Table of contents

We develop a formalism to analyse the behaviour of pulse-coupled identical phase oscillators with specific attention devoted to the onset of partial synchronization. The method, which allows describing the dynamics both at the microscopic and macroscopic level, is introduced in a general context, but then the application to the dynamics of leaky integrate-and-fire (LIF) neurons is analysed. As a result, we derive a set of delayed equations describing exactly the LIF behaviour in the thermodynamic limit. We also investigate the weak coupling regime by means of a perturbative analysis, which reveals that the evolution rule reduces to a set of ordinary differential equations. Robustness and generality of the partial synchronization regime is finally tested both by adding noise and considering different force fields.

A useful kind of continuity of quantum states functions in asymptotic regime is so-called asymptotic continuity. In this letter, we provide general tools for checking if a function possesses this property. First we prove equivalence of asymptotic continuity with so-called robustness under admixture. This allows us to show that relative entropy distance from a convex set including a maximally mixed state is asymptotically continuous. Subsequently, we consider arrowing—a way of building a new function out of a given one. The procedure originates from constructions of intrinsic information and entanglement of formation. We show that arrowing preserves asymptotic continuity for a class of functions (so-called subextensive ones). The result is illustrated by means of several examples.

In this letter we demonstrate the occurrence of first-order transitions in temperature for some recently introduced generalized XY models, and also point out the connection between them and annealed site-diluted (lattice-gas) continuous-spin models.

We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. Instanton–vortex systems, monopole–vortex–wall systems, and webs of walls in Abelian gauge theories are found to admit negative energy objects with the instanton charge (called intersectons), the monopole charge (called boojums) and the Hitchin charge, respectively. We characterize the total moduli space of these elementary as well as composite solitons. In particular the total moduli space of walls is given by the complex Grassmann manifold SU(N_F)/[SU(N_C) × SU(N_F − N_C) × U(1)] and is decomposed into various topological sectors corresponding to boundary condition specified by particular vacua. The moduli space of k vortices is also completely determined and is reformulated as the half ADHM construction. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices and walls. Review parts contain our works on domain walls (Isozumi Y et al 2004 Phys. Rev. Lett.93 161601 (Preprint hep-th/0404198), Isozumi Y et al 2004 Phys. Rev. D 70 125014 (Preprint hep-th/0405194), Eto M et al 2005 Phys. Rev. D 71 125006 (Preprint hep-th/0412024), Eto M et al 2005 Phys. Rev. D 71 105009 (Preprint hep-th/0503033), Sakai N and Yang Y 2005 Comm. Math. Phys. (in press) (Preprint hep-th/0505136)), vortices (Eto M et al 2005 Phys. Rev. Lett.96 161601 (Preprint hep-th/0511088), Eto M et al 2006 Phys. Rev. D 73 085008 (Preprint hep-th/0601181)), domain wall webs (Eto M et al 2005 Phys. Rev. D 72 085004 (Preprint hep-th/0506135), Eto M et al 2006 Phys. Lett. B 632 384 (Preprint hep-th/0508241), Eto M et al 2005 AIP Conf. Proc.805 354 (Preprint hep-th/0509127)), monopole–vortex–wall systems (Isozumi Y et al 2005 Phys. Rev. D 71 065018 (Preprint hep-th/0405129), Sakai N and Tong D 2005 J. High Energy Phys. JHEP03(2005)019 (Preprint hep-th/0501207)), instanton–vortex systems (Eto M et al 2005 Phys. Rev. D 72 025011 (Preprint hep-th/0412048)), effective Lagrangian on walls and vortices (Eto M et al 2006 Phys. Rev. D (in press) (Preprint hep-th/0602289)), classification of BPS equations (Eto M et al 2005 Preprint hep-th/0506257) and Skyrmions (Eto M et al 2005 Phys. Rev. Lett.95 252003 (Preprint hep-th/0508130)).

The wormlike chain model of stiff polymers is a nonlinear σ-model in one spacetime dimension in which the ends are fluctuating freely. This causes important differences with respect to the presently available theory which exists only for periodic and Dirichlet boundary conditions. We modify this theory appropriately and show how to perform a systematic large-stiffness expansion for all physically interesting quantities in powers of L/ξ, where L is the length and ξ the persistence length of the polymer. This requires special procedures for regularizing highly divergent Feynman integrals which we have developed in previous work. We show that by adding to the unperturbed action a correction term , we can calculate all Feynman diagrams with Green functions satisfying Neumann boundary conditions. Our expansions yield, order by order, a properly normalized end-to-end distribution function in arbitrary dimensions d, its even and odd moments and the two-point correlation function.

The compressed delta atom is a one-dimensional version of the compressed hydrogen atom where the finite-range Coulomb potential is replaced by a zero-range delta function. The spectral equation of the compressed delta atom is transcendental. Nevertheless, using recently developed quadrature and absolutely convergent periodic-orbit expansion techniques, it can be solved analytically, which yields its energy levels explicitly in the form E_n = f(n; p), where n is the quantum number, p is a set of parameters characterizing the atom and f, a function expressed as a quadrature or as an absolutely convergent sum over periodic orbits, is the same for all n. The compressed delta atom may serve as a template for the explicit, exact and analytical solution of other one-dimensional quantum problems with potentials consisting of a superposition of delta-function potentials and piecewise constant potentials.

Tunnelling rates are qualitatively different for integrable and near-integrable systems. Here a uniform result is derived which interpolates between the two regimes and is applied successfully to two-dimensional double-well and pendulum potentials. When the underlying symmetry is reflection in a single coordinate, the splitting remains positive after perturbation, but for potentials whose underlying symmetry is reflection through the origin, the splitting is predicted to oscillate as a function of a perturbation parameter, with regularly spaced zeros where tunnelling switches off.

A nonlinear realization of the nonstandard (super-Jordanian) version of is given, for all N.

Twist and writhe measure basic geometric properties of a ribbon or tube. While these measures have applications in molecular biology, materials science, fluid mechanics and astrophysics, they are under-utilized because they are often considered difficult to compute. In addition, many applications involve curves with endpoints (open curves); but for these curves the definition of writhe can be ambiguous. This paper provides simple expressions for the writhe of closed curves, and provides a new definition of writhe for open curves. The open curve definition is especially appropriate when the curve is anchored at endpoints on a plane or stretches between two parallel planes. This definition can be especially useful for magnetic flux tubes in the solar atmosphere, and for isotropic rods with ends fixed to a plane.

Novel integrable systems of coupled first-order autonomous PDEs in 1 + 1 dimensions (space x and time t) are presented. Integrable covariant 2-vector and 3-vector PDEs, as well as higher-order integrable PDEs for a single, or a couple, of scalar-dependent variables (including an extension of the sine-Gordon equation and a remarkably neat, highly nonlinear third-order PDE), are also obtained by appropriate reductions of the basic matrix equations. The Lax pairs that characterize the integrable character of the basic matrix PDEs are also exhibited, as well as their single-soliton solutions. These solitons generally exhibit the boomeronic (and, less generically, the trapponic) phenomenology, namely they do not move uniformly, but rather (in an appropriate reference system) come in from one end in the remote past and boomerang back to that same end in the remote future (boomerons), or are trapped to oscillate around a value fixed by the initial data (trappons).

The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schrödinger operators, square-root Klein–Gordon operators and Dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity. The present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schrödinger operators: (1) operators with slowly oscillating at infinity potentials, (2) operators with periodic and semi-periodic potentials, (3) Schrödinger operators which are discrete quantum analogues of the acoustic propagators for waveguides, (4) operators with potentials having an infinite set of discontinuities and (5) three-particle Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice .

One-dimensional Ginzburg–Landau equations with derivatives of noninteger order are considered. Using psi-series with fractional powers, the solution of the fractional Ginzburg–Landau (FGL) equation is derived. The leading-order behaviours of solutions about an arbitrary singularity, as well as their resonance structures, have been obtained. It was proved that fractional equations of order α with polynomial nonlinearity of order s have the noninteger power-like behaviour of order α/(1 − s) near the singularity.

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler–Lagrange equations are derived. Fractional equations are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives.

An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived. Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated. A hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is obtained via hodograph transformation. Furthermore, the dispersionless Gelfand–Dickey hierarchy with sources (dGDHWS) is presented.

Suppose we are given an entangled pair and asked how well we can produce two entangled pairs starting from a given entangled pair using only local operations. To respond to this question, we study broadcasting of entanglement using a state-dependent quantum cloning machine as a local copier. We show that the length of the interval for probability-amplitude-squared α² for broadcasting of entanglement using a state-dependent cloner can be made larger than the length of the interval for probability-amplitude-squared for broadcasting entanglement using a state-independent cloner. Further we show that there exists a local state-dependent cloner which gives better quality copy (in terms of average fidelity) of an entangled pair than the local universal cloner.

We consider the hydrogen molecular ion H⁺₂ in the fixed nuclear approximation, in the presence of a strong homogeneous magnetic field. We determine the leading asymptotic behaviour for the equilibrium distance between the nuclei of this molecule in the limit when the strength of the magnetic field goes to infinity.

Nash equilibria are found for some quantum games with particles with spin-1/2 for which two spin projections on different directions in space are measured. Examples of macroscopic games with the same equilibria are given. Mixed strategies for participants of these games are calculated using probability amplitudes according to the rules of quantum mechanics in spite of the macroscopic nature of the game and absence of Planck's constant. A possible role of quantum logical lattices for the existence of macroscopic quantum equilibria is discussed. Some examples for spin-1 cases are also considered.

Under certain constraints on the parameters a, b and c, it is known that Schrödinger's equation −d²ψ/dx² + (ax⁶ + bx⁴ + cx²)ψ = Eψ, a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this paper we show that the exact wavefunction ψ is the generating function for a set of orthogonal polynomials {P^(t)_n(x)} in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced, by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E) = P⁽⁰⁾_n(E) recently discovered by Bender and Dunne.

We explore the task of optimal quantum channel identification and in particular, the estimation of a general one-parameter quantum process. We derive new characterizations of optimality and apply the results to several examples including the qubit depolarizing channel and the harmonic oscillator damping channel. We also discuss the geometry of the problem and illustrate the usefulness of using entanglement in process estimation.

We show that an nth root of the Walsh–Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubit lines is needed for the application of the operations described here. A quantum algorithm based on this set can be interpreted as a discrete diffusion of a quantum particle on a de Bruijn graph, corrected on-the-fly by auxiliary modifications of the phases associated with the arcs.

Using the SU(N) representation of the group theory, we derive the general form of the spin swapping operator for the quantum Heisenberg spin-s systems. We further prove that such a spin swapping operator is equal to the spin singlet pairing operator under the partial transposition. For SU(2) invariant states, it is shown that the expectation value of the spin swapping operator and its generalizations, the permutations, can be used as an entanglement witness, especially, for the formulation of observable conditions of entanglement.

We construct a weak formulation of the wave equation in curved spacetime for solutions which are regularly discontinuous across a hypersurface. We adopt the framework of distributions and tensor-distributions, and allow the presence of discontinuity for the first and second derivatives of the spacetime metric. We thus find the corresponding compatibility conditions to hold at the interface, as replacement for the differential equation, which is undefined there. In particular, we find out that if discontinuity of the first derivatives of the metric is present, such compatibility conditions also involve the mean values of the field, and not only its jump across the discontinuity hypersurface. We also consider the case of singular solutions with support on a hypersurface, and derive the corresponding compatibility conditions. Applications to electromagnetism are presented.

Dirac's approach to incorporate the radiation into the equation of motion for a point charge in classical electrodynamics is based on three structural components: the point model for the electron, the Maxwell equations and the principle of relativity. These fundamental components lead to an equation of motion that involves an undetermined 4-vector B_µ. The Lorentz–Dirac equation corresponds to the case in which B_µ = 0, but in general there is a large family of 4-vectors B_µ consistent with the above three basic components. This paper deals with the study of these equations of motion in the case of the three simplest permissible choices for B_µ. We show that these equations admit as exact solutions the motion of an arbitrary number of identical charges that are equally spaced in a circumference and that rotate at constant angular velocity. These solutions show that the rate of radiation emitted by the system of charges is completely independent of the 4-vector B_µ. We also study the restrictions over the dimensionless parameters that appear in the four-vectors B_µ, in order that the trajectories of the corresponding equations cannot be discriminated from the trajectory determined by the Lorentz–Dirac equation in a practical case, as for instance the design and operation of a synchrotron.

We show a class of localized instabilities to nonlinear global states of very general form of amplitude equations. The localized instabilities are special in their spatial structure and can come in even and odd parity classes. The importance of such instabilities in the dynamics and formation of domains of a global state in 1D have been discussed.

In our recently published paper 'Chaos in Bohmian quantum mechanics' we criticized a paper by Parmenter and Valentine (1995 Phys. Lett. A 201 1), because the authors made an incorrect calculation of the Lyapunov exponent in the case of Bohmian orbits in a quantum system of two uncoupled harmonic oscillators.

After our paper was published, we became aware of an erratum published by the same authors (Parmenter and Valentine 1996 Phys. Lett. A 213 319) that recognized the error made in their previous calculations. The authors realized that, when correctly calculated, 'aperiodic trajectories with well defined boundaries...have vanishing Lyapunov exponents', i.e., they are not chaotic.

We want to supplement our paper with a reference to this erratum. The generic calculation of Lyapunov exponents in Bohmian quantum systems remains an original contribution of our paper (section 2).

In the denominators of the last terms of (2.16), (2.17), (2.18) and of the line after (2.16), the parameter μ should be replaced by μ-μ₀.