We have developed a novel simulation method that combines a multigrid technique with a stochastic blocking procedure. Our algorithm eliminates critical slowing down completely, as demonstrated by simulations of the two-dimensional Ising model at criticality.

We explicitly provide, in a simple way, the form of the conserved quantities of the one-dimensional Hubbard model. The method developed is based in a newly found factorization property of the local transition matrices, which is satisfied by several spin and fermion chain models. In this way, we have shown a systematic method for deriving the conserved quantities of integrable quantum systems.

The groups Pin(n,m) and Pin(m,n) are not isomorphic, and the obstruction classes to their respective bundles are different. It follows that for nonorientable superstring theories the contributions to a Polyakov path integral from surfaces with positive metric are different from the contributions from those with negative metrics.

Topological entanglements play an important role in the physical properties, such as viscosity, of macromolecular structures. We investigate the likelihood of the appearance of entanglements in the bond percolation problem. We show that below the percolation threshold ${\mathrm{p}}_{\mathrm{c}}$ (but extremely close to it) there exists an entanglement threshold ${\mathrm{p}}_{\mathrm{e}}$. Between ${\mathrm{p}}_{\mathrm{e}}$ and ${\mathrm{p}}_{\mathrm{c}}$, there exists an infinite spanning group of interlocked (linked) clusters. The achieved numerical resolution of ${\mathrm{p}}_{\mathrm{c}}$-${\mathrm{p}}_{\mathrm{e}}$=(1.8±0.4)×${10}^{\mathrm{-}7}$ required averaging over an extremely large number of configurations.

A hierarchy of L-state cyclic solid-on-solid lattice models, related to the Andrews-Baxter-Forrester solid-on-solid models, is solved exactly. In addition to the usual anisotropy parameter (spectral parameter) u and the temperaturelike variable (elliptic nome) p, the exact solution admits a phase angle ${\mathrm{w}}_0$ that interpolates between heights. Results for the free energy and local height probabilities are obtained. The exact solution exhibits first-order coexistence manifolds which coalesce at the multicritical point (p=0). For L=3, our results extend Baxter’s results on the three colorings of the square lattice.

We have found that the usual measuring procedure for preselected and postselected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value. A description of the measurement of the weak value of a component of a spin for an ensemble of preselected and postselected spin-(1/2 particles is presented.

We propose and test a new molecular-dynamics method to study fragmentation of condensed matter under homogeneous adiabatic expansion. Our atomistic simulations give significant insight into the nature of fragment distributions. We also find that a simple continuum model based on energy balance gives a reasonably good estimate of the average fragment mass.

In periodically driven systems, the intersection of stable and unstable manifolds of saddle orbits forms two distinct (topological) horseshoes: The first horeshoe is associated with the destruction of a chaotic attractor, while the second horseshow creates a new chaotic attractor. A laser model is used to illustrate how sequential horseshoe formation controls the birth and death of chaotic attractors.

We report a direct experimental observation of Pancharatnam’s phase, which is closely related to Berry’s topological phase. The experiment involves measurement of the phase change in one beam of a laser interferometer as the polarization state of light is taken along a closed circuit on the Poincaré sphere. The phase change is found to be equal to half the solid angle subtended by the circuit at the center of the Poincaré sphere. Apart from providing a striking demonstration of the topological phase, the experiment demonstrates that unitary time evolution of a system is not essential for the appearance of the topological phase.

We report the direct observation of a topological phase, i.e., an Aharonov-Bohm-type phase, as a fringe shift in an optical interferometer, which consisted of a modified Mach-Zehnder interferometer, in which the light traveled along nonplanar paths in its two arms. These arms were arranged symmetrically so as to have nearly equal path lengths, but opposite senses of handedness. The relationship between the phase acquired by a circularly polarized light beam and the solid angle subtended by the circuit of the spin vector of a photon in this beam was found to be a linear one with a slope of unity. The sign of the fringe shift also agreed with theory.

Aharonov and Anandan have recently reformulated and generalized Berry’s phase by showing that a quantum system which evolves through a circuit C in projective Hilbert space acquires a geometrical phase β(C) related to the topology of the space and the geometry of the circuit. We present NMR interferometry experiments in a three-level system which demonstrate the Aharonov-Anandan phase and its topological invariance for different circuits.

An infinite number of ways are developed for representing a function in terms of the eigenfunctions of a three-dimensional scattering problem and simple known auxiliary functions. The utility of the new expansions, which generalize both the Fourier and Radon transforms, is shown by derivation of a new representation of the scatterer for the near- (far-) field inverse problem. Further, the scattering amplitude and potential are shown to be a generalized Fourier-transform pair.

A new class of uncertainty relations is derived for pairs of observables in a finite-dimensional Hilbert space which do not have any common eigenvector. This class contains an ‘‘entropic’’ uncertainty relation which improves a previous result of Deutsch and confirms a recent conjecture by Kraus. Some comments are made on the extension of these relations to the case where the Hilbert space is infinite dimensional.

The fully directed compact lattice-animal model on the square lattice is solved exactly. An explicit expression is obtained for the cluster-number generating function which has a complicated complex-plane singularity pattern including a natural boundary of essential singularities. However, the cluster numbers are controlled by a simple pole singularity and grow proportionally to ${\lambda }^{\mathrm{N}}$, where N is the number of sites, and λ≃2.66.

We study the significance of a pronounced Cantori structure in classical phase space for the level statistics of the associated quantum system. As an example we choose the anisotropic Kepler problem. For high excitations, we find that the level statistics tend to the predictions of random matrix theory, but the convergence is rather slow.

An adaptation of lattice-gas cellular automata to the simulation of solid-fluid suspensions is described. The method incorporates both dissipative hydrodynamic forces and thermal fluctuations. At low solid densities, theoretical results for the drag force on a single disk and the viscosity of a suspension of disks are reproduced. The zero–shear-rate viscosity has been obtained over a range of packing fractions and results indicate that simulations of three-dimensional suspensions are feasible.

An increase in fractal dimension is shown to occur in filtered chaotic signals. The dependence of the Lyapunov dimension on the filter parameters is used to predict the behavior of the information dimension which is directly evaluated for two experimental systems: the NMR laser and Rayleigh-Bénard convection. Good quantitative agreement with the theoretical predictions is found. The understanding of the role of filtering not only clarifies aspects relevant in the calculation of fractal dimensions, but also yields an indirect but precise way of evaluating Lyapunov exponents.

A scalar model of two-dimensional Navier-Stokes turbulence first proposed by Gledzer is shown to realize the power law E(k)∼$k^{\mathrm{-}3}$ in its chaotic state, which is found to obey the same scaling law as that of the enstrophy-cascade theory. All the Lyapunov exponents are calculated for several values of viscosity, and they are found to have a scaling property in the interior of attractor. The calculated distribution function of the Lyapunov exponents appears to have a singularity at null Lyapunov exponent.

A two-dimensional generalization of the one-dimensional kicked-rotor model is introduced. As in one dimension, this model can be mapped on a two-dimensional Anderson model for localization. It is found that all the states are localized in momentum space and that the localization length grows exponentially with the mean free path, as expected from the scaling theory for localization. This suggests that the correspondence between the quantum dynamics of chaotic systems and Anderson localization that was found in one dimension holds in two dimensions as well.

This paper presents a new method for the solution of diffusion-reaction problems in one dimension. The method is used to derive some new exact results for the polymerization (cl-cl aggregation) and annihilation processes on openR and openZ. Through well-known dualities, these results have implications for the T=0 limit of the kinetic Ising model and for two interacting particle processes, the invasion and voter models. Prospectively, the method may be useful in providing one-dimensional verification for speculations in the theory of diffusion reaction.