We present a study of the antiferromagnetic Potts model in two and three dimensions, using a new method of Monte Carlo simulations, which enables us to perform simulations with greatly improved efficiency. Illustrating the method for the three-state model, we have obtained new results for the entropy and critical exponents in two and three dimensions. The low-temperature phase in three dimensions is shown to have long-range order with a finite-size dependence similar to that of the XY model.

The chaotic dynamics from a nonlinear electronic circuit is shown to exhibit the universal topological structure of maps on an annulus. This suggests that the corresponding universality class is large enough to include physical systems. We suggest that low-dimensional strange attractors fall into a few classes, each characterized by distinct universal topological features.

A Monte Carlo study is carried out for the three-state Potts model in three dimensions. The correlation length is shown to remain finite and to be discontinuous at the transition point. It is also shown that the transition exhibits genuine first-order characteristics in all aspects.

We suggest a new variational approach for quantum spin chains, based on the stochastic behavior of the quantization axes, which describes a chain in terms of a classical and a quantum contribution to the free energy. For an easy-plane ferromagnetic chain with the external field in the plane it reveals that the nonlinear excitations are predominantly described by the classical degrees of freedom. Hence it offers a fundamental basis for the description of the elementary excitations of such systems in terms of classical solitons and quantum spin waves, as suggested by various experiments.

We present a simple recursive iteration of the leapfrog discretization of Newton’s equations which leads to a removal of the finite-step-size error to any desired order. This is done in a manner that preserves phase-space areas and reversibility, as required for use in the hybrid Monte Carlo method for simulating fermionic fields. The resulting asymptotic volume dependence is exp[(lnV${)}^{1/2}$]. We test the scheme on the (2+1)-dimensional Hubbard model.

A theorem which connects exact thermal averages in a quantum system (fermions or bosons) with microcanonical averages in the presence of only one extra classical variable is proven. The theorem is a nontrivial extension of that introduced by Nosé for a classical system. It provides the basis for new canonical algorithms based on solving the time-dependent Schrödinger equation with a parametric dependence upon the extra variable. One such algorithm is constructed, and the canonical distribution is demonstrated for fermions in a harmonic potential.

The quantum signature of chaos in dissipative systems is cubic repulsion of their generalized energies (eigenvalues of the generators of the dynamics) in the complex plane. As in the Hamiltonian case the degree of repulsion is the same under temporally homogeneous conditions and periodic driving. Somewhat surprisingly, however, cubic repulsion prevails irrespective of whether or not the Hamiltonian embedding of the dissipative system obeys time-reversal invariance. More-over, even antiunitary symmetries of the dissipative generator itself cannot modify the repulsion exponent. In establishing the universality in question we find, as a useful byproduct, the generalization of detailed balance for periodically driven systems with damping.

We study the spectral statistics of systems of two-dimensional pseudointegrable billiards. These systems are classically nonergodic, but nonseparable. It is found that such systems possess quantum spectra which are closely simulated by the Gaussian orthogonal ensemble. We discuss the implications of these results on the conjectured relation between classical chaos and quantum level statistics. We emphasize the importance of the semiclassical nature of any such relation.

A closed-form expression is proposed for the critical frontier, or the critical line, of the antiferromagnetic Ising model on the honeycomb lattice in a nonzero external magnetic field. We formulate the Ising model as an 8-vertex model and identify the critical frontier as a locus invariant under a generalized weak-graph transformation. In its simplest form the locus is an expression containing two unknown constants which we determine numerically. The resulting critical frontier lies very close to results of a finite-size analysis.

We study the roughening of a growing surface near to a morphological transition within a general scaling framework. For a class of systems, where the transition can be related to directed percolation, the anomalous roughness at the critical point is at most logarithmical. In two-dimensional simulations we find ${\mathrm{w}}^2$∼logL where w is the width of the surface and L is the substrate size.

The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.

Detailed statistical analyses of chaotic signals have considered, so far, flows with one expanding and one contracting direction. Here, by studying filtered signals, we provide evidence that the existence of a further stable direction can break the self-similarity of the attractors and generate phase transitions (according to the thermodynamic formalism). In a hyperbolic case the dimension spectrum is computed exactly as a function of the filter cutoff, while in a nonhyperbolic case (NMR laser) the existence of phase transitions is evidenced numerically.

We describe the statistical mechanics of a D-dimensional Gaussian crumpled manifold embedded in d-dimensional space, and interacting by excluded volume with an Euclidean D’-dimensional subspace of ${\mathrm{R}}^d$. The effective theory describes a self-interacting manifold. An ε expansion [ε=D-(2-D)(d-D’)/2] is devised. Its one-loop renormalizability is proven to all orders, giving the first step towards that of self-avoiding manifolds.

We consider the relativistic interaction of a charged particle with obliquely propagating waves of arbitrary polarization. We show that an arbitrarily small wave packet with parallel phase velocity comparable to the speed of light can, under certain conditions, accelerate particles to unlimited energy through a process of Arnol’d diffusion in two-dimensions.

The simplest random evolution is the motion of particle on a straight line with constant velocity suffering random collisions, which reverses the velocity. The position x(t) of the particle at time t is the stochastic process that defines Kac’s path-integral solution of the telegrapher equation. We view Kac’s prescription as a path-dependent time reparametrization, which associates to the time t spent by the particle in going from ${\mathrm{x}}_0$ to x along a path ω, the time r(ω) the particle would have taken to go from ${\mathrm{x}}_0$ to x without reversal. We compute explicitly the probability distribution of the random variable r. It is then possible to compute by simple quadrature any path integral with an integrand function of r(ω).

The quantum-mechanical uncertainty in the position of a particle or the time of its emission is shown to produce observable effects that are inconsistent with any local hidden-variable theory. A new experimental test of local hidden-variable theories based on optical interference is proposed.

An exposition is given of the fundamental ideas of the recently opened field of two-particle interferometry, which employs spatially separated, quantum mechanically entangled two-particle states. These ideas are illustrated by a realizable arrangement, in which four beams are selected from the output of a laser-pumped down-converting crystal, with two beams interferometrically combined at one locus and two at another. When phase shifters are placed in these beams, the coincident count rates at the two loci will oscillate as the phases are varied, but the single count rates will not.

Diffusion is explored in a two-dimensional phase space in which a connected separatrix layer (web) of intrinsic stochasticity bounds regions of regular motion (tiles). In the presence of weak extrinsic noise, if the web diffusion dominates, the noise slows the web diffusion rate; if the extrinsic diffusion dominates, the diffusion is enhanced. The diffusion is characterized by two space and time scales. For a local-equilibrium model analytic calculations agree well with numerical results.

The cubic nonlinear Schrödinger equation and some of its discretizations, one of which is integ- rable, are studied. Apart from the integrable version the discretizations produce chaotic solutions for intermediate levels of mesh (mode) refinement. Chaos disappears when the discretization is fine enough and convergence to a quasiperiodic solution is obtained. Details are given for finite difference calculations, although similar results are also obtained by Fourier spectral methods. Results regarding a forced nonlinear Schrödinger equation are briefly described.

We present a method to study the existence of effective Hamiltonians for lattice, Ising-type model systems with competing dynamics, and find explicit expressions for some relevant nonequilibrium situations. We also define dynamical versions of the random-field and spin-glass models.