Table of contents

Using the Bohr-Sommerfeld quantisation rule, the authors obtain the correct energy spectrum of the Hartmann potential. They also also analyse the degeneracies found for this problem in the classical context.

The authors study symmetry properties of some Fokker-Planck (FP) equations. In the one-dimensional case, when symmetry groups turn out to be six-parameter ones, this allows us to find changes of variables to reduce such FP equations to the one-dimensional heat equation. They present seven well known FP equations which are reduced to the heat equation. They also study the symmetry and obtain some exact solutions of the Kramers equation.

A new concept of quantitative measurement of chiral structure is introduced and labelled as chiral coefficient chi . A distinction is made between geometric and physical chiralities and chi is defined for various physical cases. Few general geometrical features are discussed and it is shown that for every chiral body there exists a natural z direction depending on the physical property treated. Possible practical applications of this concept are discussed. A few simple examples of the chiral coefficient of mass chi _m for chiral molecules are treated.

For the integrable XXX antiferromagnetic ring of N spins s=1 or s=1/2 the numerical solutions to the Bethe-ansatz equations are found, which involve non-string configurations, namely multiplets. The results up to sN=150 are compared with higher-level Bethe-ansatz predictions. The absolute difference between the predicted and finite-N energies of the spin-zero states with a multiplet and four holes is of O(1/N). The coefficient is not the same as for the vacuum and depends on the positions of the holes. As has been expected, the multiplets are of an exponential accuracy in N, while sea strings are much more strongly deformed.

The basic variables in discrete Lax equations can be represented as moments of the one-particle distribution function satisfying certain Vlasov-type kinetic dynamics. These kinetic equations are Hamiltonian, and the representation map is canonical.

The authors consider a complex Hamiltonian map, and show how to approximate the critical value of the perturbation parameter at which a given KAM circle disappears by means of a function (the Brjuno function) which only depends on the continued fraction expansion of the rotation number.

The authors study a tight-binding model given by Psi _n+1+ Psi _n-1+V cos(2 pi Qn+ theta ) Psi _n=E Psi _n, where Q is a Liouville number. It is demonstrated that all eigenstates are critical with singular continuous spectra for all values of V. These critical states appear to be composed of 'connected extended states'. They also find a band-centre anomaly: the resistance of the band centre displays coupled oscillations and is much larger than that of other states.

The Hamiltonian of a 1D quantum chain corresponding to Belavin's z_n × z_n symmetric model is derived. This Hamiltonian is a multicomponent generalisation of the XYZ model with n² coupling constants in general. In the limit tau to i infinity it reduces to the generalised XXZ model with (n-1) n-fold-degenerate coupling constants and n non-degenerate ones. The Hamiltonian is Hermitian only for n=2 or for n > 2 with a crossing parameter w restricted to integers. There exists a domain of parameters in which the Boltzmann weights are positive but the corresponding Hamiltonian is non-Hermitian. The relations of the Hamiltonian to other models are discussed.

The path integral for the free quantum motion on an arbitrary homogeneous space M is considered. The author expands the short-time propagator in unitary irreducible representations of the transformation group on M. The path integral is performed explicitly by using the orthogonality of the representations. The correct normalised wavefunctions are given by associate spherical functions and the energy spectrum is obtained from the time derivative of the Fourier coefficients of the expansion.

The following theorem is proved. The spin J_g of the ground state of a system of any two particles of spins s₁, s₂ interacting through rotationally invariant but otherwise arbitrarily spin-dependent potentials fulfils the inequality J_g<or=s₁+s₂.

The mapping of critical dynamics onto a static (D+1)-dimensional system is analysed in the large-n limit. At criticality Lifshitz tricritical behaviour (with upper critical dimensionality D*=4) occurs in the neighbourhood of an ordinary Lifshitz point (with D*=6). The violation of universality is a consequence of the very special path along which the critical point is approached.

By using a quantitative version of the c-theorem in conformal theories, the authors determine some universal geometrical features of two-dimensional critical systems, with emphasis on the ratios of mean square distances for polymers.

In this letter the author explores the knottedness of self-avoiding walks. He was not able to represent the data as a power law.

The partition function for a model system of N vicious walkers is expressed as a determinant when the walkers are confined to a half-line or a finite interval. For the walkers on the half-line, the probability of survival, the probability of a reunion and the conditional probability of a reunion are obtained. The two-point correlation near and parallel to a boundary of fixed like spins in the magnetised phase of the two-dimensional Ising model is conjectured to be proportional to the probability of a reunion near the boundary for N=2 and thus decay as e^{- xi} zeta //r⁵ for large distances r.

The influence of trap diffusion on the fluctuation slow-down of death of Brownian particles, discovered earlier in the case of stationary traps, is analysed. It is shown that fluctuation slow-down also takes place with movable traps if the diffusion is slow enough.

The authors study the asymptotic scaling of the rupture strength threshold S(W), on the width W, of a quasi-one-dimensional structure, the 'bubble' model. This model can mimic many features of the percolation transition on Euclidean lattices, while still being simple enough to be exactly soluble. The dependence of the system length L on W can be tuned to give rise to different behaviours of the rupture strength S. Three regimes are found: for L<<exp( alpha W), S(W) diverges as W to infinity while for L>>exp( alpha W), S(W) vanishes as W to infinity . In the transition regime, L approximately exp( alpha W), the dependence of S(W) depends on microscopic details.

Periodic and chaotic behaviour of the Bonhoeffer-van der Pol model of a nerve membrane driven by a periodic stimulating current a₁ cos omega t is investigated. Results show that there exist ordinary and reversed period-doubling cascades and a mode-locking state. At low driving amplitudes a₁, there are period-doubling and chaotic states, but no impulse solutions. When a₁ is larger than a₀=0.749, there are chaotic, reversed period-doubling, and mode-locking states and there also exist impulse trains. A mode-locking state with period 4 over a very large range of amplitudes is also found. At a₁=1.7059 the system goes back to a one-period state.

A simple Hamiltonian system with two degrees of freedom is investigated numerically. Although the Hamiltonian is not everywhere differentiable but continuous, the authors nevertheless find KAM tori, showing that not all the assumptions of the KAM theorem are necessary. Furthermore there exist islands in phase space, embedded in a chaotic region, where the system seems to be exactly integrable.

An exact functional relation is found for the susceptibility of the generalised square lattice Ising model. This result contains, as a particular case, a Fisher relation between the susceptibilities of the triangular and honeycomb lattices models. It is also shown that the closed-form expression for the susceptibility of the generalised square lattice, proposed by Syozi and Naya (1960), satisfies the functional relation presented in this letter.

The time evolution of two ferromagnetic Ising cubic systems, which differ by the orientation of a single spin at t=0, has been studied in a magnetic field B by computer simulation. The critical field B_c(T), above which the previous initial damage does not spread, has been determined. The magnetisation for B=B_c is very well approximated by M(B_c)=1-exp(-T_c/T) in the range of T values investigated. The concentration of minority spins at B_c(T) differs strongly from the corresponding concentration for percolation at T.

The authors study the time evolution of the Hamming distance of two configurations by Monte Carlo simulation, differing only by the central spin, of the +or-J 3D Ising spin glass. They observe two temperature regimes: a high-temperature one where the damage readily spreads and a low-temperature one where the spreading of damage is hindered. The critical temperature numerically coincides with the spin-glass temperature found by Monte Carlo simulation.

The author compares recent results on finite-size corrections for the isotropic (XXX) Heisenberg chain (and its integrable generalisations) with former numerical and analytic calculations. For the ground state and its lowest excitations, leading and non-leading terms are considered. The analysis confirms the predictions of conformal invariance and sheds some new light on the fine structure of finite-size corrections.

A theory for a classical D=2+1 string with a distributed spinor field is suggested. This field is defined by real Majorana spinors which are spinors both in initial three-dimensional spacetimes and in corresponding touching planes. It is shown that the dynamics of that object in terms of Poisson brackets is defined by the pair of algebras of (so(1,2)(X)(t^-1,t))(+)C_z type. A one-to-one correspondence between this model and the conformal-invariant model of two-dimensional scalar and spinor fields with non-trivial interaction (Thirring*Liouville model) is determined.

An exact expression for the quantum mechanical propagator for a particle moving on a group manifold is shown to arise from the application of an infinite-dimensional version of the Duistermaat-Heckman integration formula to a suitable path integral over the based loop group of the group. In an appendix the equivalence of this expression to a spectral representation of the propagator is demonstrated by means of a Poisson resummation.

The symmetry properties of outer product coupling coefficients for the symmetric group and of Clebsch-Gordan coefficients for the unitary group and of the corresponding isoscalar factors are discussed. A simple rule is given for determination of a phase factor of outer reduction symmetry of S_n and U(N). The rule is provided with a relationship between the phase factor of a Young diagram with its subdiagrams.

If a gauge configuration has a non-trivial holonomy group Phi in the vacuum, then the gauge symmetry is broken to the centraliser of Phi in the gauge group. This approach to symmetry breaking has found important applications recently. Here the author studies this mechanism from a general point of view: given a compact Lie group G and a compact subgroup H, he finds conditions for the existence of a group J with H=CJ (centraliser of J), studies the uniqueness of J, and asks whether J can be represented as the holonomy group of some connection.

Two independent algorithms are presented, which together allow the determination of branching rules from an irreducible representation of a compact Lie algebra to those of a subalgebra (or subjoined algebra). The first gives the subalgebra Weyl orbits contained in an algebra orbit. The second gives the irreducible representations of an algebra contained in an orbit, and by inversion of a triangular matrix, the orbits contained in an irreducible representation.

A one-parameter family of approximations of the initial Cauchy problem for the Dirac equation in (3+1) spacetime dimensions is given. The approximation is defined in terms of the mean translation operator G(t,y)f=(4 pi rho )^-1 A integral _{mod nu mod = rho} gf(y-t_v) d nu S. The family is indexed by the 'speed' rho . The main result is that the approximation converges to the solution (in the L₂-norm) if rho >or= square root 3c,c being the speed of light.

In this paper the author presents some new similarity solutions of the modified Boussinesq equation, which is a completely integrable soliton equation. These new similarity solutions include reductions to the second and fourth Painleve equations which are not obtainable using the standard Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method which involves no group theoretical techniques.

The singular part of the hydrogen dipole matrix element is exactly calculated. The result can be of interest for both quantum electrodynamics (bremsstrahlung, Rayleigh scattering) and quantum optics (above-threshold ionisation). Comparison with approximate methods is performed.

Sufficient conditions for a zero-curvature equation U_t-V_x+(U,V)=0 being Liouville integrable are investigated. In the case that the equation is integrable an explicit formula of the Poisson bracket (H( lambda ),H( mu )) for Hamiltonians H is proposed. The Yang hierarchy is derived and shown to be Liouville integrable.

Numerical hammagraphy is used to determine the statistical distribution of knots which are confined to a thin layer. The statistics used are based on more than 2*10⁶ knots. Among various striking features is a marked regularity in the occurrence of the prime knots.

A canonical formalism based on forward and backward propagators is developed for problems described by systems of general non-linear equations. These propagators are shown to yield the problem's solution by propagating exactly the bulk/surface/initial sources. They naturally generalise to non-linear problems the Green functions of linear theory. Unlike the customary Green functions, though, the forward and backward propagators depend parametrically and non-linearly on the problem's solution; however, the propagators themselves satisfy linear equations that can, in principle, be solved by methods of linear theory. Three examples, comprising both scalar and vector problems, are presented to highlight the main points underlying the application of this formalism.

A new method is proposed in evaluating the Bloch density matrix for a three-dimensional charged harmonic oscillator placed in a constant magnetic field. The method first reduces the operator exp (- beta H) to a product of several factors of a simple nature by using the commutation relations among the Hamiltonian components and by performing some transformations on the Hamiltonian, and then calculates the matrix element. The method is especially efficient when the system has a rotational symmetry around the axis parallel to the magnetic field, but it is also useful in other cases. A generalised problem in which a uniform electric field coexists is also discussed. To make the whole discussion consistent, the density matrix for a one-dimensional harmonic oscillator, which plays an essential role in the whole problem, is recalculated in the spirit of this method, and it is shown that the present method derives the well known form of the Bloch density matrix for this system in a quite elementary way, without reference to any advanced knowledge of eigenfunctions.

Block diagonalisation of the Hamiltonian by an unitary transformation is an important theoretical tool, e.g., for deriving the effective Hamiltonian of the quasidegenerate perturbation theory or for determining diabatic molecular electronic states. There are infinitely many different unitary transformations which bring a given Hermitian matrix into block diagonal form. It is, therefore, important to investigate under which conditions the transformation becomes unique. The explicit construction of such a transformation and its properties is discussed in detail. An illustrative example is presented. The non-Hermitian case is briefly discussed as well.

The Kustaanheimo-Stiefel (1965) and Levi-Civita (1956) transformations, used to regularise the three- and two-dimensional Kepler problem respectively, are generalised to the n-dimensional case. Explicit formulae are given for n=2, 3, 5, thus covering in a more transparent way those given by Lambert and Kibler (1988).

In this paper, the circular symmetric two-dimensional sine-Gordon equation omitting the origin is investigated by using the numerical integral method. There exists a ring-shaped quasisoliton solution and this ring-shaped wave firstly travels outward and then at certain positions it returns. A simple analytical treatment of return time is made and is in agreement with the numerical one. Finally, an interpretation of the return effect is also presented.

Extreme subsets of the set P_n^p of all p-body matrices are discussed. The concept of p-body rank of an n-electron wavefunction is defined and its physical meaning is investigated.

From the consideration of quasiparticles in thermal situations, the authors derive a universal expression of the self-consistent renormalisation condition in thermofield dynamics. This condition is valid even when the renormalised energy becomes time dependent, and provides us with four independent self-consistent equations for four real parameters, i.e. renormalised energy, dissipative coefficient, number density and a new degree of freedom chi , which is a phase in thermal doublet space.

The logarithm of the transfer matrix of the two-dimensional, three state chiral clock model is shown to be equivalent to a one-dimensional non-Hermitian quantum Hamiltonian. In the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite-size scaling. A single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy. Both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase. The critical exponents nu _x, nu _tau for the correlation lengths obtained from finite-size scaling point towards the existence of a Lifshitz point at finite chirality. The results of the series expansion for the exponent nu _x, however, are inconsistent with finite-size scaling. For larger chirality the authors' finite-size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase. They analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations.

The author considers one-dimensional n-state quantum chains having as symmetry an Abelian group A of order n (generalised Ising chain). These Hamiltonians depend on 2(n-1) coupling constants. For special values of the coupling constants the symmetry of the quantum chains is larger than A corresponding to a (in general non-Abelian) group G with m elements (m)n). For each element of G he finds explicitly a boundary condition which leaves the Hamiltonian translationally invariant. This problem is relevant in finite-size scaling studies at the critical point of a second-order phase transition.

The asymptotic dynamics of the spin autocorrelation function is studied in the Griffiths phase of bond-dilute Ising and Heisenberg ferromagnets by Monte Carlo simulation, for simple relaxational dynamics (model A). Systems above, at and below the percolation threshold are studied. Relaxation is non-exponential in all cases. Agreement with theoretical predictions based on clustering arguments is excellent for the Heisenberg systems, less so for the Ising systems.

The microscopic model of Ullersma (1966) for a harmonic oscillator in contact with a thermal reservoir is quantised in the framework of Nelson's stochastic mechanics. Eliminating the degrees of freedom of the thermal reservoir the stochastic process of the quantum Brownian oscillator is obtained as the sum of the thermal contribution and the quantum (zero-temperature) contribution. The properties of the quantum fluctuations are studied in detail and the equation of motion is derived, obtaining the first example of a non-Markovian Nelson process.

A self-avoiding walks model is studied in which the walker is, in addition to the usual self-avoiding condition, restricted not to make any turn which will put the walker in a direction rotated more than Phi _max from any of the directions previously taken. Here, Phi _max( Phi _R where Phi _R=2 pi - (the smallest exterior angle of the unit cell of the lattice under consideration). The generating function for the square lattice is obtained, and various properties such as the number of N-step walks, the total number of steps to given directions and the mean-square end-to-end distance are evaluated analytically. The model exhibits characteristics reminiscent of one-dimensional self-avoiding walks in all directions, while retaining the anisotropic effect of the direction of the first step.

A polymer chain in equilibrium in a disordered environment is studied using a Flory theory and by mapping the problem onto random walks in an environment with traps. The asymptotic behaviour of the size of the polymer, R, as a function of the number of monomers, N, is obtained. If the disorder is weak in comparison with the self-repulsion of the chain, the self-avoiding random walk result is found. The random environment leads to effective attractive forces which, for sufficiently strong disorder, lead to the collapse of the chain. The properties of the collapsed chain depend upon the type of disorder and on the self-repulsion of the chain. If the self-repulsion increases sufficiently fast as the density increases then the collapsed chain has a finite density (N/R^d to constant as N to infinity ); otherwise several other interesting scaling forms are possible.

A recursive algorithm previously developed for carrying out renormalisation calculations for the lambda -state Potts model is generalised to the Z(λ) model. The relations used are based on an expression the authors derive for the pair correlation function in terms of mod- lambda flows, which represents an extension of a similar result for the partition function previously obtained by Biggs (1976). The use of flows enables them to prove and extend the formulae which appear in the break-collapse method of Mariz and co-workers (1989). It is argued that the use of fixed-flow bonds rather than the precollapsed bonds used by the latter authors leads to a more efficient algorithm.

Corner transfer matrices are used to calculate the order parameters of the self-dual Ashkin-Teller model on the square lattice. In the non-critical regime, it is found that the model is partially ordered, i.e. the magnetisations (s₁) and (t₁) vanish while the polarisation (s₁t₁) is non-zero. The polarisation, which is given simply in terms of elliptic functions, exhibits an essential singularity at criticality.

The authors study the phase diagram of the isotropic three-state chiral Potts model. Monte Carlo simulations performed on twisted square lattices with different sizes up to 64*64 indicate different types of critical curves. In particular, a floating phase seems to occur. Finite-size scaling analyses are performed at some specific critical points. The phase diagram they propose is discussed using exact results.

The authors show that long polymer chains may bind to surfaces which are curved or rough if the equivalent planar interface is sufficiently close to supporting bound states. Analogous results apply to the binding of point quantum mechanical particles. Attractive interactions favour binding preferentially on the invaginations of the surface, but they find that entropy/kinetic energy favours binding at the outward extremities. Which effect dominates depends on details of the interaction potential and both cases should be observable in practice.

The inclusion of the anisotropic vertex is shown to influence the character of random walks. The isotropic fixed point will be unstable. The correction to the diffusion law at large timescales is calculated in the framework of the in -expansion method.

The author studies the renormalisation aspects of a generalised Coulomb gas by means of a novel method of renormalisation. The model corresponds to an XY system with N-fold symmetry breaking perturbation and is cast in a field-theoretical framework. The method uses the operator product expansion and the renormalisation constants are obtained from the singular terms in this expansion as the ultraviolet cutoff is removed. This method clearly exposes the multiplicative renormalisations involved. An expansion in the departure of the Kosterlitz-Thouless temperature, fugacities, and in in =N-4 to third order is performed. He finds a non-trivial self-dual fixed point at which the vortex and symmetry breaking fugacities are of order in . The method also allows the identification of a new set of 'fixed-point' theories that do not undergo renormalisation.

Models solvable by the hierarchy of Bethe ansatze (i.e. by the multicomponent Bethe ansatz) are considered. The spectrum of conformal dimensions which determines the long-distance asymptotics of correlations is calculated. This asymptotics in a general case is described by the direct sum of conformal theories, each possessing a central charge equal to one.

The authors give a new definition dim_H(A) for the dimension of an arbitrary subset of the lattice ^d. They establish elementary properties, and calculate the dimension for some examples. Finally, they announce a result which states that, if dim_H(A)< d - 2, then A is transient for the simple random walk on ^d, and that if dim_H(A) > d-2 then A is recurrent.

It is pointed out that the different definitions of the fractional dimension of an arbitrary subset A of ^d, given by Naudts (1988) and by Barlow and Taylor (1989), are interrelated but obtained using completely different criteria.

The authors present results of Monte Carlo simulations on the kinetics of wall-wall wetting transitions in the two-dimensional three-state chiral clock and anisotropic next-nearest-neighbour Ising models. They confirm the existence of an algebraic growth law for the width of the wetting layer (W(t)) approximately t^1/4 in the limit of large systems and long enough times at non-zero temperatures. They also discuss finite-size and temperature dependence of the effective growth exponents in these two models.

The authors calculate numerically by a transfer-matrix algorithm the critical exponent of the surface order parameter, β_s, for the case of the ordinary transition. Their result, β_s/ν = 0.98 ± 0.02, agrees with recently published series expansion results, β_s/ν = 1.04 ± 0.05 and ε-expansion results with Pade approximants, β_s/ν = 1.01 ± 0.06.