Ising model

About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.

Performance limitations of flat-histogram methods.

Abstract: Monte Carlo methods are well-suited for the simulation of large many body problems, since the complexity for a single Monte Carlo update step scales only polynomially and often linearly in the system size, while the config- uration space grows exponentially with the system size. The performance of a Monte Carlo method is then deter- mined by how many update steps are needed to efficiently sample the configuration space. For second order phase transitions in unfrustrated systems the problem of "crit- ical slowing down" - a rapid divergence of the number of Monte Carlo steps needed to obtain a subsequent un- correlated configuration - was solved more than a decade ago by cluster update algorithms (1). At first order phase transitions and in systems with many local minima of the free energy such as frustrated magnets or spin glasses, there is the similar problem of long tunneling times be- tween local minima. With energy barriersE scaling lin- early with the linear system size L, the tunneling times � at an inverse temperature � = 1/kBT scale exponentially with the system size, � � exp(��E) / exp(const × L). Several methods were developed to overcome this tun- neling problem, such as the multicanonical method (2), broad histograms (4), simulated and parallel tempering (3), and Wang-Landau sampling (5). The common aim of all these methods is to broaden the range of energies sam- pled within Monte Carlo simulations from the sharply peaked distribution of canonical sampling at fixed tem- perature in order to ease the tunneling through barriers. Ideally, all relevant energy levels are sampled equally often during a simulation, thus producing a "flat his- togram" in energy space. Some methods approach this goal by variations and generalizations of canonical dis- tributions (2, 3), while others (4, 5) discard the notion of temperature completely and instead are formulated in terms of the density of states. With a probability p(E) for a single configuration with energy E, the probability of sampling an arbitrary configuration with energy E is given as PE = �(E)p(E), where the density of states �(E) counts the number of states with energy E. Upon choos- ing p(E) / 1/�(E) instead of p(E) / exp( �E) one ob- tains a constant probability PE for visiting each energy level E, and hence a flat histogram. Wang and Landau (5) proposed a simple and elegant flat histogram algorithm that iteratively improves approximations to the initially unknown density of states �(E). Once �(E) is determined with sufficient accuracy, the Monte Carlo algorithm just performs a random walk in energy space. Within two years of publication this algorithm has been applied to a large number of problems (6, 7, 8) and extended to quantum systems (9). In this Letter we investigate the performance of flat histogram algorithms in general, and the Wang-Landau algorithm in particular, for three systems for which the density of states �(E) is known exactly on finite two- dimensional (2D) lattices: the Ising ferromagnet as the simplest example, the fully frustrated Ising model as a prototype for frustrated systems, and the ±J Ising spin glass. For each of these models we construct a perfect flat histogram method by simulating a random walk in configuration space where we employ the known density of states for these models to set p(E) / 1/�(E). As a measure of performance we use the average tun- neling timeto get from a ground state (lowest energy configuration) to an anti-ground state (configuration of highest energy), which is the relevant time scale for sam- pling the whole phase space (10). Since the number of energy levels in a d-dimensional system with linear size L scales with the number of spins N = L d , the tunneling time for a pure random walk in energy space is

Properties of one-dimensional quasilattices.

Abstract: We study the properties of one-dimensional quasilattices numerically and analytically. The geometrical properties of general one-dimensional quasilattices are discussed. The Ising model on these lattices is studied by a decimation transformation: The critical temperature and critical exponents do not differ from those for a regular periodic chain. The vibrational spectrum in the harmonic approximation is analyzed numerically. The system exhibits characteristics of both a regular periodic system and a disordered system. In the low-frequency region, the system behaves as a regular periodic system; wave functions appear extended. In the high-frequency region, the spectrum is self-similar and there is no unique behavior for the wave functions. The spectrum shows many gaps and Van Hove singularities. The gaps in the spectrum are also obtained analytically by examining the convergence of a continued-fraction expansion plus decimation transformation. The energy spectrum of a tight-binding electron Hamiltonian on the Fibonacci chain is also analyzed; it shows similar characteristics to those of the lattice vibration spectrum.

Percolation thresholds in Ising magnets and conducting mixtures

Abstract: The effect of the percolation threshold on conduction in mixtures and on the magnetic ordering temperature in alloys of high- and low- ${T}_{C}$ material is described using a real-space rescaling approximation (introduced by Kadanoff, following an idea of Migdal) which is applicable to systems of arbitrary spatial dimensionality $d$. We calculate exponents describing the regimes just above and just below ${p}_{c}$ for the case of bond dilution in resistor networks or Ising ferromagnets. The results are accurate for $d=2$ in those cases where comparison is possible and qualitatively correct for $d=3$, but are invalid in the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Ising spin glasses do not show order in 2 dimensions, in this approximation, but may do so in 3 dimensions.