Table of contents

The authors give the operator formalism for the transfer matrix in Z_q symmetric models. They give the structure and irreducible representations of the associated operator algebra. They show that the Potts model limit gives a subalgebra which is the unitarisable quotient of the Temperley-Lieb algebra, and give the inclusion of this subalgebra. The authors discuss other subalgebras in the context of a conformal field theory limit of the Z_q symmetric models.

A new Holstein-Primakoff type realisation of the non-compact Lie algebra sp(4,R) is given in a completely explicit, analytic and closed form. The construction is related to the representations where the Casimir invariants of the subalgebras sp_1.2(2,R) are diagonal, rather than the invariants of the maximal compact subalgebra u(2).

The tangent bundle geometry is used to obtain a coordinate-free derivation of the Euler-Lagrange equation.

An infinite hierarchy of solvable systems of purely differential non-linear equations is introduced within the framework of asymptotic modules. Each system consists of (2+1)-dimensional evolution equations for at most four complex functions and of quite strong differential constraints. It may be interpreted formally as an integro-differential equation in (1+1) dimensions.

Using the factorisation method in supersymmetric quantum mechanics the author determines new potentials from the Morse oscillator. This method is applied although the ladder operators are not used.

The authors discuss the generator of Baxter's corner transfer matrix (1982) for a critical Ising model of finite size. They perform its diagonalisation in terms of special orthogonal polynomials and give an asymptotic expression of the eigenvalues for large size, which agrees with the conformal predictions.

The authors report analytical studies of the adsorption-desorption transition of polymers grafted at a penetrable surface, in the 2D and 3D partially directed SAW models. The fraction of adsorbed monomers and the phase diagram of the transition are calculated by the transfer matrix method.

The stability of the steady state of a completely ionised plasma under an external electric field is analysed from extended irreversible thermodynamics. Fourth-order terms in the electric current are included in the generalised entropy and fluctuation theory is used to obtain numerical values for the coefficients. The final result is in good agreement with evaluations from a Fokker-Planck equation in plasma theory.

The spectrum-generating functions for a variety of strings and superstrings are derived. These are expressed in terms of characters of irreducible representations of the appropriate transverse spacetime symmetry group and, in the case of heterotic strings, the relevant gauge group E₈*E₈ or Spin(32)/Z₂. The expansions involve principal specialisations of characters of both GL(N) and GL(M/N) in the limit as M and N tend to infinity. By using Schur function methods, modification rules and various theta function identities a number of spectrum-generating functions are derived and tabulated. The equivalence of the GSO-projected Neveu-Schwarz-Ramond superstring and the Green-Schwarz superstring is proved, and the connection between the lattice formulation and the GSO-projected Neveu-Schwarz-Ramond formulation of heterotic strings is spelled out.

The authors consider the motion of a charged quantum particle on a loop with two external leads which is placed into an electrostatic field. The loop Hamiltonian is chosen in the simplest possible way; in order to join it to the free Hamiltonians describing the leads, they employ a method based on self-adjoint extensions. Under a symmetry requirement, the resulting full Hamiltonian contains four free parameters; each junction is characterised by a pair of them. The system under consideration represents a model of metallic or semiconductor structure that can be fabricated by presently available technologies. Assuming the ballistic regime for electrons in such a structure, the authors calculate the resistance dependence on intensity of the external field. The results suggest the possibility of constructing quantum interference transistors whose size and switching voltage would be much smaller than in current microchips.

Considers the effect of slowly varying the parameters X_i of a finite-sized quantum mechanical system. The system is excited to higher energies by Landau-Zener transitions at avoided crossing; since this increases the energy of the system, it has the effect of dissipation of the driving motion. The rate of dissipation depends on the level spacing distribution of the system. When the spectral statistics are those of the Gaussian unitary ensemble, the rate of dissipation is proportional to X_i², i.e. there is viscous or ohmic damping. When the spectral statistics are of those of the Gaussian orthogonal ensemble, the rate of dissipation is proportional to X_i³2/.

The relative motion of N impenetrable mass points in one dimension is shown to be equivalent to a billiard in a simplex of N-1 dimension, the orientation of the confining hyperplanes being fixed by the mass ratios. The discrete group generated by reflections in these planes and its closure in SO(N-1) are used to show that the only invariant functions of the particle velocities are total momentum and total energy, if n)3 and at least one mass is different from the others.

The three-state Potts model with nearest-neighbour random Gaussian interaction on the simple cubic lattice is investigated by Monte Carlo simulation. Both static quantities (e.g. glass ordering susceptibility and the correlation function) and dynamic quantities (the analogue of the time-dependent Edwards-Anderson order parameter q(t) for Potts spins) are obtained. As for related models for orientational glasses, it is found that q(t) is consistent with the Kohlrausch law q(t) varies as exp(-(t/ tau )^y) for a wide range of temperatures, with a strongly temperature-dependent exponent y, with y becoming very small as the temperature T to 0. The relaxation time tau increases dramatically as T is lowered; thus the system could only be equilibrated for temperatures where the correlation length is rather small. Since the critical region has not been reached, it cannot be distinguished whether the critical temperature T_c is non-zero or at T=0. If T_c=0, the divergences of tau and the ordering susceptibility are probably exponential, i.e. the system is then at its lower critical dimensionality.

Site percolation on the ordinary hypercubic lattice in four and five dimensions is studied using block cluster theory. The exact analytic renormalisation group equations of block cluster theory are obtained by a computer algorithm. They are solved numerically and yield, for the susceptibility exponent, gamma =1.277 (d=4), gamma =1.031 (d=5), and for the correlation length exponent nu =0.566 (d=5).

The percolation characteristics of hard-sphere (HS) and Lennard-Jones (LJ) fluids are determined by molecular dynamics computer simulations. The LJ percolation threshold is raised in the hard-core limit and lowered in the soft-core limit when compared with the equivalent hard-sphere fluid, using a temperature-dependent effective hard-sphere diameter. This difference in the soft-core limit increases as T to T_c. The values for two percolation exponents are close to the values for random static percolation.