Table of contents

The static Abelian Higgs model in an arbitrary two-dimensional geometry is considered. The consistency of a Bogomol'nyi bound to the energy functional is discussed and it is shown that the Bogomol'nyi equations are equivalent to a prescribed curvature problem.

A closed analytic form is obtained for the Bloch or canonical density matrix of a localised oscillator in a magnetic field of arbitrary strength, by solving the Bloch equation with the completeness boundary condition.

An exact solution is obtained for the root-mean-square end-to-end distance R_N of spiral self-avoiding walks with N steps on a triangular lattice. For N to infinity , R_N increases as (2 pi )^-1(6N)^1/2 log N.

The fractal dimension d_G of cluster perimeters generated by a recently proposed 'butterfly' growth walk is considered. In the long-range limit of the walk on a percolation cluster, d_G appears to be equal to the fractal dimension of the singly connected bonds: d_G=1/ nu . The new relation for chemical dimension d_l is proposed: d_l=d_f/(d_f-d_G). In the short-range limit the 'butterfly' walk on a Euclidean lattice appears to be in the same universality class as a random walk. The dynamic aspect of the growth walk is discussed and the continuously tunable spectral dimension is obtained. Both short- and long-range limits of this diffusion process are different from a random diffusion on percolation.

The average length of the shortest paths along the backbone of the incipient infinite cluster is expected to diverge near the percolation threshold p_c as mod p-p_c mod ^{- nu s}, with nu _s different from the correlation length exponent nu . The author applies the position space renormalisation group (PSRG) technique to obtain nu _s both for square and cubic lattices and for a triangular lattice in two dimensions. The results obtained ( nu _s=1.49 for d=2 and 1.21 for d=3) are in very good agreement with the existing results obtained by different methods.

A relatively large number (10²-10³) of reasonably large (about 10⁵ particles or sites) two-dimensional Witten-Sander aggregates are simulated using new algorithms. This enables the reduction of statistical errors by an order of magnitude and provides an opportunity to determine some new quantities describing in more detail the structure of Witten-Sander clusters. A quantitate description of the distortion of large lattice-based Witten-Sander clusters into a diamond-like shape is provided. The orientational correlation in the bond directions for off-lattice Witten-Sander clusters is also measured. This quantity has a power law dependence on distance with an exponent of about -0.37.

A modification of the classical Heisenberg model is proposed in which the continuously rotating spin variables are replaced by spins which are only allowed an isotropically distributed discrete set of directions. The change is motivated by a need to simplify the calculations associated with Monte Carlo simulation of vector spin systems, a goal accomplished by the present model. An efficient table-based simulation technique is described and preliminary results for the critical region presented; various extensions of the model are discussed.

The author discusses the similarities between random-field Ising ferromagnets and two-dimensional Ising spin glasses.

The authors prove that an Ising model with even-spin interactions has no spontaneous magnetisation at high temperatures when the ranges of interactions are finite.

The Parisi overlap functions P(q) and P(q₁, q₂, q₃), and the function Pi (Y), which measures sample to sample fluctuations, are calculated for the one-dimensional Ising spin glass in the temperature range in which the correlation length is of the order of the length of the system. The functions P(q) and Pi (Y) are strikingly similar to those of the SK model but the topology of the space of states is not ultrametric.

The quantisation of the nonAbelian charge of a point-like particle is shown to arise from the BWS condition in the pre-quantisation scheme of Kostant and Souriau.

The dimensionalisation hypothesis given in a previous paper (1981) is modified and established in a way which is as independent of arbitrary choices as possible. The process of group contraction is also analysed as well as the relation between dimensions in the original and contracted group. The theory is illustrated with the example of Cayley-Klein geometries and an ultrarelativistic limit of the Poincare group.

The infinite-dimensional Abelian group of general Backlund-Calogero transformations is constructed for the evolution equations which can be integrated by a Gelfand-Dikij-Zakharov-Shabat spectral problem of an arbitrary order. The structure of the recursion operator and transformation properties of the Backlund-Calogero transformations under the gauge group are considered.

A canonical orthonormal basis is given for generic representations of the group chain SU(3) contains/implies SO(3).

For pt.I see ibid., vol.18, p.1891 (1985). A simple algorithm is given for the calculation of reduced matrix elements of the generators of the SU(3) algebra in the canonical SU(3) contains/implies SO(3) basis introduced in I.

A class of unitary transformations of reflective character is proposed for multi-mode quantum transport systems. The simplest of these is discussed in detail. To optimise the transformation parameters, a new variational principle is introduced which is based on the definition of a positive definite 'measure of non-diagonality' for each state. The optimised transformation removes the leading order of non-diagonality in the phonon coupling strength D (from D⁰ to D^-2 for D²>>1 and from D² to D⁴ for D²<<1). In this manner the basic shortcoming of quantum diffusion systems, which is the dominating non-diagonality between energetically far-off states, is drastically reduced.

Backlund transformations for the non-potential MKdV equation and its Calogero-Degasperis-Fokas modifications are presented and used for finding particular, wave-type solutions to the equations.

The numerical solution of the inverse spectral transform for the Korteweg-de Vries equation is rather time consuming and can give rise to instabilities. The authors propose a formalism which allows one to derive in closed form the solution of the Cauchy problem for the Korteweg-de Vries equation for short times. Such a solution is obtained by going over to a Pade approximant of the time-evolved reflection coefficient.

Functional integrals over complex and real auxiliary fields for propagators relating to two-time (memory) actions are constructed. The integrals over fields result from partial replacement of the path variable, via functional integration, in the path integral form of the propagator by a field variable in a way that allows the remaining path integration to be performed, thus leaving a functional integral. Two new cases of actions involving memory, a quadratic and a quartic are treated explicitly. A functional integral identity is also developed. Furthermore, the method of coupling an auxiliary dynamical field for generating memory actions is also employed and formulae are developed in the case of certain quadratic actions giving the propagator exactly. The method is flexible enough embracing all explicitly known cases of two-time quadratic actions and in addition enabling further cases to be treated. An example of this nature is given explicitly.

An integrable dynamical problem of a classical neutral particle interacting with an inhomogeneous magnetic field via a magnetic dipole moment is considered; this has been studied quantum mechanically (1978). The equation for the centre-of-mass velocity is of second order resembling that of an anharmonic oscillator with velocity replacing position in the equation. The velocity undergoes curious oscillations, where the spin and centre-of-mass degrees of freedom exchange energy, although there is also, in general, a uniform component to the velocity. An application to the motion of a classical 'neutron' in a magnetic domain wall is discussed, and comparison with previous quantum treatments made.

This is the first of a series of papers concerning the group theoretical approach to the classical particle systems, related to a definite special relativity principle and which are explicitly expressed in a Lagrangian formalism. In the Galilei case, the free classical particle can be characterised by its mass, internal energy and spin, similarly in the quantal version. The spin appears as a derived observable related to the orientation and to the internal motion of the system and has in general three parts; one of intrinsic nature, one, like a spherically symmetric body, related to the angular velocity and a third which is the (anti)orbital part of the relative motion of the system around its centre of mass. Galilei photons are massless particles, with spin and linear momentum lying along the direction of motion, travelling at infinite speed and carrying no energy. The Lagrangian of a free particle with intrinsic spin is no longer time-reversal invariant.

The authors present two cases of a quantum harmonic oscillator with a time-dependent nonzero mass. These are considered to be more realistic than the previously studied variable masses. In particular, periodic solutions for the kinetic and potential energies are investigated by means of stability charts. The general periodic mass is discussed briefly.

By an analysis of the current and dipole mean square fluctuations per unit volume for a quantum mechanical jellium in thermal equilibrium at temperature beta ^-1, the authors give a new proof of the sum rule integral dx mod x mod ²S(x)=³/₄(h(cross) omega _p/ pi ) coth( beta h(cross) omega _p/2) for the second moment of the charge-charge correlation function S(x), where omega _p is the plasma frequency. No use is made of perturbation theories. The proof relies on correlation inequalities and sum rules which are valid in phases of charged systems having good screening properties. Up to second order in Planck's constant, the current-current correlations have a non-integrable spatial decay even when the corresponding classical state shows Debye screening.

The phase diagram of the Z(5) model on a square lattice is studied using Monte Carlo simulation techniques. The authors confirm the existence of three phases in the ferromagnetic Z(5) model and find a critical phase for the antiferromagnetic model. The phase boundaries are determined using Monte Carlo renormalisation group methods.

The constraint that extended molecules such as polymers cannot pass through each other is formulated in terms of a topological invariant. A simple model of the motion of a labelled loop entangled in a uniform background of amorphous loops is studied. A dynamical weighting factor is found which effectively removes configurational changes that would lead to the labelled loop passing through the strands of the background material. A perturbation treatment of this weighting factor is presented which describes the effects of the entanglements in terms of random environmental forces possessing considerable space and time correlations. A detailed analytic calculation is presented for the case of a labelled molecule with a rigid-ring configuration. This calculation illustrates the limited validity of perturbation theory in dealing with entanglements and suggests the existence of a transition in time to a random 'tube'-like behaviour, where the centre-of-mass motion of the ring depends on the time as t^1/4 instead of the usual diffusive t^1/2. The origin of this behaviour can be located in the long-time persistence of the bond vector correlation function of the labelled ring.

Non-equilibrium reaction-diffusion models show an exciting array of critical phenomena ranging from the continuous transitions reminiscent of equilibrium systems to the spectacular chemical oscillators. Transforming the underlying stochastic master equations using 'Poisson' techniques the author obtains for the first time a comprehensive description of the dynamics, including in particular the problematical multitime correlation functions. The formalism is ideally adapted for the use of scaling and renormalisation group ideas.

The Migdal-Kadanoff renormalisation scheme, which becomes exact on hierarchical lattices, is used to investigate electrical properties of 2D and 3D percolation clusters. The author studies in detail the frequency dependence of the conductivity and the loss angle, and the amplification of resistance (flicker) noise. This type of noise exhibits critical amplification, with its own exponents X and Y, which are related to the exponent b recently defined by Rammal et al. (1985).

Using a functional Fokker-Planck equation for a scalar phi ⁴ model, it is shown that the initial decay rate out of a metastable state after an instantaneous quench to an unstable situation is proportional to the corresponding imaginary part of the free energy, as calculated by Langer (1969).

The problem of the two-dimensional neutral Coulomb gas is analysed in two separate approximations: the one-atom approximation at low temperatures and the Thomas-Fermi approximation at high temperatures. A possible connection between the Kosterlitz-Thouless transition and the thermal ionisation of a single atom is discussed. An expression for the position dependence of the dielectric constant of the gas is obtained in the Thomas-Fermi approximation.

The authors present an analysis of N=4 supersymmetric Yang-Mills theory using a construction involving additional bosonic variables in the coset space SU(4)/H. No choice of H can be shown to lead to an analytic formulation of the theory. By introducing an analysis on dual planes the theory (including the reality constraint) is reduced to one involving N=2 symmetry. This approach has to be extended to include truly harmonic derivatives. For the typical case of SU(4)/SU(2)*U(1) prepotentials are introduced which solve the constraints. It has not been possible, however, to construct an action which leads to the equation of motion for the original N=4 supersymmetric Yang-Mills theory (at the linearised level).

The author considers the lattice Abelian Higgs model with frozen radial degrees of freedom using the mean-field approximation with one-loop corrections. In the weak-coupling region the behaviour of the frequencies arising in the expansion of the action allows the Higgs and Coulomb phases to be distinguished. Analytical results are presented for all phase transition lines. The phase structure obtained is in qualitative agreement with Monte Carlo calculations for Higgs charges q=1, 2 and 6.

The probability density of detecting a relativistic particle has been found by Newton and Wigner (1949) for a point in space. This probability density must, however, be the zero component of the same vector field. This (unique) vector field-the probability current-is constructed and discussed.

The authors consider in two-dimensional non-relativistic quantum mechanics an arbitrary static magnetic field confined within an inaccessible region D of arbitrary geometry. It is shown that global gauge invariance implies that the wavefunction psi (x, y) for a charged particle outside D is single-valued in a single-valued gauge.