Showing papers in "Journal of Physics A in 1987"
TL;DR: In this article, the conformal anomaly and surface exponents of the critical quantum Ashkin-Teller and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.
Abstract: Eigenspectra of the critical quantum Ashkin-Teller and Potts chains with free boundaries can be obtained from that of the XXZ chain with free boundaries and a complex surface field. By deriving and solving numerically the Bethe ansatz equations for such boundaries the authors obtain eigenenergies of XXZ chains of up to 512 sites. The conformal anomaly and surface exponents of the quantum XXZ, Ashkin-Teller, and Potts chains are calculated by exploiting their relations with the mass gap amplitudes as predicted by conformal invariance.
496Â citations
TL;DR: The authors motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix (Jij) and numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns.
Abstract: To ensure large basins of attraction in spin-glass-like neural networks of two-state elements xi imu =+or-1. The authors propose to study learning rules with optimal stability Delta , where delta is the largest number satisfying Delta
398Â citations
TL;DR: In this article, an integrable generalisation of the XXZ Heisenberg model with arbitrary spin and with light plane type anisotropy is studied, and integral equations describing the thermodynamics of the model are found.
Abstract: An integrable generalisation of the XXZ Heisenberg model with arbitrary spin and with light plane type anisotropy is studied. Integral equations describing the thermodynamics of the model are found. The antiferromagnetic ground state, the excitation spectrum, the quantum numbers and scattering amplitudes of the excitations are determined.
347Â citations
TL;DR: In this article, it was shown that the maximisation of the entropy in non-equilibrium is equivalent to the exploitation of entropy inequality in extended thermodynamics, and that this is the case also in the case of stochastic systems.
Abstract: This author proves that the maximisation of the entropy in non-equilibrium is equivalent to the exploitation of the entropy inequality in extended thermodynamics.
221Â citations
TL;DR: The leading finite-size corrections to the ground-state energies of these chains were derived using the methods of de Vega and Woynarovich (1985) and Eckle as discussed by the authors.
Abstract: Exact equivalences between the critical quantum Potts and Ashkin-Teller chains and a modified XXZ Heisenberg chain have recently been derived by Alcaraz et al (1987). The leading finite-size corrections to the ground-state energies of these chains are derived using the methods of de Vega and Woynarovich (1985) and Eckle. Exact results are then obtained for the conformal anomaly of each model, and for the surface energy in the case of free boundaries.
162Â citations
TL;DR: In this paper, the authors apply the Painleve test to the generalised derivative non-linear Schrodinger equation, i.e., i = u+iauu*ux+ibu2ux+cu3u*2 where u denotes the complex conjugate of u, and a, b and c are real constants.
Abstract: The authors apply the Painleve test to the generalised derivative non-linear Schrodinger equation, iut=uxx+iauu*ux+ibu2ux+cu3u*2 where u* denotes the complex conjugate of u, and a, b and c are real constants, to determine under what conditions the equation might be completely integrable It is shown that, apart from a trivial multiplicative factor, this equation possesses the Painleve property for partial differential equations as formulated by Weiss, Tabor and Carnevale (1983) only if c=1/4b(2b-a) When then this relation holds, this is equivalent under a gauge transformation to the derivative non-linear Schrodinger equation (DNLS) of Kaup and Newell, which is known to be completely integrable, or else to a linear equation
162Â citations
TL;DR: The statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common as mentioned in this paper, and it is shown that the probability distributions of the largest piece W, P2(W) of the second largest piece, and n(Y) of Y = 1, Wi always have singularities at W, = l/n, W2 = l n, W 2 = l /n and Y = Ijn, n = 1.
Abstract: The statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common. For all these problems, if W, denotes the weight of the sth piece, we show that the probability distributions PI( W,) of the largest piece W,, P2( W,) of the second largest piece, and n( Y) of Y = 1, Wi always have singularities at W, = l/n, W2 = l/n and Y = Ijn, n = 1,2,3,. . . .
155Â citations
TL;DR: In this article, the probability of finding the largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation, based on the stress (voltage) enhancement at the tip of the most critical defect in the network.
Abstract: The probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation. From this result and the stress (voltage) enhancement at the tip of the most critical defect in the network, the probability of failure in percolation models of breakdown is calculated. For defect fractions less than the percolation point, this distribution is found to be of the form exponential of an exponential. Numerical simulations on the two-dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the Weibull (1951) form most often used in the fitting of breakdown data.
148Â citations
TL;DR: In this article, leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region.
Abstract: Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.
142Â citations
TL;DR: In this paper, a method for the estimation of the topological dimension of a manifold from time series data is presented based on the approximation of the manifold near a point chi by its tangent space at chi.
Abstract: A method for the estimation of the topological dimension of a manifold from time series data is presented. It is based on the approximation of the manifold near a point chi by its tangent space at chi . The dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition. The method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state.
130Â citations
TL;DR: In this article, the odd-order isospectral flows admit both a KdV and MKdV type reduction, and the non-linear terms are related to the curvature tensor of the corresponding Hermitian symmetric space.
Abstract: The authors extend previous results on the linear spectral problem introduced by Fordy and Kulish (1983). The odd-order isospectral flows admit both a KdV and MKdV type reduction. The non-linear terms are related to the curvature tensor of the corresponding Hermitian symmetric space. Their KdV equations are themselves reductions of known matrix KdV equations. They discuss the conserved densities and Hamiltonian structure associated with these equations.
TL;DR: In this article, the authors discuss properties of the set of scattering singularities in regions of irregular scattering and show how a symbolic organization of the singularities can be used to determine the fractal dimension and the scaling function.
Abstract: We discuss properties of the set of scattering singularities in regions of irregular scattering. We show how a symbolic organisation of the set can be used to determine the fractal dimension and the scaling function. This yields information on the distribution of Lyapunov exponents of bounded orbits. The specific model studied is the motion of a particle in a plane, elastically reflected by three circular discs centred on the corners of an equilateral triangle.
TL;DR: In this article, the scaling dimensions of order parameters in ADE lattice models were computed and a connection was made between the lattice algebra and the operator algebra in conformal invariant theories.
Abstract: The author computes the scaling dimensions of order parameters in ADE lattice models. Some connection is made between the lattice algebra and the operator algebra in conformal invariant theories.
TL;DR: In the semiclassical limit, the sum S(E, Delta E)= Sigma nm mod Anm mod 2 delta (E-1/2(En+Em)) delta ( Delta E-(En-Em)) of matrix elements of an arbitrary operator -A can be related to the classical correlation function of the Weyl symbol A(q, p) of -A: CA(e, t).
Abstract: In the semiclassical limit, the sum S(E, Delta E)= Sigma nm mod Anm mod 2 delta (E-1/2(En+Em)) delta ( Delta E-(En-Em)) of matrix elements of an arbitrary operator -A can be related to the classical correlation function of the Weyl symbol A(q, p) of -A: CA(E, t)= integral d alpha delta (E-H( alpha ))A( alpha )A( alpha t). S(E, Delta E) is proportional to the Fourier transform of CA(E, t) over t, plus a set of correction terms associated with periodic trajectories in phase space. If the system has a chaotic classical limit, the matrix elements are independently Gaussian distributed with mean value zero, and S(E, Delta E) gives the variance of this distribution.
TL;DR: In this article, it was shown that no pure thermodynamic states other than the paramagnetic state and a pair of states in zero magnetic field which are related by a global spin flip can exist in short-range Ising spin glass models in any dimension.
Abstract: Based on a simple scaling ansatz, the authors argue that no pure thermodynamic states other than the paramagnetic state and a pair of states in zero magnetic field which are related by a global spin flip can exist in short-range Ising spin glass models in any dimension. An analogous result should hold for XY and Heisenberg spin glasses, as well as for square-integrable long-range interactions.
TL;DR: In this paper, the q-state zero-temperature antiferromagnetic Potts model was extended to the full complex q plane, giving the limiting distribution of the zeros of the chromatic polynomial.
Abstract: Evaluating the q colourings of a lattice is equivalent to solving the q-state zero-temperature antiferromagnetic Potts model. This has recently been done exactly for an infinite triangular lattice with q real. Here the results are extended to the full complex q plane, giving the limiting distribution of the zeros of the chromatic polynomial. The results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the Beraha numbers is noted.
TL;DR: The use of Feynman integrals enables the derivation of new properties of hypergeometric series including new analytic continuation formulae for a generalised hypergeometrical series and for a Kampe de Feriet function as discussed by the authors.
Abstract: For pt. I, see ibid., vol.20, p.4109 (1987). Further examples of the use of Feynman integrals enable the derivation of new properties of hypergeometric series including new analytic continuation formulae for a generalised hypergeometric series and for a Kampe de Feriet function. This motivates the derivation of two new summation formulae for a generalised hypergeometric series and furthermore leads to a natural generalisation of the H function. While the latter, as is well known, contains as particular cases most of the special functions of applied mathematics, it does not contain some of importance, for instance the Riemann zeta function nor indeed any polylogarithm. Our generalisation of the H function does contain the polylogarithm; it also contains the exact partition function of the Gaussian model from statistical mechanics. Another new result is the simple summation formula 3F2(1,1,3/2;2,2; x)=-(-)-1 ln((1+(1-x)12/)/2).
TL;DR: In this paper, the authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary.
Abstract: The authors present an approach to dilute Ising and Potts models, based on the Fortuin-Kasteleyn random cluster representation, which is simultaneously rigorous, intuitive and surprisingly simple Their analysis yields, with no dimensional restrictions or other caveats, the following asymptotic form of the phase boundary For the regular dilute model in which bonds have constant ferromagnetic coupling J with probability p and are vacant with probability 1-p, the critical temperature scales as exp(-J/(kTc(p))) approximately mod p-pc mod , implying that the crossover exponent is Phi =1 If the constant couplings are replaced by a distribution F(J) with mass near J=0, quite different crossover behaviour is observed For example, if F(J) approximately Jalpha then, for p near pc, Tc(p) approximately mod p-pc mod 1 alpha /
TL;DR: In this paper, the authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof, including Coulomb interactions.
Abstract: The authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof First they treat the pure delta sphere model; self-adjointness of the Hamiltonian, spectral properties, stationary scattering theory, approximation by scaled short-range Hamiltonians Next they extend the model by adding a point interaction at the centre of the sphere or, alternatively, a Coulomb interaction Finally the whole analysis is extended to the case of a delta ' sphere interaction, first taken alone, then superimposed on a point interaction or a Coulomb potential
TL;DR: In this article, a dynamical invariance algebra is constructed for the ring-shaped potential V = eta sigma 2r-1+1/2q eta 2sigma 2(r sin theta )-2.
Abstract: The 'accidental' degeneracy occurring in the quantum mechanical treatment of the ring-shaped potential V=- eta sigma 2r-1+1/2q eta 2 sigma 2(r sin theta )-2 is explained by constructing an su(2) dynamical invariance algebra. The Schrodinger equation is solved in parabolic coordinates written in the framework of the Kustaanheimo-Stiefel transformation and the Hamilton-Jacobi equations are solved in ordinary parabolic coordinates. All finite trajectories are found to be periodic.
TL;DR: In this article, a generalised definition for invariance of partial differential equations is proposed and exact solutions of the equations with broken symmetry are obtained, where broken symmetry is defined as a special case of symmetry.
Abstract: A generalised definition for invariance of partial differential equations is proposed. Exact solutions of the equations with broken symmetry are obtained.
TL;DR: In this paper, the authors studied conformal variance properties of the polymer and percolation problems in two dimensions by analysing the transfer matrix spectrum of these models at criticality, their series of thermal and magnetic exponents are identified.
Abstract: The author studies some conformal variance properties of the polymer and percolation problems in two dimensions. By analysing the transfer matrix spectrum of these models at criticality, their series of thermal and magnetic exponents are identified. The results for percolation agree with the recent conjectures of Dotsenko and Fateev (1984) while some of the results for polymers are different. In the case of polymers, these series are interpreted as a new set of geometrical exponents. In each case the question of corrections to scaling is discussed.
TL;DR: In this paper, it is shown that a basis of weak eigenstates can always be chosen such that WTUW has a certain real standard form, and that such a basis can be chosen in such a way that the form U, W has the same real form.
Abstract: The investigation of general CP transformations leads to transformations of the form U to WTUW with unitary matrices U, W. It is shown that a basis of weak eigenstates can always be chosen such that WTUW has a certain real standard form.
TL;DR: In this paper, the Dirac matrices and tensors were implemented in R4 without difficulties, and a new regularisation method was proposed, which is gauge invariant, covariant and differs from dimensional regularisation in some aspects.
Abstract: Spacetime is modelled as a fractal subset of Rn. Analysis on homogeneous sets with non-integer Hausdorff dimensions is applied to the low-order perturbative renormalisation of quantum electrodynamics. This new regularisation method implements the Dirac matrices and tensors in R4 without difficulties, is gauge invariant, covariant and differs from dimensional regularisation in some aspects.
TL;DR: In this article, the authors show that the complicated scattering behavior is caused by unstable periodic orbits having homoclinic and heterocliic connections, leading to horseshoe chaos in the flow.
Abstract: A classical mechanical system is analysed which exhibits complicated scattering behaviour. In the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular. Though in the complement of this Cantor set the deflection function is regular, one can choose impact parameter intervals leading to arbitrarily complicated trajectories. The authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections. Thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow. This invariant set contains infinitely many unstable localised orbits (periodic and aperiodic ones). The stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function.
TL;DR: The effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined in this article, where it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system.
Abstract: The effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined. It is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system. It is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means. This allows the calculation of effective permeability, the mean pressure and the pressure variance. The method, as well as recovering familiar results, gives a formal means of improving the approximation and approaching more complex systems.
TL;DR: In this article, the authors derived the partition functions of the ADE lattice models on the torus in terms of partition function of six-vertex models and which can be computed in the continuum limit.
Abstract: The author expresses the partition functions of the ADE lattice models on the torus in terms of partition function of six-vertex models and which can be computed in the continuum limit. Thus the author recovers the partition function of minimal models with C<1 and new ones with C=1.
TL;DR: In this article, a non-compact case is applied to solve a family of second Poschl-Teller Morse-Rosen and Eckart equations with quantised coupling constants.
Abstract: The method of an earlier paper (see ibid., vol.20, p.4075 (1987)) is applied to the non-compact case to solve a family of second Poschl-Teller Morse-Rosen and Eckart equations with quantised coupling constants. Both discrete and continuous spectra, bound state and scattering wavefunctions (transmission coefficients) are found from the matrix elements of group representations,.
TL;DR: In this article, the authors proposed non-linear normal forms for periodic orbits, which describe how satellite periodic orbits coalesce with the central one as resonance is approached (in to 0), and expressed the resonant contributions as diffraction integrals.
Abstract: The semiclassical density of states depends, according to the periodic-orbit sum formula, on the linear stability of the orbits. This means, however, that contributions from the marginally stable or 'resonant' orbits, which necessarily accompany stable ones, diverge unphysically. The remedy for a system of two degrees of freedom is found to lie in the classical non-linear normal forms for periodic orbits, which describe how satellite periodic orbits coalesce with the central one as resonance is approached ( in to 0). Through these forms the resonant contributions are expressed as diffraction integrals (the first few being 'diffraction catastrophes') uniformly valid in in and h(cross), and finite even for in to 0 provided h(cross)=0. An extension is proposed to incorporate, jointly, multiple resonances found in repetitions of orbits.
TL;DR: In this paper, the authors derived the operator dispersion equation of the multilayer gyroanisotropic waveguide and its reflection and transmission operators by means of the characteristic matrix of such a medium Fresnel's reflection operator.
Abstract: For the general case of an inhomogeneous anisotropic and gyrotropic medium a differential tensor equation, expressing the evolution of the tangential component of the field vectors of an electromagnetic wave is obtained. A fundamental solution of this equation is given by a multiplicative integral. A plane-stratified system of anisotropic and gyrotropic layers is considered. By means of the characteristic matrix of such a medium Fresnel's reflection and transmission operators are derived. These operators have wide utility because they describe exactly the interaction of light with any plane-stratified gyroanisotropic structure. The conservation of the normal component of the Poynting vector in such a structure allows the authors to find a correlation between the operators of reflection and transmission. The operator dispersion equation of the multilayer gyroanisotropic waveguide is presented. All the calculations in this paper are based on the direct manipulation of tensors and their invariants, eliminating the use of coordinate systems. This facilitates solutions and provides results of great generality which are suitable for computer use.