We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to what seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a "bubbling" behaviour, as also happens for dilution.
Combinatorial optimization problems can be mapped onto Ising models, and their ground state is generally difficult to find. A lot of heuristics for these problems have been proposed, and one promising approach is to use continuous variables. In recent years, one such algorithm has been implemented by using parametric oscillators known as coherent Ising machines. Although these algorithms have been confirmed to have high performance through many experiments, unlike other familiar algorithms such as simulated annealing, their computational ability has not been fully investigated. In this paper, …
Various kinds of Ising machines based on unconventional computing have recently been developed for practically important combinatorial optimization. Among them, the machines implementing a heuristic algorithm called simulated bifurcation have achieved high performance, where Hamiltonian dynamics are simulated by massively parallel processing. To further improve the performance of simulated bifurcation, here we introduce thermal fluctuation to its dynamics relying on the Nosé-Hoover method, which has been used to simulate Hamiltonian dynamics at finite temperatures. We find that a heating proce…
We consider the superstable cycles of the Q-state Potts (QSP) and the three-site interaction antiferromagnetic Ising (TSAI) models on recursive lattices. The rational mappings describing the models' statistical properties are obtained via the recurrence relation technique. We provide analytical solutions for the superstable cycles of the second order for both models. A particular attention is devoted to the period three window. Here we present an exact result for the third order superstable orbit for the QSP and a numerical solution for the TSAI model. Additionally, we point out a non-trivial …
A spatially one dimensional coupled map lattice possessing the same symmetries as the Miller Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behaviour are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Is…
A parallel version of the invaded cluster algorithm is described. Results from large scale (up to 4096^2 and 512^3) simulations of the Ising model are reported. No evidence of critical slowing down is found for the three-dimensional Ising model. The magnetic exponent is estimated to be 2.482 \pm .001 (beta/nu = 0.518 pm .001) for the three-dimensional Ising model.
It is discussed how the equilibrium properties of the Ising model are described by an Hamiltonian with an antiferromagnetic low temperature behavior if only an heat bath dynamics, with the characteristics of a Probabilistic Cellular Automaton, is assumed to determine the temporal evolution of the system.
Critical bifurcation and deconfined quantum criticality in an interacting cluster Ising chain
Indian Institute of Technology Madras（马德拉斯理工学院）；Center for Quantum Information, Communication and Computation (CQuICC)（量子信息、通信与计算中心）；Karlstad University（卡尔斯特德大学）；Heinrich-Heine-Universität（海因里希·海涅大学）；Ilia State University（伊利亚州立大学）；Andronikashvili Institute of Physics（安德罗尼卡什维利物理研究所）；The Abdus Salam International Centre for Theoretical Physics (ICTP)（阿卜杜斯·萨拉姆国际理论物理中心）
We study persistence in one-dimensional ferromagnetic and anti-ferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) \sim 1/t^theta_p with theta_p \simeq 0.75 numerically. A mapping to the dynamics of two decoupled A+A \to 0 models yields theta_p = 3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.
The dynamics of non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising model…
Finding an energy minimum in the Ising model is an exemplar objective, associated with many combinatorial optimization problems, that is computationally hard in general, but occurs in all areas of modern science. There are several numerical methods, providing solution for the medium size Ising spin systems. However, they are either computationally slow and badly parallelized, or do not give sufficiently good results for the large systems. In this paper, we present a highly parallel algorithm, called Mean-field Annealing from a Random State (MARS), incorporating the best features of the classic…
Kinetic Ising models on the square lattice with both nearest-neighbor interactions and self-interaction are studied for the cases of random sequential updating and parallel updating. The equilibrium phase diagrams and critical dynamics are studied using Monte Carlo simulations and analytic approximations. The Hamiltonians appearing in the Gibbs distribution describing the equilibrium properties differs for sequential and parallel updating but in both cases feature multispin and non-nearest-neighbor couplings. For parallel updating the system is a probabilistic cellular automaton and the equili…
The magnetization probability density in d=2 and 3 dimensional Ising models in slab geometry of volume $L_{\parallel}^{d-1} \times L_{\perp}$ is computed through Monte-Carlo simulation at the critical temperature and zero magnetic field. The finite-size scaling of this distribution and its dependence on the system aspect-ratio $ρ=\frac{L_{\perp}}{L_{\parallel}}$ and boundary conditions is discussed. In the limiting case $ρ\to 0$ of a macroscopically large slab ($L_{\parallel} \gg L_{\perp}$) the distribution is found to scale as a Gaussian function for all tested system sizes and boundary cond…
Network reliability is the probability that a dynamical system composed of discrete elements interacting on a network will be found in a configuration that satisfies a particular property. We introduce a new reliability property, Ising feasibility, for which the network reliability is the Ising model s partition function. As shown by Moore and Shannon, the network reliability can be separated into two factors: structural, solely determined by the network topology, and dynamical, determined by the underlying dynamics. In this case, the structural factor is known as the joint density of states. …
An extension of the synchronous parallel kinetic Monte Carlo (pkMC) algorithm developed by Martinez {\it et al} [{\it J.\ Comp.\ Phys.} {\bf 227} (2008) 3804] to discrete lattices is presented. The method solves the master equation synchronously by recourse to null events that keep all processors time clocks current in a global sense. Boundary conflicts are rigorously solved by adopting a chessboard decomposition into non-interacting sublattices. We find that the bias introduced by the spatial correlations attendant to the sublattice decomposition is within the standard deviation of the serial…
We perform a numerical investigation of the \emph{shaken dynamics}, a parallel Markovian dynamics for spin systems with local interaction and whose transition probabilities depend on two parameters, $q$ and $J$, that tune the geometry of the underlying lattice. We determine a phase transition curve, in the $(q, J)$ plane, separating the disordered phase from the ordered one, study the mixing time of the Markov chain and evaluate the spin-spin correlations as $q$ and $J$ vary. Further, we investigate the relation between the equilibrium measure of the shaken dynamics and the Gibbs measure for t…
Excited state quantum phase transitions (ESQPTs) are generalizations of quantum phase transitions (QPTs) to excited levels. They are associated with local divergences in the density of states. Here, we investigate how the presence of an ESQPT can be detected from the analysis of the structure of the Hamiltonian matrix, the level of localization of the eigenstates, the onset of bifurcation, and the speed of the system evolution. Our findings are illustrated for a Hamiltonian with infinite-range Ising interaction in a transverse field. This is a version of the Lipkin-Meshkov-Glick (LMG) model an…
We introduce a new update scheme to systematically improve the efficiency of parallel tempering simulations. We show that by adapting the number of sweeps between replica exchanges to the canonical autocorrelation time, the average round-trip time of a replica in temperature space can be significantly decreased. The temperatures are not dynamically adjusted as in previous attempts but chosen to yield a 50% exchange rate of adjacent replicas. We illustrate the new algorithm with results for the Ising model in two and the Edwards-Anderson Ising spin glass in three dimensions
We use a phase-separated driven two-dimensional Ising lattice gas to study fluid interfaces exposed to shear flow parallel to the interface. The interface is stabilized by two parallel walls with opposing surface fields and a driving field parallel to the walls is applied which (i) either acts locally at the walls or (ii) varies linearly with distance across the strip. Using computer simulations with Kawasaki dynamics, we find that the system reaches a steady state in which the magnetisation profile is the same as that in equilibrium, but with a rescaled length implying a reduction of the inte…
We study a bilayer Ising spin system consisting of antiferromagnetic (AF) and ferromagnetic (FM) triangular planes, coupled by ferromagnetic exchange interaction, by standard Monte Carlo and parallel tempering methods. The AF/FM bilayer is found to display the critical behavior completely different from both the single FM and AF constituents as well as the FM/FM and AF/AF bilayers. Namely, by finite-size scaling (FSS) analysis we identify at the same temperature a standard Ising transition from the paramagnetic to FM state in the FM plane that induces a ferrimagnetic state with a finite net ma…
We present exact expressions for hysteresis loops in the ferromagnetic random field Ising model in the limit of zero temperature and zero driving frequency for an arbitrary initial state of the model on a Bethe lattice. This work extends earlier results that were restricted to an initial state with all spins pointing parallel to each other.
The dynamic evolution at zero temperature of a uniform Ising ferromagnet on a square lattice is followed by Monte Carlo computer simulations. The system always eventually reaches a final, absorbing state, which sometimes coincides with a ground state (all spins parallel), and sometimes does not (parallel stripes of spins up and down). We initiate here the numerical study of ``Chaotic Time Dependence'' (CTD) by seeing how much information about the final state is predictable from the randomly generated quenched initial state. CTD was originally proposed to explain how nonequilibrium spin glasse…
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 pro…
We study from tempered Monte Carlo simulations the magnetic phase diagram of a textured dipolar Ising model on a face centered cubic lattice. The Ising coupling of the model follow the dipole-dipole interaction. The Ising axes are distributed with a uniaxial symmetry along the $\hat{z}$ direction with a gaussian probability density of the polar angles. This distribution provides a quenched disorder realization of the dipolar Ising model making a continuous link between the parallel axes dipoles and the random axes dipole models. As expected the phase diagram presents three distinctive phases: …
Networks of coupled parametric resonators (parametrons) hold promise for parallel computing architectures. En route to realizing complex networks, we report an experimental and theoretical analysis of two coupled parametrons. In contrast to previous studies, we explore the case of strong bilinear coupling between the parametrons, as well as the role of detuning. We show that the system can still operate as an Ising machine in this regime, even though careful calibration is necessary to ensure that the correct solution space is available. Apart from the formation of split normal modes, new stat…
The multicanonical method has been proven powerful for statistical investigations of lattice and off-lattice systems throughout the last two decades. We discuss an intuitive but very efficient parallel implementation of this algorithm and analyze its scaling properties for discrete energy systems, namely the Ising model and the 8-state Potts model. The parallelization relies on independent equilibrium simulations in each iteration with identical weights, merging their statistics in order to obtain estimates for the successive weights. With good care, this allows faster investigations of large …
We consider the three-dimensional Ising model in a $L_\perp \times L_\parallel \times L_\parallel$ cuboid geometry with finite aspect ratio $ρ= L_\perp/L_\parallel$ and periodic boundary conditions along all directions. For this model the finite-size scaling functions of the excess free energy and thermodynamic Casimir force are evaluated numerically by means of Monte Carlo simulations. The Monte Carlo results compare well with recent field theoretical results for the Ising universality class at temperatures above and slightly below the bulk critical temperature $T_\mathrm{c}$. Furthermore, th…
We have simulated, using parallel tempering, the three dimensional Ising spin glass model with binary couplings in a helicoidal geometry. The largest lattice (L=20) has been studied using a dedicated computer (the SUE machine). We have obtained, measuring the correlation length in the critical region, a strong evidence for a second-order finite temperature phase transition ruling out other possible scenarios like a Kosterlitz-Thouless phase transition. Precise values for the $ν$ and $η$ critical exponents are also presented.
We study parametrically driven quantum oscillators and show that, even for weak coupling between the oscillators, they can exhibit various many-body states with broken time-translation symmetry. In the quantum-coherent regime, the symmetry breaking occurs via a nonequilibrium quantum phase transition. For dissipative oscillators, the main effect of the weak coupling is to make the switching rate of an oscillator between its period-2 states dependent on the states of other oscillators. This allows mapping the oscillators onto a system of coupled spins. Away from the bifurcation parameter values…
Using the parallel tempering algorithm and GPU accelerated techniques, we have performed large-scale Monte Carlo simulations of the Ising model on a square lattice with antiferromagnetic (repulsive) nearest-neighbor(NN) and next-nearest-neighbor(NNN) interactions of the same strength and subject to a uniform magnetic field. Both transitions from the (2x1) and row-shifted (2x2) ordered phases to the paramagnetic phase are continuous. From our data analysis, reentrance behavior of the (2x1) critical line and a bicritical point which separates the two ordered phases at T=0 are confirmed. Based on…
We study the critical behavior and the out-of-equilibrium dynamics of a two-dimensional Ising model with non-static interactions. In our model, bonds are dynamically changing according to a majority rule depending on the set of closest neighbors of each spin pair, which prevents the system from ordering in a full ferromagnetic or antiferromagnetic state. Using a parallel-tempering Monte Carlo algorithm, we find that the model undergoes a continuous phase transition at finite temperature, which belongs to the Ising universality class. The properties of the bond structure and the ground-state en…
