Table of contents

Analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp(3,R). It is shown that the expressions are exact for representations of the type ( sigma ₁= sigma ₂= sigma ₃) and for all states of an arbitrary representation that are multiplicity free with respect to the u(3) subalgebra. Furthermore they are remarkably accurate in general.

For the discrete series irreducible representations (( lambda +n/2)^d) of Sp(2d,R), the determination of the Sp(2d,R) generator matrix elements in an Sp(2d,R) contains/implies U(d) basis is reduced to the much simpler calculation of boson operator matrix elements between U( nu ) contains/implies U(d) boson states, where nu =d(d+1)/2. The key of this reduction is the previously derived Holstein-Primakoff boson representation of the Sp(2d,R) generators. As an illustration, the case of Sp(6,R) is worked out in detail.

It is shown that if a classical system has an integral motion which is a homogeneous polynomial in the velocities then it is also a polynomial in the position variables. The case of two degrees of freedom is investigated more closely and it is shown that the only systems which have such integrals are those with familiar momentum integrals.

For a subclass of the set of reversible cellular automata, the author gives the form of an exact time invariant quantity, which can be seen as a kind of energy. With the ergodic assumption, this is a model of two interacting Ising spin systems.

The description of a quantum Brownian oscillator based on stochastic quantisation is applied to the non-equilibrium relaxation process. The quantum fluctuation-dissipation theorem is derived in a form which relates only the thermal component of the fluctuations to the response function. In the zero-temperature limit, it is shown that the transient is described by processes related to the solution of the nonlinear Schrodinger equation.

Scaling and renormalisation-group concepts are used to derive phenomenological Langevin equations for non-equilibrium steady states. For the Schlogl model the authors unambiguously identify the critical Langevin equation in 4- epsilon dimensions simply by requiring that the correct mean-field limit be reproduced. The approach is applicable to models for which only the mean-field behaviour is well established.

Percolation properties, including the total cluster mass S, the shell mass B, and the linear geometrical size R, are studied as a function of the topological 'chemical distance' parameter L. All critical exponents are shown to be related to an apparently new exponent nu , defined by R approximately L^v. Critical exponents are calculated exactly for percolation clusters on the Cayley tree (a model for 6D percolation), for which S approximately L², B approximately L, and R² approximately L. For diffusion on such clusters one finds that L³ approximately t. Numerical estimates of the exponents are obtained for other dimensions. A conjecture which relates nu to beta and nu is discussed.

A kinetic gelation model is developed for which the number of active initiator molecules is conserved. For this reason this model exhibits a gelation threshold even for very small values of c_I, the concentration of initiator molecules, in contrast to the conventional model of kinetic gelation with non-conserved initiators. This model is studied by Monte Carlo analysis for a two-dimensional triangular lattice with a sequence of lattice sizes up to 900*900 sites. The authors use finite-size scaling to study the dependence of y_h (the fractal dimension) and y_T (=1/ nu ) upon c_I. They find that y_T depends weakly on c_I, but that the fractal dimension y_h is independent of c_I. This is in accord with the most recent work for the conventional kinetic gelation model with non-conserved initiators.

The author uses a position space renormalisation group transformation to show that directed diffusion limited aggregation and directed animals are in different universality classes.

A generalised model, intended to describe a number of quantum systems in the presence of short-range correlated quenched impurities, is studied by the renormalisation group approach. To first order in epsilon , it is shown that the tricritical behaviour is unstable towards impurity perturbations for both finite and zero temperatures. A brief discussion of the zero-temperature tricriticality in pure systems is also presented.

The density-density correlation function matrix H(z₁,z₂;x) and its Fourier transform H(z₁,z₂;k) in the interface of a lattice gas in the solid-on-solid (SOS) approximation were computed numerically with the aid of the transfer matrix. Strips z_max* infinity were studied in a weak external field. The matrices H and H were diagonalised for small k_{perpendicular to} and for k_{perpendicular to} =0. As postulated by Wertheim, the largest (single) eigenvalue of H(k_{perpendicular to} =0) was well separated from the others, but its field dependence was represented by lambda _H^-1=A( beta V₀)^delta with delta =0.75 or delta =1 for alpha =2 or alpha =1 respectively, with alpha specifying the power law of the external field.

The authors study the kinetics of a diffusion-controlled reaction in which perfect traps (scavengers) diffuse and consume randomly distributed particles. They find that the number of particles remaining after time t decays as exp(- alpha ct^d2/), for spatial dimensions d<2, and as exp(- alpha ct) for d>or=2, where alpha is a constant, and c is the trap concentration. These predictions are supported by Monte Carlo data in d=1 and d=2. They also discuss the differences between the scavenger reaction and the reaction of particles diffusing and being consumed by randomly distributed static traps. Finally, they treat the situation where both the particles and the traps diffuse.

The theory of symmetric functions is used to obtain simple expressions for the characters and dimensions of the covariant irreps of the supergroup U(N/M). Schur function expressions are given for a number of important branching rules.

A construction of the Casimir operators of higher degree of the Lie superalgebras osp(1,2n) by a method which represents a generalisation of the method proposed by Micu for the construction of the Casimir operators of the Lie algebras, namely the algebras Sp(2n), is suggested. The method applied has a close connection with Backhouse's method of construction of Casimir operators for the semisimple Lie superalgebras. Above all the fourth-order coefficients of the Casimir operators of superalgebras osp(1,2n) are shown to play a determining role for the Casimir operators of higher degrees.

The properties of Thomas-Fermi-von Weizsacker energies as functions of the degree of ionisation are studied. A concavity result is established, which can be interpreted as a bound to the decrease of the ionisation energies. A numerical calculation of Hartree energies is described and the critical ionisation is numerically calculated as 1.21.

For pt.I see ibid., vol.16, no.17, p.4001-11 (1983). The general Schrodinger equation is converted into equivalent (2t+1)-term recurrences by the generalised Lanczos construction of the basis. Then, the exact bound states are given by the t-parametric determinantal formula. Finally, the values of the free parameters are shown to follow from the Hill type normalisability requirement.

For pt.II see ibid., vol.17, no.8, p.1603-10 (1984). In the generalised Lanczos basis, the author partitions the Hamiltonian to tridiagonal form and, mutatis mutandis, apply the ideas of I and II. It solves the Schrodinger eigenvalue problem again. The resulting method modifies and complements the formalism and simplifies its use and interpretation. As an example of application, the unique asymptotical effective Hamiltonian is derived. Also, for the anharmonic oscillators, the convergence of its continued fractional definition is proved and the compact formula for the higher-order asymptotic corrections is found.

Using perturbation theory the author proves self-adjointness of a non-relativistic Hamiltonian H with a wide class of matrix potentials. They also give a simple criterion based on the Molchanov theorem which guarantees that H has a purely discrete spectrum.

Coherent states of an electron embedded in a constant homogeneous magnetic field are constructed. The centres of the probability distributions belonging to these states gyrate along possible classical trajectories. Suitable packets of such coherent states are defined which reduce to properly normalised free electronic states in the zero magnetic field limit. A simple example is given to illustrate the dynamics of free electron localisation due to the presence of a magnetic field.

The energy function E( lambda ) of a quantum system with Hamiltonian H( lambda )=H₀+ lambda H₁ is determined by an analytical iterative procedure which yields a rich variety of functional forms. The wavefunction iterates include the solutions of Rayleigh-Schrodinger perturbation theory as well as some additional functions, all of which may be obtained sequentially by standard analytical techniques. At each step, an arbitrary ( lambda -dependent) constant may be chosen freely so as to improve the results of low-order iterates, thereby enhancing the convergence of the procedure.

The cluster expansion methods of Nickel are applied to calculate high-temperature series for the vacuum energy and specific heat, the susceptibility, and the mass gap in the (2+1)D Ising model. Critical points and critical exponents are estimated for the square and triangular lattices. The results demonstrate universality with the 3D Ising model, within errors. Exact linked cluster expansions are formulated for the quantities above, and their convergence and scaling properties are investigated.

A transfer matrix approach is used to calculate the partition function of the Ising model on the Koch curve. In zero magnetic field it is possible to obtain an exact analytic result. In the presence of a field the problem becomes more difficult. A procedure, based on an expansion about H=0, is developed which shows that the spontaneous magnetisation vanishes identically and which makes it possible to give the zero field susceptibility as a power series in a small temperature dependent parameter.

The lattice model for water, previously investigated by Bell using a Guggenheim-McGlashan first-order type of approximation (FOA), has been examined at three reduced temperatures T*=(kT/w)=0.2, 0.5 and 0.9 using grand ensemble Monte Carlo simulations. The parameters of the model are epsilon , w, u associated with first-neighbour energies, a hydrogen bond increment to this, and a 'penalty' incurred by molecules in triad sets on the two intertwined cubic ice lattices of the model. These parameters were given values suggested to be optimum by Bell ( epsilon /w=2.0, u/w=1.25). Results indicate that the cooperativity of the model is significantly underestimated by the FOA, although the general behaviour of the model is correctly represented. The authors have also made a preliminary examination of the possibility of introducing values of epsilon , w and u into the model which are directly related to H₂O dimer potentials, rather than being estimated a posteriori from thermodynamic criteria. It is shown that this approach is feasible and that the advantage of extremely rapid calculation offered by the lattice model would not be lost.

Corrections to scaling for percolation cluster numbers in two dimensions are studied by Monte Carlo simulations of very large systems (up to 17*10⁹ lattice sites) and by series analysis. Both series and Monte Carlo work suggests that the value of the correction-to-scaling exponent is slightly lower at the percolation threshold than away from it. Moreover, the corrections to scaling observed at p_c ( Omega equivalent to 0.64) might be due to the mixing of scaling fields rather than to the irrelevant scaling fields. The Monte Carlo results are compatible with finite-size scaling, and finite-size scaling corrections are estimated. Technical problems associated with Monte Carlo simulation of very large systems are discussed in an appendix.

The phase diagram of the q-state Potts lattice-gas (or site-diluted Potts model) is studied in this paper. After listing some rather simple results for different limiting cases of the interaction parameters, an exact mapping between the two-state model, which is equivalent to the Blume-Emery-Griffiths model, subjected to a special constraint and the pure Ising model is given for lattices with coordination number three. Thus, an exact intersection of the critical surface is obtained on the honeycomb lattice. The model has a complex phase diagram on the Cayley tree (or Bethe lattice): two types of critical endpoints, tricritical points and a tetracritical point are located for 1<or=q<2. The nature of the phase transition at multicritical points changes abruptly at q=2. The authors argue that the tetracritical point might be found also in lower dimensions if q is less than some q_m(d). The three-dimensional Blume-Emery-Griffiths and correlated site-bond percolational models are the important cases conjectured to have tetracritical points. The divergence of the susceptibility in spite of the discontinuity of the order parameter at the pure (q+1)-state Potts transition point is found on the Cayley tree (1<q<2). This phenomenon, which is due to the fact that the line of tricritical points terminates at the (q+1)-state Potts transition point, might also be present in lower dimensions in accordance with series expansion and Monte Carlo results.

The crossover between the random walk and the self-avoiding walk limits of a large ring polymer is described using a direct renormalisation theory. The methods of field theory renormalisation are applied directly to the ring polymer system to calculate the mean square radius of gyration (R_G²)_ring through first order in epsilon =4-(dimension of space). The expansion factor alpha ² for (R_G²)_ring is not a universal function of the natural scaling variable x that characterises the strength of the self-repelling (excluded volume) interaction. The crossover exhibited by the radius of gyration ratio R(x)=(R_G²)_ring/(R_G²)_line is described by a universal scaling function of x and has a range space (¹/₂<or=R(x)<or=¹/₂(1+¹³/₉₆ epsilon )) that is bounded below and above by finite universal numbers representing the random and self-avoiding limits, respectively.

The spinless fermion model used by some authors for a description of a metal-insulator transition is proven to develop a long-range order at sufficiently low temperatures if the dimension of the underlying simple cubic lattice nu >or=2. The Hamiltonian of the model consists of the lattice kinetic energy term and a nearest-neighbour repulsion term. The method used in the proof is that of Frohlich and Lieb, which combines a Peierls argument with chessboard estimates. For nu >or=3 infrared bounds are applied to find a simple expression for the critical temperature lower bound.

It is shown that, for hopping systems, even in the presence of a magnetic field, the Kubo approach for zero frequency gives the same DC conductivity tensor as the generalised stationary Miller-Abrahams approach. In general, the result relies on certain assumptions of macroscopic homogeneity about the system, which for periodic lattices turn out to hold exactly.

The results of earlier work by Butler and Reid for the pure-rotation double group D_infinity are used to obtain algebraic formulae for the 3jm factors of the branching D_{infinity h} contains/implies C_{infinity v}. Other branchings involving these groups are trivially related to branchings involving D_infinity . The transformations of the group chains O₃ contains/implies D_{infinity h} contains/implies C_{infinity h} and O₃ contains/implies D_{infinity h} contains/implies C_{infinity v} contains/implies C_infinity to the standard angular momentum basis are also given.

The authors use 1/ sigma -expressions ( sigma =2d-1) to investigate systematically a number of conjectures and approximations for the bond and site percolation thresholds of a d-dimensional simple hypercubic lattice. The technique provides convincing evidence that none of the conjectures are correct and gives a quantitative understanding of the quality of the approximations.

Using a primitive droplet model in conjunction with recent exact results for the surface free energies of Ising droplets on a square lattice with anisotropic couplings, the author obtains an expression for the principal non-universal amplitude appearing in the imaginary part of the small field metastable state energy of the Hamiltonian field theory version of the Ising model in (1+1)D. The calculation is in very good numerical agreement with the results of a recent finite lattice study.

The resistive susceptibility chi _R(p) of Harris and Fisch is expanded graphically for the directed simple cubic bond problem as a power series in the diode density p up to p¹¹. Standard Pade analysis shows that chi _R(p) diverges with the exponent gamma _R=2.70+or-0.04 and using a scaling relation the exponent t for the conductivity of the infinite cluster is estimated to be t=1.29+or-0.02.