubbles or stripes are thermodynamically stable. We observe, in particular, two distinct lamellar phases-the less ordered one with global orientational order and the more ordered one with both orientational and translational order. Our results concern spontaneous pattern formation on solid surfaces, fluid interfaces or membranes that is driven by competing interactions between adsorbing particles or molecules.

Modeling hydrogen transport in large disordered systems

The diffusion of hydrogen in disordered structures (i.e. polycrystalline materials) can be modeled in terms of a network of transitions between neighbored local energetic minima. These networks have been generated for the tungsten–hydrogen system based on results of classical molecular dynamics simulations. The sparsity of the resulting energy-landscape-based networks enables within the transition-state theory approximation an efficient computation of the transport properties of hydrogen in large systems. The results confirm that grain boundaries in polycrystalline tungsten can provide a fast and extended transport channel. Finally, the relation of transition network approach with kinetic Monte Carlo is exploited to simulate break-through curves in a simple two-dimensional lattice model system.

Periodic ordering of clusters and stripes in a two-dimensional lattice model. I. Ground state, mean-field phase diagram and structure of the disordered phases

The short-range attraction and long-range repulsion between nanoparticles or macromolecules can lead to spontaneous pattern formation on solid surfaces, fluid interfaces, or membranes. In order to study the self-assembly in such systems we consider a triangular lattice model with nearest-neighbor attraction and third-neighbor repulsion. At the ground state of the model (T = 0) the lattice is empty for small values of the chemical potential μ, and fully occupied for large μ. For intermediate values of μ periodically distributed clusters, bubbles, or stripes appear if the repulsion is sufficiently strong. At the phase coexistences between the vacuum and the ordered cluster phases and between the cluster and the lamellar (stripe) phases the entropy per site does not vanish. As a consequence of this ground state degeneracy, disordered fluid phases consisting of clusters or stripes are stable, and the surface tension vanishes. For T > 0 we construct the phase diagram in the mean-field approximation and calculate the correlation function in the self-consistent Brazovskii-type field theory.

Percolation modeling of self-damaging of composite materials

We propose the concept of autonomous self-damaging in “smart” composite materials, controlled by activation of added nanosize “damaging” capsules. Percolation-type modeling approach earlier applied to the related concept of self-healing materials, is used to investigate the behavior of the initial material’s fatigue. We aim at achieving a relatively sharp drop in the material’s integrity after some initial limited fatigue develops in the course of the sample’s usage. Our theoretical study considers a two-dimensional lattice model and involves Monte Carlo simulations of the connectivity and conductance in the high-connectivity regime of percolation. We give several examples of local capsule–lattice and capsule–capsule activation rules and show that the desired self-damaging property can only be obtained with rather sophisticated “smart” material’s response involving not just damaging but also healing capsules.

Flocking with minimal cooperativity: The panic model

We present a two-dimensional lattice model of self-propelled spins that can change direction only upon collision with another spin. We show that even with ballistic motion and minimal cooperativity, these spins display robust flocking behavior at nearly all densities, forming long bands of stripes. The structural transition in this system is not a thermodynamic phase transition, but it can still be characterized by an order parameter, and we demonstrate that if this parameter is studied as a dynamical variable rather than a steady-state observable, we can extract a detailed picture of how the flocking mechanism varies with density.

A statistical model of aggregate fragmentation

A statistical model of fragmentation of aggregates is proposed, based on the stochastic propagation of cracks through the body. The propagation rules are formulated on a lattice and mimic two important features of the process—a crack moves against the stress gradient while dissipating energy during its growth. We perform numerical simulations of the model for two-dimensional lattice and reveal that the mass distribution for small- and intermediate-size fragments obeys a power law, F(m)∝m−3/2, in agreement with experimental observations. We develop an analytical theory which explains the detected power law and demonstrate that the overall fragment mass distribution in our model agrees qualitatively with that one observed in experiments.

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed, which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.

Theory of network contractor dynamics for exploring thermodynamic properties of two-dimensional quantum lattice models

Based on the tensor network state representation, we develop a nonlinear dynamic theory, coined network contractor dynamics (NCD), to explore the thermodynamic properties of two-dimensional quantum lattice models. By invoking the rank-1 decomposition in the multilinear algebra, the NCD scheme makes the contraction of the tensor network of the partition function be realized through a contraction of a local tensor cluster with vectors on its boundary. An imaginary-time-sweep algorithm for implementation of the NCD method is proposed for practical numerical simulations. We benchmark the NCD scheme on the square Ising model, which shows great accuracy. Also, the results on the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice are disclosed to be in good agreement with the quantum Monte Carlo calculations. The quasientanglement entropy $S$, Lyapunov exponent ${I}^{\text{Lya}}$, and loop character ${I}^{\text{loop}}$ are introduced within the dynamic scheme, which are found to display ``nonlocality'' near the critical point, and can be applied to determine the thermodynamic phase transitions of both classical and quantum systems.

On the spatial dynamics and oscillatory behavior of a predator-prey model based on cellular automata and local particle swarm optimization

A two-dimensional lattice model based on Cellular Automata theory and swarm intelligence is used to study the spatial and population dynamics of a theoretical ecosystem. It is found that the social interactions among predators provoke the formation of clusters, and that by increasing the mobility of predators the model enters into an oscillatory behavior.