Ising Dynamics Research Articles

A note on Edwards' hypothesis for zero-temperature Ising dynamics

We give a simple criterion for checking the so called Edwards' hypothesis in certain zero-temperature, ferromagnetic spin-flip dynamics and use it to invalidate the hypothesis in various examples in dimension one and higher.

Direct versus measurement-assisted bipartite entanglement in multiqubit systems and their dynamical generation in spin systems

We consider multi-qubit systems and relate quantitatively the problems of generating cluster states with high value of concurrence of assistance, and that of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-1/2 systems that model qubits, with different couplings and possible presence of magnetic field to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an XY Hamiltonian, and their generation can be controlled by the initial state of one of the spins. The same Hamiltonian is capable of creating states with high concurrence of assistance with suitably chosen initial state. We show that the production of graph states using the Ising Hamiltonian is controllable via a single-qubit rotation of one spin-1/2 subsystem in the initial multi-qubit state. We shown that the property of Ising dynamics to convert a product state basis into a special maximally entangled basis is temporally enhanced by the application of a suitable magnetic field. Similar basis transformations are found to be feasible in the case of isotropic XY couplings with magnetic field.

Cooperative freezing in spin glasses and magnetic nanostructures

Relaxation processes in disordered nanostructures and spin glasses are investigated by examining the role of the magnetic anisotropy. To abstract from freezing processes associated with phase transitions, the consideration is restricted to a one-dimensional bond-disordered Ising spin glass. Using a Mattis-type gauge transformation, the spin glass is mapped onto a disordered ferromagnet. Based on the behavior of the spin chain we argue that spin-glass freezing is not necessarily related to a phase transition. The magnetic anisotropy, which is largely ignored in the Glauber approach to Ising dynamics, gives rise to cooperative deviations from the Arrhenius behavior and mimics Vogel–Fulcher relaxation.

Universal Ising dynamics in two dimensions

We explore several dominant eigenvalues of the spectra of Markov matrices governing the dynamics of models in the universality class of the two-dimensional Ising model. By means of a variational approximation, we determine autocorrelation times of progressively rapid relaxation modes. The approximation of one eigenstate, associated with the slowest mode, is employed in a variance-reducing Monte-Carlo method. The resulting correlation times, for which statistical errors exceed the systematic errors associated with the variational approximation, are used for a finite-size scaling analysis which corroborates universality of the dynamic critical exponent z for three distinct Ising models on the square lattice. Tentative, variational results for subdominant states strongly suggest that the amplitudes of the divergent time scales associated with different relaxation modes differ solely by metric factors, setting a single non-universal time scale for each model. A by-product of our analysis is a highly accurate confirmation of static universality.

Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness

We study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood. These are equivalent, for a restricted set of initial conditions, to nonzero probability transitions in single spin-flip dynamics of the Ising model at zero temperature. We show that in three or more dimensions these systems can simulate Boolean circuits of AND and OR gates, and are therefore P-complete. That is, predicting their state t time-steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer. Therefore, unless a widely believed conjecture in computer science is false, it is impossible even with parallel computation to predict majority-vote cellular automata, or zero-temperature single spin-flip Ising dynamics, qualitatively faster than by explicit simulation.

An algorithm for series expansions based on hierarchical rate equations

We propose a computational method to obtain series expansions in powers of time for general dynamical systems described by a set of hierarchical rate equations. The method is generally applicable to problems in both equilibrium and nonequilibrium statistical mechanics such as random sequential adsorption, diffusion-reaction dynamics, and Ising dynamics. New result of random sequential adsorption of dimers on a square lattice is presented.

Damage spreading and critical exponents for “model A” Ising dynamics

Using damage spreading and heat bath dynamics, we study the Ising model in 2 and 3 dimensions with non-conservative dynamics. Our algorithm differs in some important points from previous ones, which makes it rather efficient. We give estimates for the exponent z which seem to be the most precise published so far (2.172 ± 0.006 for d = 2, 2.032 ± 0.004 for d = 3). We also give precise estimates of the exponent θ′ introduced by Janssen et al. (Z. Phys. B 73 (1989) 539) and of analogous but in principle independent exponents. We find surprisingly that some of the latter agree with θ′, and give an explanation for this.

Magnetic hysteresis loops as Lissajous plots of relaxationally delayed response to periodic field variation

We have studied the nature of the response function (magnetisation) arising in response to a periodic perturbation (magnetic field) on Ising systems, using the Monte Carlo simulation (for a three dimensional Ising system) and also solving the mean field equation of Ising dynamics. The hysteresis loop is identified here as the Lissajous plot of the response function versus the perturbing field. It is observed that the response function is always periodic with the same frequency as that of the perturbing field and gets delayed in general due to relaxation. This equality in the frequency gives rise to the quadratic equation of the Lissajous figures (double valuedness and stability of magnetisation in the hysteresis loop) and the delay gives rise to the nonvanishing width of the Lissajous figures (nonvanishing hysteresis loss). We have also studied the scaling behaviour for the hysteresis loop area A, obtained from the mean field solution of the Ising dynamics and from the Monte Carlo study of a three dimensional Ising system. The scaling function is observed to be Lorentzian in form in the mean field case, while the scaling function is an exponentially decaying one multiplied by a power law function, in the case of (Monte Carlo results for) the three dimensional Ising systems. Additionally, we have studied the dynamic phase transition for the three dimensional Ising system and the tricritical point in the phase diagram has been located.

Monte Carlo study of hysteretic response and relaxation in Ising models

We have studied, using Monte Carlo (MC) simulations for ferromagnetic Ising systems in two to four dimensions, the variations of “hysteretic” loop area A with frequency f and amplitude h 0 of the applied external magnetic field and temperature T of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter ( f, h 0, T) values and the best fit exponents are obtained in all three dimensions ( d = 2, 3, 4). We also compared the scaling behaviour for A, obtained from the mean field (MF) solution of the Ising dynamics. Our observations, both from the MC simulations (in d = 2 to 4) and from the MF solutions, indicate nonvanishing paramagnetic loop area A. We confirmed these MC observations by checking the finite size variation and also studied the same for a one-dimensional chain for comparison. We additionally studied the dynamic phase transition for the time averaged magnetisation (over a cycle), which acquires a nonzero value below a critical field amplitude h c 0( T, f). The tricritical point in the phase diagram is also located.

A parallel multigrid algorithm for percolation clusters

A new parallel cluster-finding algorithm is formulated by using multigrid relaxation methods very similar to those used for differential equation solvers. For percolation clusters, this approach drastically reduces critical slowing down relative to local or scan relaxation methods. Numerical studies of scaling properties with system size are presented in the case of the 2D percolation clusters of the Swendsen-Wang Ising dynamics running on the Connection Machine.

Cellular automata as microcanonical simulators

The deterministic Q2R cellular automation has been tested by many authors as a fast algorithm to simulate the Ising model in the microcanonical ensemble. However, the magnetic susceptibility curve, measured from the fluctuations of the magnetization, is found to be far below the expected results inside the ordered Ising phase. The non-ergodicity degree of this automaton seems to be inadequate to simulate the Ising dynamics. In this work, the authors introduce two modified automata, and test their performance for the square lattice. For the first one, they found improved results concerning the Ising transition. For the second modified automaton the Ising transition does not occur, the magnetization vanishing even for energies far below the normal critical threshold. On the other hand, another transition appears at a different critical energy value that seems to be the same as that of the periodic-chaotic transition already observed in the normal Q2R dynamics. The critical indices of this new transition are measured from finite-size scaling, the results indicating that it is at the same universality class as the Ising transition.

Dynamics in a long-range exchange model

A new Ising dynamics with long-range exchange is proposed in which the exchange probability is a power law function of the distance between exchanged spins. Using a coarse-grained picture, a generalized time-dependent Ginzburg-Landau (TDGL) equation is derived. We analyze the dynamics of a first-order phase transition and a second-order phase transition, and we obtain the scaling function for droplet size distribution and the exponent of critical slowing down, respectively.

Embedded dynamics for lattice field theory

A new approach to simulating lattice field theory with continuous fields is suggested and it is tested on φ 4 field theory. Discrete variables are embedded into the field theory, so that large scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents z ED = 0.07±0.07, 0.29±0.09, 0.87±0.20 in d = 1,2,3 dimensions respectively are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.

Embedded dynamics for phi4 theory.

A new Monte Carlo dynamics is proposed for ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory that reduces critical slowing down. Discrete Ising variables are embedded into the field theory, so that large-scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents ${\mathrm{z}}_{\mathrm{EL}=0.07\ifmmode\pm\else\textpm\fi{}}$0.07, 0.29\ifmmode\pm\else\textpm\fi{}0.09, and 0.87\ifmmode\pm\else\textpm\fi{}0.20 in one, two, and three dimensions, respectively, are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.

Critical slowing down in microcanonical Ising dynamics.

We describe measurements of autocorrelation times in a version of Creutz's microcanonical dynamics for the nearest-neighbor Ising system on a 128\ifmmode\times\else\texttimes\fi{}129\ifmmode\times\else\texttimes\fi{}128 lattice. The dynamical critical exponent z is found to be z=2.26\ifmmode\pm\else\textpm\fi{}0.05, which is comparable to the dynamical critical exponent for Monte Carlo algorithms.