Table of contents

The gauge equivalence between the Davey-Stewartson (DS) equation and the (2+1)-dimensional continuous Heisenberg ferromagnetic model is shown explicitly. Through the gauge transformation, solutions of the DSII are obtained from the vortex type solutions of the former equation.

An SO(3) gauge theory of the spin-1/2 Heisenberg antiferromagnet (HA) on a two-dimensional square lattice is developed. The effective action Γ_SC of the system is described by an anisotropic SO(3) σ model supplemented by Berry's phase terms, featuring a possible instability of the regular Neel ground state. This problem is discussed in connection with the singlet nature of the ground state of the spin-1/2 HA and in terms of a generalised chiral SO(3)² σ model.

The Bethe-ansatz equations for the integrable spin-s isotropic Heisenberg antiferromagnet are solved numerically at a finite number of sites N. The vacuum and the lowest singlet and triplet solutions are presented for s=1/2, . . ., 2 up to sN=240. Through extrapolation of the finite-size data, the central charge and anomalous dimensions of the scaling operators for the underlying conformal field theory are calculated and found to agree with the expected theoretical values. The status of the Bethe string hypothesis about the structure of the solutions is discussed based on the obtained computer data up to s=9/2.

The authors generalise the well known connection between critical exponents and finite size effects in conformal field theories to non-local operators and find the long-wave asymptotics of vacuum expectation values of some non-local operators in one-dimensional quantum systems.

A general stochastic model of cluster growth is proposed. In two special limits the model is reduced to the diffusion-limited aggregation (DLA) and the Eden models. By varying a physical parameter, the model interpolates smoothly between the DLA and the Eden model, and is also capable of simulating miscible viscous fingers in a porous medium, when dispersion effects are absent, in which the ratio of the viscosities of the displaced and displacing fluids is finite. The predictions agree with previous results obtained by a deterministic model. The model also has close connections with a variety of other models of growth phenomena.

The staggered quadratic dimer model of a biomembrane is considered in the case of a lattice with modified activities of the vertical bounds in a selected row. An exact expression for the excess surface free energy is obtained and its dependence on the bulk parameters which control the density, flexibility and ordering of the polymer chains is investigated.

A model describing an ensemble of disjoint clusters on a square lattice is introduced. In addition to the usual surface energy terms, in _x and in _y, the model includes a bending energy in _c per corner in the interface. It is formulated as a seven-vertex model which is in general not solvable. For beta in _c=ln(2)/2, however, this seven-vertex model becomes free fermionic one and an exact expression for its free energy is derived. In the extreme anisotropic ('Hamiltonian') limit the operator content coincides with that of the 1D Ising model in a transverse field. Three critical lines with Ising-like behaviour are found.

The authors show that the most physically interesting solutions of the SU(2) Yang-Mills equations, the well known meron and instanton solutions (and some others), are generated by corresponding solutions of nonlinear Dirac-Gursey spinor equations.

The author has performed a classification of the Fokker-Planck-type equations according to the maximal symmetry groups that keep the equations invariant. It is found that there are only four classes of such equations and the relations between the 'transition probabilities' of the equations have been obtained. He has also obtained complete functional bases of the invariants of the corresponding four groups.

The author proposes a reversible and local rule that allows large-scale objects called 'strings' to move with adjustable speed and energy in a three-dimensional space. He shows that the motion of these strings is governed by discrete Hamiltonian equations and mediated by one longitudinal and two transverse sound waves that propagate along the string. This rule is a first attempt to model a solid body with a cellular automaton. It also provides interesting possibilities for simulating new physical situations.

The author uses a generalised Von Karman-Heisenberg-von-Weizsacker-type model for the inertial transfer to give a generalised spectral law for the enstrophy cascade in a two-dimensional turbulence that exhibits a steeper energy spectrum for large wavenumbers and reduces to the well known k^-3 spectrum at the other end of the spectrum. For very high wavenumbers, this spectrum is, in fact, an arbitrarily steep power law. Nonetheless, it is possible to give an even more rapidly decaying spectrum for this range, using a continuous spectral cascading model. He then discusses the intermittency aspects of the departures from the Batchelor-Kraichnan scaling law and shows that while the intermittency corrections within the framework of the beta model of Frisch et al. (1984) are in qualitative agreement with the predictions made by the generalised spectral law given in this paper, intermittency by itself is unable to account fully for the steeper spectra at large wavenumbers observed in the numerical experiments. The author discusses further fractal aspects of the enstrophy cascade, and shows that for the enstrophy cascade, the fractal dimension rules not only the manner in which the cascading proceeds but also the point where it stops.

Wave propagation across potential barriers is an important aspect of theoretical physics. In this paper, the waves are described by fourth-order ordinary differential equations, giving two modes in which wave-energy can be carried in the non-dissipative medium. A conserved quantity controlling the relative distribution of energy into the different channels of propagation is derived, using only the properties of the governing equations.

An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.

The reflection coefficients of one-dimensional disordered conductors is asymptotically a pure phase, when the sample length is much larger than the localisation length. The authors study the stationary distribution of this reflection phase for tight-binding chains with diagonal disorder, i.e. random site potentials. They derive the Dyson integral equation obeyed by this distribution and relate it to the Lyapunov exponent and the integrated density of states. This general approach allows them to consider then more specifically two types of disorder. In the case of a binary alloy, where the potentials take two values, the distribution of the reflection phase is shown to have generically an infinity of power-law singularities related to the occurrence of periodic patterns of potentials. This phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy-dependent threshold. They also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials. This non-trivial solvable model allows them to examine analytically various properties of the phase distribution, such as the anomalies which occur at weak disorder.

The direct correlation function for a fluctuating interface in a two-dimensional solid-on-solid lattice model is studied. They obtain for a weak gravitational field an asymptotic expression for the direct correlation function that agrees well with previous numerical results. In particular two different vertical length scales appear.

The authors use the generalised coherent state distribution to study in phase space the eigenfunctions of non-integrable Hamiltonians. The methods available in the Weyl group (the Husimi distribution) are extended to the more complicated phase spaces that arise from other Lie groups. They demonstrate the feasibility of numerical calculations in two models with W₂ and SU(3) symmetries, emphasising the similarities and showing the emergence of classical invariant structures in the semiclassical limit.

The quantisation of the two-dimensional toric and spherical phase spaces is considered in analytic coherent state representations. Every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its Husimi function. For eigenstates of quantised systems, this description explicitly reflects the nature of the underlying classical dynamics: in the semiclassical regime, the distribution of the zeros in the phase space becomes one-dimensional for integrable systems, and highly spread out (conceivably uniform) for chaotic systems. This multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems.

Using efficient cluster expansion methods, the known high-temperature series for the vacuum energy, specific heat, susceptibility and mass gap of the (2+1)-dimensional Ising model on the square and triangular lattices have been extended by several terms. Estimates of the critical indices demonstrate very convincing universality with the Euclidean version of the model.

Recently the author suggested studying the scaling properties of complex two-dimensional phase transitions by calculating the universal amplitudes of interfacial free energies on a torus geometry employing Monte Carlo simulations. He tests this method at the 3*3 phase transition in the Ising lattice gas on a triangular lattice belonging to the chiral three-state Potts universality class. The interfacial free energies for this lattice gas model can be written in terms of free energy differences between systems with different sizes. The algorithm to calculate these free energy differences is presented. Using lattice sizes up to 18*18 and modest computing effort, he finds that the universal amplitude A=1.220.02, in accordance with the exact analytical value for the non-chiral three-state Potts model, A=1.203291 . . ., which he determined earlier.

The authors study the dynamics of a strongly diluted and a non-diluted hierarchical model. The first one is solved exactly. The phase diagram and the size of the attraction basins of the fixed points of the dynamics are calculated and their behaviour is compared.

The author writes down explicit expressions for the high-temperature expansion of several spin and gauge models with logarithmic action, defined on certain lattices. Such lattices exist in any dimension for spin models, but in the gauge case their existence is only proved modulo a technical assumption. The partition function of the spin models is a gas of closed loops, weighted by N^L, where N is the number of spin components and L is the total length of the loops. The gauge model is a gas of closed, self-avoiding surfaces, weighted by N^chi , where N is the dimension of the representation and chi is the Euler characteristic.

See ibid., vol.22, p.L191 (1989). In a recent letter on the generalised exponents of the Lie algebra sl(3,C), Scutaru refers to an integrity basis of the enveloping algebra proposed by the authors as a hypothesis and then proceeds to establish its validity. The purpose of this comment is to stress the fact that this integrity basis was not a hypothesis but a proven result.

The authors present a local functional integral representation for the random diffusion equation studied by Kardar, Parisi and Zhang (1986).

The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.

The author shows how one can give a quantum description for systems having (2n+1)-dimensional anticommuting phase space. The Hilbert space related to these systems has 2ⁿ dimensions. Among the applications of those systems are spin in three dimensions and the d-dimensional relativistic spinning particle.

For low concentrations, the occupied sites in Q2R cellular automata on the square lattice are known to be confined to the smallest rectangles surrounding them. However, they do not necessarily fill out the whole rectangle allowed for them. The effective confinement transition is found numerically to agree well with that of bootstrap percolation.