Table of contents

The boson realisation of the two-dimensional symplectic Lie algebra sp(2) for a given irreducible representation (irrep) of the Sp(2) group has been known for a long time. More recently the boson realisation of the 2d-dimensional symplectic Lie algebras sp(2d) have been derived for particular irreps of the Sp(2d) group. The author outlines the corresponding result for an arbitrary irrep of Sp(2d).

Using elementary methods such as Dirichlet decoupling, the computation of essential spectra of hard-core type Schrodinger operators H_{Omega 2}^D in L²( Omega ₂) ( Omega ₂=Rⁿ/ Omega ₁, Omega ₁ the compact hard core, n in N) is reduced to that of ordinary Schrodinger Hamiltonians in L²(Rⁿ) whose potentials coincide with that of H_{Omega 2}^D sufficiently far out.

An expression for the four-potential of an em field is derived as a path integral involving the fields, the formula being analogous to one given in elementary vector analysis.

Calculation of critical exponents on a simple class of hierarchical lattice reveals that lambda _s>or= lambda _t, where lambda _s is the self-avoiding walk fixed point eigenvalue and lambda _t the Ising thermal eigenvalue. High-dimensional limits of some families of hierarchies obey lambda _t to lambda _s as D to infinity ; this convergence replaces the Euclidean concept of upper critical dimension on these lattices. However, families of hierarchies for which D to infinity but with constant connectivity do not show this convergence.

The correlation length of boundary spins in the Ising model, defined on strips of triangular lattice with free boundary conditions, is determined with an efficient numerical procedure based on the star-triangle transformation. In the case of isotropic critical interactions, the extrapolated amplitude of the correlation length is in excellent agreement with the value 2/( pi eta /sub ///) predicted by conformal invariance. An analytical formula for the amplitude in strips with anisotropic interactions is proposed. Fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor ¹/₂. The convergence of phenomenological renormalisation with free boundary conditions is studied.

The authors give exact results for the energy spectrum of a chain of N Ising spins in a transverse field with periodic, free, and antiperiodic boundary conditions. The dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two-dimensional strips. The feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large N are discussed.

Improved Monte Carlo methods have been used to study the asymptotic behaviour of self-avoiding walks on two-dimensional lattices. The mean-square end-to-end distance R_N² and radius of gyration S_N² are both found to scale as N^2v with exponent values very close to the expected v=³/₄ and an amplitude ratio that is universal. Small-N deviations of R_N² from the asymptotic results indicate a correction to scaling exponent Delta =1, a value that differs from previous estimates.

The epsilon -expansion for the critical exponents of a vector spin glass is calculated to O( epsilon ³) ( epsilon =6-d). A general cubic field theory is used.

The author verifies the truth of a conjecture of Hammersley and Whittington (1985) concerning bond percolation on certain subsets of the simple cubic lattice Z³. Let f and g be non-decreasing, non-negative functions on (0, infinity ) and let Z³(f, g) denote the (f,g)-wedge of Z³, being the set of points (x, y, z) such that 0<or=y<or=f(x), O<or=z<or=g(x) and x>or=0. The author shows that the condition (1+f(x))(1+g(x)) to infinity as x to infinity is sufficient for the critical probability of the bond percolation process on Z³(f,g) to be less than or equal to ¹/₂.

The authors discuss the condition necessary for the Harris criteria to be valid on hierarchical lattices. They prove that disorder is always relevant when the specific heat exponent, alpha _p, is positive and show how to construct lattices for which disorder becomes relevant while alpha _p is still negative.

The author extends a recent construction of isotropic spin-coherent states to unitary groups. Expectation values in such a quasi-coherent state can be obtained from a generating functional which is given explicitly for the fundamental and the adjoint representations of the unitary groups SU(N) and U(N). The close connection between the corresponding generating functional and the external field problem in QCD is pointed out. Free quantum fields in such a condensed, colour singlet, coherent state are formally related to two-dimensional lattice gauge theories with, in general, a mixed action. In the large-N limit, the author exhibits a phase transition for a free quantum field in a singlet quasi-coherent state. A path integral representation of the transition amplitude in terms of quasi-coherent states is also given. The corresponding effective action describes dynamics on a complex Kahler manifold.

A semiclassical connection is established between quantal and classical properties of a system whose Hamiltonian is slowly cycled by varying its parameters round a circuit. The quantal property is a geometrical phase shift gamma _n associated with an eigenstate with quantum numbers n=(n_l); the classical property is a shift Delta theta _l(I) in the lth angle variable for motion round a phase-space torus with actions I=(I_l); the connection is Delta theta _l=- delta gamma / delta n_l. Two applications are worked out in detail: the generalised harmonic oscillator, with quadratic Hamiltonian whose parameters are the coefficients of q², qp and p²; and the rotated rotator, consisting of a particle sliding freely round a non-circular hoop slowly turned round once in its own plane.

Weyl's rule of association is applied to the multinomial basis set for classical phase space function and, using an appropriate symplectic notation, the corresponding basis set in quantum-mechanical phase space is found to consist of averages of multinomial operators. Groenewold's rule for the Wigner equivalent of a product of two operators is generalised to an arbitrary number of multipliers. An explicit expansion is derived for general multinomial operators in terms of symmetric multinomials (i.e. the basis set averages) and powers of h(cross). This makes it simple to apply Weyl's correspondence rule to both classical and quantum-mechanical functions with multinomial expansions.

Using a symplectic notation which allows the equations to apply equally to a wavepacket representing a spin-less quantum particle or to a cluster of identical classical particles without mutual interactions, the study of the time-evolution of moments is extended to arbitrary orders for arbitrary Hamiltonians. Appropriately symmetrised moments for the quantum case are seen to play a special role and correspond very closely to the classical moments.

The propagator for a charged, anisotropic harmonic oscillator in a constant magnetic field is computed. Making use of this an explicit expression is given for the path integral to an action with generalised memory recently considered in the literature.

This paper describes a Monte Carlo study of the 2D kinetic Ising model. The long-time behaviour of various time-delayed correlation functions is investigated by renormalisation group methods. This long-time behaviour can be described in terms of a simple exponential relaxation of the magnetisation which enables a reliable estimate of the dynamical critical exponent, z, to be made.

Results of computer studies of the geometries produced by two distinct decay models are present. While one of the models (diffusion-limited decay) results in compact clusters, the other (random-walk decay) produces ramified clusters with the Hausdorf dimension (d_f) equal to 1.75+or-0.03 and 2.34+or-0.03 in two and three dimensions respectively. The spectral dimension (d_s) is found to be 1.68+or-0.13 and 1.31+or-0.04 for d=2 and 3.

Bond animals with a constraint of a given winding direction on the square lattice are enumerated up to 14 bonds. Numerical evidence further confirms the previous conjecture using the position space renormalisation group approach, that they belong to a new universality class.

The authors consider natural extensions of the Ashkin-Teller model to six- and eight-states spin systems. The phase diagrams corresponding to different symmetries are obtained using mean-field and Monte Carlo analysis combined with finite-size scaling. The critical indices corresponding to two new universality classes are obtained.

The author studies the properties of the backbone of, and conduction in, random clusters which have homogeneous interior structure. These include the largest percolation cluster at p_c, lattice animals and the Witten-Sander (WS) aggregates. For lattice animals the fractal dimension d_a of the backbone is estimated, the for first time, and is found to be about 1.14 at d=2 and about 1.39 at d=3. These values are much lower than the corresponding values for the animals themselves. The fractal and spectral dimensions of the backbone of the WS aggregates are estimated to be about 1.25 and 1.06 at d=2, respectively. A recent hypothesis which relates the fractal dimension of random walks on these clusters to those of the clusters and their backbones is also discussed. It is shown that for the WS aggregates d_s=d_s=2 on a Bethe lattice (i.e. at d= infinity ), in contrast with d_s=4/3 and d_s=1 for percolation clusters and lattice animals. The conductivity exponent t_a of lattice animals is found to be t_a(d=2) approximately=0.73 and t_a(d=3) approximately=1.19 and t_a(d>or=8)=2, whereas for the WS aggregates the author finds t_WS(d=2) approximately=0.67 and t_WS(d=3) approximately=0.94 and t_WS(d to infinity )=1.

The isotropic-nematic transition is studied using the Lebwohl-Lasher lattice model of liquid crystal behaviour. The details of the transition are investigated using the Migdal-Kadanoff potential moving version of the real space renormalisation group. The method gives good results when applied to the closely related Heisenberg model of ferromagnetism. A second-order transition is predicted, in disagreement with mean field and some computer simulation results.

The authors consider the number of self-avoiding walks confined to a subset Z^d(f) of the d-dimensional hypercubic lattice Z^d, such that the coordinates (x₁,_x2, . . .,x_d) of each vertex in the walk satisfy x₁>or=0 and 0<or=x_k<or=f_k(x₁) for k=2,3, . . .,d. They show that if f_k(x) to infinity as x to infinity , the connective constant of walks in Z^d(f) is identical to the convective constant of walks in Z^d. They also explore conditions on f_k which lead to a smaller connective constant for walks in Z^d(f) and, in particular, consider walks between two parallel (d-1)-dimensional hyperplanes. Finally they contrast some of these results with recent work by Grimmett on percolation on subsets of the square lattice.

Long self-avoiding walks of up to 2400 steps have been generated on the SC and BCC lattices using an improved Monte Carlo technique. Analysis of the asymptotic length dependence of the end-to-end distance and radius of gyration of the walks leads to values in the range 0.591-0.593 for the critical exponent nu . The simple power-law dependence on length provides an excellent fit to the data over walk lengths between 120 and 2400 with fluctuations around the asymptotic results averaging to a mere 0.13%. Systematic deviations from asymptotic behaviour in short self-avoiding walks have also been examined using Monte Carlo generated walks ranging in length from 12 to 60; the results do not support the existence of the non-analytic correction predicted by the renormalisation group. In the light of this unexpected result, available series expansions for walks on the FCC lattice have been re-examined and previous claims to have observed non-analytic behaviour questioned; equal, if not better convergence of the extrapolated series can be obtained without resorting to non-analytic correction terms. Finally, an analysis has been made of the radius of gyration series to which several new terms have been added.

For pt.I see ibid., vol.16, no.17, p.4155-70 (1983). The authors study the static correlations of monomer density fluctuations in semi-dilute polymer solutions in a good solvent. By means of the conformation space renormalisation theory, the universal scaling form of the scattering functions is obtained up to order epsilon =(4-d) with d the dimensionality of space. In a dilute limit agreement with the experiment is improved over the mean field theory. The correlation length and the osmotic compressibility obtained previously are also compared with the recent light scattering experiment. These theoretical results exhibit a very good agreement with the experimental observation without any adjustable parameters.

The microstructure of two-phase disordered media can be characterised in terms of a set of n-point matrix probability functions S_n which give the probability of finding n points all in the matrix phase. The authors obtain, for the first time, an exact analytical expression for S₂ for a distribution of equi-sized rigid rods in a matrix at any density and for all values of its argument. They evaluate, also for the first time, S₂ for a distribution of equi-sized rigid discs in a matrix, for a wide range of densities. Using these results for S₂ and rigorous upper and lower bounds on S₃, one may obtain bounds on S₃ for distributions of rigid rods and discs. The one- and two-dimensional results obtained here are compared to the three-dimensional results of Torquato and Stell (1983) at certain particle volume fractions.

The authors present a new formulation of the two-fluid model, the thermal coherent state for handling the relativistic quantum field theory at finite temperature. Three models in one-dimensional space, the phi ⁴, sine-Gordon and the Schwinger model, are discussed.

The authors relate the dual symmetry of the chiral field to the dual symmetry of the first and second forms on pseudospherical surfaces in asymptotic coordinates. Every given (up to conformal similarity) solution of chiral field on O(3)/O(2) sphere has been put into one-to-one correspondence with the Gauss image of a definite (up to homothetic transformations) pseudospherical surface. They establish a gauge covariant formulation which unifies the chiral field and the differential forms on the pseudospherical surface. Then they get the explicit geometrical pictures and the covariant relations for Backlund transformations, Riccati equations and an infinite number of conserved currents in both cases. The SO(3) and SO(2,1) sigma -model are also related by dual symmetry.