Table of contents

Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.

For pt. I, see ibid., vol. 20, no.8, p.L479-85 (1987). The authors present the operator content of the Ashkin-Teller quantum chain with eight boundary conditions corresponding to the eight elements of the group D₄ which gives the global symmetry of the system. This operator content is a conjecture based on an extensive numerical study and was proven to be correct at the Ising decoupling point. For the values of the coupling constant where N=2, superconformal invariance was seen for free boundary conditions. The situation is now more complex. One finds sectors which have N=2 and N=1 superconformal invariance and sectors which have no superconformal invariance at all. If, however, they combine the 'wrong' sectors of the spectra which correspond to two different values of the coupling constant, a new symmetry shows up. They also analyse the operator content at the Fateev-Zamolodchikov points. Here the 'wrong' sectors can be described by a few new primary fields.

It is well known that there is no strict universality of the spectral fluctuations of quantum Hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos. The author discusses the level spacings distribution P(S), and explains why the semiclassical formulae of Berry and Robnik cannot be correct for small S. There is no global universality of P(S) for nearly integrable systems, but the approach to the integrability as the perturbation parameter in goes to zero can be universal. This is reflected in the fact that the slope of dP/dS at S=0 for small in is universally inversely proportional to in . The author gives two models in terms of two-dimensional random matrices, one of them being based on maximum entropy considerations. The author also points out the connection to the statistics of zeros of random functions, and discusses the numerical evidence.

A simple real space renormalisation scheme is proposed for the statistics of the polymer coil-globule transition in two dimensions. This approach describes the coil, theta point and globular states of the macromolecule within a single two-parameter transformation. Results from a small-cell calculation include the theta -point size exponent, v_theta =0.494, and crossover exponent phi =0.486.

An experiment measuring the transition to a convective pattern in a Rayleigh-Bernard cell is shown to furnish evidence supporting the fact that the onset of a centre manifold determines scaling equations relating the intensity of intrinsic fluctuations to the other characteristic small parameters of the system.

The authors use exact enumeration and Monte Carlo techniques to test some recent predictions by Duplantier and Saleur (1986) of the values of the critical exponent gamma for uniform star-branched polymers in a wedge geometry in two dimensions. Results support their predictions. The authors have also estimated the exponent v and amplitude governing the n dependence of the mean square radius of gyration and the mean square end-to-end branch length. In some cases the branches are distinguished and have different mean lengths but the exponent v is equal to the bulk self-avoiding walk value in every case.

The author defines a model for information propagation through the oriented bonds of a square lattice. The source of information is the origin, and the propagation takes place in only one quadrant, along the diagonal t (time) direction. The information reaches increasing-t layers of lattice points (r, t) according to a probability distribution P(x-x_c<<, r, t) dependent only on the previous P(x-x_c, r, t-1), where x is the concentration of present bonds. The model shows two distinct behaviours for large values of t: the information will disappear if xx_c , or survive forever if xx(inf) c . Taking advantage of the Markovian behaviour and assuming that P is homogeneous, the author gets the values x_c=1/2 for the critical concentration, v=1 for the t-correlation length critical exponent, theta =2 for the t/r crossover exponent, and beta =1/2 for the order parameter exponent. Along directions other than t, from the same origin, we get v₁=1. The homogeneity assumption is supported by numerical calculations of the time evolution. This evolution is a deterministic cellular automaton, each cell r retaining a real (instead of discrete) value for P. The similarities and differences between the present model and directed bond percolation are also discussed.

Viscous fingering in porous media is an instability which occurs when a less viscous fluid displaces a more viscous one. An interface between the fluids is unstable against small perturbations and gives rise to a fingered configuration. In the oil industry viscous fingering can be a serious problem when displacing viscous oil by a more mobile fluid because it leads to poor recovery of the hydrocarbon. Recent work suggests an analogy between viscous fingering at an infinite viscosity ratio and diffusion-limited aggregation (DLA) and hence that the fingers may be fractal with a fractal dimension of around 1.7 (in two dimensions). This leaves unanswered the question of the nature of the fingered patterns at a finite viscosity ratio. To answer this a network model of the porous medium has been used. The rock is modelled as a lattice of capillary tubes of random radius through which miscible displacement occurs. At a high viscosity ratio and in the presence of a large amount of disorder the model reproduces DLA fingering patterns. The results of this model provide evidence that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and a DLA result of 1.7 with increasing viscosity ratio.

The author constructs a three-dimensional generalisation of the brick lattice and shows that the partition function for a suitably modified SO(n) or SU(n) lattice gauge theory can be written exactly as a sum over closed surfaces on this lattice. For Ising lattice gauge theory, the author derives a formal expression for the partition function.

The author shows how to write down the irreducible representations of the Temperley-Lieb algebra. The author shows that these representations satisfy Jones' trace condition in the infinite limit.

The authors calculate exactly the number generating function for partially directed compact lattice animals on the square lattice. For the fully directed model, they report results of new numerical studies.

The author determines the bulk, thermal and magnetic exponents at the conformal invariant point in the Z₅ model with a Monte Carlo based finite-size calculation. The results are in good agreement with the predictions of Zamolodchikov and Fateev (1985). The author also argues that the (N-1) surface magnetic exponents x_h(j) in the Z_n model are given by j(N-j)/N (j=1, . . ., N-1). This prediction is also verified in a Monte Carlo finite-size calculation for the Z₅ model.

Symmetry properties of the density of states in the Brillouin zone for the Heisenberg model of a finite one-dimensional magnetic crystal are investigated using a general prescription of Weyl, which consists here in an analysis of the action of the ring of endomorphisms of a cyclic group in the space of quantum states of the magnet. It is shown that the density of states is constant on orbits of the group of automorphism in the Brillouin zone. Each such orbit can be thus interpreted as a generalised star of a wavenumber. It is shown that the distribution of states in the discrete one-dimensional Brillouin zone is governed by some selection rules on the lattice (i.e. partially ordered set with unique maximal and minimal elements) of subgroups of the cyclic group.

Prolongation structures of the supersymmetric sine-Gordon equation are discussed. It is shown that an infinite-dimensional superalgebra is associated with these structures and that a linear representation of the algebra gives the super Lax pairs of the equation.

Two sets of infinite number of symmetries, their Lie algebra properties and constants of motion for a class of non-linear evolution equations associated with non-isospectral deformations of the AKNS spectral problem are constructed.

Using the indefinite metric, the author studies the coupling of three irreps of the OSP(1, 2) algebra, and gives the general definition, orthogonality conditions, symmetry properties and calculating formulae of the Racah coefficients of the OSP(1, 2) algebra. Eight kinds of Racah coefficients exist for the OSP(1, 2) algebra and they are all expressed in terms of the Racah coefficients of the SO(3) algebra.

The author studies the one spatial dimensional, 6-velocity Broadwell model with four identical densities and three independent ones. The author determines 'solitons' (one-dimensional shock wave solutions) and 'bisolitons' (two-dimensional, space plus time solutions) which are rational fractions with one or two exponential variables. The author obtains three classes of positive exact solutions in 1+1 dimensions (space x, time t). The first one is periodic in the space variable and for large time the solutions correspond to propagating damped linear waves. The second is positive only along one semi x axis while the third, positive along the whole x axis, represents non-planar damped shock waves. Using the same tools in a companion paper, for the discrete 2-velocity models, the author obtains in a two-dimensional space the first two classes of solutions mentioned above. This suggests that, for the discrete Boltzmann models, general methods exist for the determination of non-trivial exact solutions.

New affinely minimal surfaces are constructed with the use of Backlund's theorem. The corresponding affine Backlund transformation is studied in some detail.

First integrals for the classical Kepler problem with drag were obtained by Jezewski and Mittleman (1982). A derivation which is more attractive from an intuitive point of view is provided and this leads naturally to the orbit equations as for the standard Kepler problem.

The authors apply the Painleve test to the generalised derivative non-linear Schrodinger equation, iu_t=u_xx+iauu*u_x+ibu²u_x+cu³u*² where u* denotes the complex conjugate of u, and a, b and c are real constants, to determine under what conditions the equation might be completely integrable. It is shown that, apart from a trivial multiplicative factor, this equation possesses the Painleve property for partial differential equations as formulated by Weiss, Tabor and Carnevale (1983) only if c=1/4b(2b-a). When then this relation holds, this is equivalent under a gauge transformation to the derivative non-linear Schrodinger equation (DNLS) of Kaup and Newell, which is known to be completely integrable, or else to a linear equation.

The modulation of a one-dimensional weakly non-linear purely dispersive quasi-monochromatic wave (the carrier) is usually governed by the non-linear Schrodinger (NS) equation. The critical wavenumber for which the carrier is marginally modulationally unstable is determined by the condition that the product of the coefficients of the non-linear and dispersive terms in the NS equation is zero. However, near this marginal state the assumptions that lead to the NS equation are invalid and a modified form of the NS equation that involves higher-order non-linearities is appropriate. This modified NS equation is here derived formally for a general system involving a single dependent variable and a revised form of the instability criterion is obtained. The results are illustrated by considering a particular system described by a generalised Korteweg-de Vries equation.

The equivalence of the Ampere and Biot-Savart force laws in magnetostatics is examined by considering the difference in the forces predicted by the two laws. The conditions under which this force difference vanishes are discussed, with special reference to the question of convergence of the integrals involved. The conclusions are that a complete current distribution exerts forces, which are the same according to both laws, when acting on a volume current element or a macroscopic part of a current distribution, irrespective of whether these are situated outside or inside the distribution exerting the force. These forces are shown to be normal to the current density vector at all points according to both force laws. Both the force distributions and stresses are predicted equal by both laws, thus proving the impossibility of devising an experiment which might differentiate between the two laws by measurements of forces exerted on a part of a circuit by the circuit itself. The equivalence of the two force laws in magnetostatistics is considered to be complete.

A survey of analytic techniques for solving the two-electron atomic Schrodinger equation is presented. The hyperspherical formalism is introduced and specialised to the case of two electrons and zero total angular momentum. Following Fock, the Schrodinger equation is then converted to an infinite set of coupled second-order differential equations by proposing an expansion including logarithmic functions of the interparticle coordinates. The equivalence of the techniques of Pluvinage and Hylleraas (1985) to the Fock expansion is demonstrated and the method for solution is illustrated. The extension to states of arbitrary angular momentum and excited states is indicated. Methods for simplifying the recurrence relation generated by the Fock expansion are used to determine the highest power logarithmic terms to sixth order. Finally, the wavefunction for S states is given to second order as a singly infinite sum of Legendre polynomials.

For pt.I, see ibid., vol.20, no.8, p.2043-75 (1987). Several coordinate systems for solving the few-electron Schrodinger equation are presented. Formal solutions corresponding to each coordinate system are given in terms of the Fock expansion and their interrelationships and general structure are examined. Attention is focused on the solutions obtained using spherical polar coordinates for a Coulomb potential of arbitrary symmetry. The wavefunction is obtained up to second order in the hyperradius r=(r²₁+r(sup)2₂)¹2/, and the special case of ¹S states is then reduced to a closed form using classical techniques. The insight gained from this reduction suggests methods for solving the wavefunction to all orders. The results hint at the existence of closed form wavefunctions for few-body systems.

The radiation reaction of electrons and positrons in a magnetic field is discussed using quantum electrodynamics based on the exact solution of Dirac's equation. One quantum number arising from this solution is found to depend on the position of the centre of gyration. The change in this quantum number under the emission or absorption of radiation is interpreted as a change in the centre of gyration, which leads to a classical current. A conservation law for momentum components in the xy plane may be derived from this interpretation. 'Quantum broadening' is reinterpreted as a change in the gyrocentre instead of a spread in the wavefunction. The concept of recoil is generalised to include processes involving pair production and annihilation.

On the basis of the theory of Lindblad (1976) for open quantum systems the author derives master equations for a system consisting of two harmonic oscillators. The time dependence of expectation values, Wigner function and Weyl operator are obtained and discussed. The chosen system can be applied for the description of the charge and mass asymmetry degrees of freedom in deep inelastic collisions in nuclear physics.

Hinton and Sejnowski (1983) have described recently a novel statistical mechanical system which they named the Boltzmann machine. The interesting property of Boltzmann machines is that they can learn to recognise the structure in a set of patterns simply by being shown an example subset of patterns. In this paper some numerical simulations of Boltzmann machines are reported. It is found that the annealing schedule proposed by Ackley, Hinton and Sejnowski (1985) is adequate to obtain a Boltzmann distribution of states, on which the key part of the algorithm depends, but it is clear that the algorithm will require massive computations for large networks. It is also found that there is a window of annealing temperatures at which learning is possible, and the sensitivity of the learning rate to temperature can be understood in terms of the density of states at low energies. Direct calculations of the partition function in small instances of Boltzmann machines are used to characterise the number of states which are thermally accessible for particular annealing schedules. Finally, since Boltzmann machines bear some resemblance to models of disordered magnetic systems, a comparison is made with results for the Sherrington-Kirkpatrick spin-glass model. Both systems support multiple metastable states (i.e. stable with respect to single spin flips), but, in contrast to the SK spin glass, Boltzmann machines exhibit a random distribution of low-energy states in terms of Hamming distance.

The master equations for bistability in the bad cavity limit in an atomic system is solved when the number of atoms N is not more than 45. A detailed comparison is made of results from Fokker-Planck equations and from the master equation. The validity of the decorrelation approximation for the master equation is examined.

The authors consider the dilute Potts model on the Sierpinski carpet. The model has a phase transition at a finite temperature and both critical and tricritical behaviour are observed for qq_c (where q is the number of Potts states). They use Migdal-Kadanoff type recursion relations to obtain the value of q_c and of critical and tricritical exponents and investigate their dependence on various geometrical characteristics of the fractal.

Diffusion-limited aggregation is a proven method for simulation of certain types of two-fluid displacements in porous media. Heterogeneity in permeability can be simulated by setting the lattice spacing equal to the local permeability.

The (111) plane to (111) plane transfer matrix of the Ising model on a simple cubic lattice is obtained in a simple product form by using a purpose-built matrix product previously defined by this author. As a first application of this theoretical result, the exact analytic expression of the partition function of an Ising model on a 2*2* infinity lattice is derived.

The Ising model with an infinite line of defects is mapped onto a strip with two defect lines. The Hamiltonian spectrum is studied at the bulk critical point. Its exact diagonal form is found for an infinite number of sites. The spectrum of physical excitations contains an infinite number of primary fields, while the leading ground-state energy correction is independent of the defect strength. A novel algebraic structure interpolating between those belonging to periodic and free boundary conditions is signalled.

The author tries to explain some common features of the spin-glass phase diagrams found so far using Nishimori's method for +or-J and Gaussian distributions in Ising models on hypercubic lattices.

The authors analyse the non-Abelian Aharonov-Bohm effect in the Feynman path integral framework generalised to pseudo-classical mechanics.

The authors develop a very simple variational procedure for obtaining quite accurate analytical expressions for the eigenvalues of quantum mechanical models. It consists of finding the minimum value of a properly built functional form for the energy in the phase space. The method enables one to make use of available information about the analytic structure of the eigenenergies. Results are shown for the linear confirming potential model.

Based on a paper by Nakamura, J. Phys. A, vol.19, p.2345 (1986). The authors correlate two methods for packing squares on a square lattice and find agreement with published computer simulation results.

The polygon exponent h associated with directed self-avoiding walks in two dimensions has been obtained by exact enumeration and series extrapolation methods. The value of h is found to be -1.40+or-0.05 and the connectivity constant associated with such polygons is seen to be less than that of the corresponding walks.

Based on a paper by Lipson et al., J. Phys. A, vol.18, p.L469 (1985). Additional terms are reported for the series of star branched polymers. Very recently, Lipson et al (1985) presented a series enumeration study of star branched polymers. The author has extended several of their series by one term (one series by two terms) in an attempt to permit a more accurate analysis of the series data. Unfortunately, this new series information appears to be primarily of academic interest, as the new terms do not alter the exponent estimates given in Lipson et al. Possibly the only modification is in the extrapolation of gamma (3) (in the notation of Lipson et al) in two dimensions. Based on data from the triangular lattice, a slightly tighter error bar for this exponent is reasonable.

A new field theory representation for the percolation problem is derived by explicitly taking the n=0 limit in the usual (n+1)-state Potts model formulation. This representation is used to investigate the validity of an analytic continuation recently introduced to determine the large-order behaviour of the perturbation expansion in this problem.