Table of contents

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to the subject of the conference `Viewing The World Through Spin Glasses', in honour of David Sherrington on the occasion of his 65th birthday, 31 August–1 September 2007 (http://www.nottingham.ac.uk/\verb.~.ppzjpg/DS2007/). Invited speakers and participants at that meeting and other researchers working in the field are invited to submit a research paper to this issue.

The Editorial Board has invited Ton Coolen, Hidetoshi Nishimori, Nicolas Sourlas and Michael Wong to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are as follows:

•The subject of the paper should relate to the subject of the conference (see the website of the conference http://www.nottingham.ac.uk/\verb.~.ppzjpg/DS2007/).

•Contributions will be refereed and processed according to the usual procedure of the journal.

•Conference papers may be based on already published work but should either contain significant additional new results and/or insights or give a survey of the present state of the art, a critical assessment of the present understanding of a topic, and a discussion of open problems.

•Papers submitted by non-participants should be original and contain substantial new results.

The guidelines for the preparation of contributions are the following:

•The DEADLINE for submission of contributions is 1 December 2007. This deadline will allow the special issue to appear in July 2008.

•There is a nominal page limit of 16 printed pages per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length.

•Further advice on publishing your work in Journal of Physics A: Mathematical and Theoretical may be found at www.iop.org/Journals/jphysa.

•Contributions to the special issue should, if possible, be submitted electronically by web upload at www.iop.org/Journals/jphysa or by e-mail to jphysa@iop.org, quoting `JPhysA Special Issue—Sherrington'. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions. All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.

•This special issue will be published in the paper and online version of the journal. Each participant at the workshop and the corresponding author of each contribution will receive a complimentary copy of the issue.

We consider a simple modification of standard phase-space path integrals and show that it leads in configuration space to Lagrangians depending also on accelerations.

The exact entanglement evolution of two two-level systems, coupled to N surrounding spin , is analysed with the help of a new criterion relying on the ability of measuring a few simple observables only.

A very particular connection between the commutation relations of the elements of the generalized Pauli group of a d-dimensional qudit, d being a product of distinct primes, and the structure of the projective line over the (modular) ring is established, where the integer exponents of the generating shift (X) and clock (Z) operators are associated with submodules of . Under this correspondence, the set of operators commuting with a given one—a perp-set—represents a submodule of . A crucial novel feature here is that the operators are also represented by non-admissible pairs of . This additional degree of freedom makes it possible to view any perp-set as a set-theoretic union of the corresponding points of the associated projective line.

A generalized zero-range process with a limited number of long-range interactions is studied as an example of a transport process in which particles at a T-junction make a choice of which branch to take based on traffic levels on each branch. The system is analysed with a self-consistent mean-field approximation which allows phase diagrams to be constructed. Agreement between the analysis and simulations is found to be very good.

We consider a superconductor/multiferromagnet/superconductor (S/I/F₁/⋅ ⋅ ⋅/F_n/I/S) ballistic junction with thin insulating layers in the interfaces. We develop a diagrammatic approach for the equation that determines the Andreev spectrum, by examining the closed loops in the intermediate ferromagnetic layers. It is expressed in terms of the S/F and F/F interface scattering amplitudes. A set of rules is determined. Several analytical formulas are obtained, and in particular we study the triple ferromagnetic layer case. We also obtain the maximum current from the scattering matrix, and show that it is consistent with the diagrammatic approach. A specific application is the asymmetric S/F/I/S junction. The diagrammatic approach is also checked with the matrix of scattering amplitudes method.

We consider three fermions with two spin components interacting on a lattice model with an infinite scattering length. Low-lying eigenenergies in a cubic box with periodic boundary conditions, and for a zero total momentum, are calculated numerically for decreasing values of the lattice period. The results are compared to the predictions of the zero-range Bethe–Peierls model in continuous space, where the interaction is replaced by contact conditions. The numerical computation, combined with analytical arguments, shows the absence of negative energy solution, and a rapid convergence of the lattice model towards the Bethe–Peierls model for a vanishing lattice period. This establishes for this system the universality of the zero-interaction range limit.

We provide a framework for building periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the required mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems, and illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.

We analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator–prey system. The individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired by the processes of the Lotka–Volterra model. Two levels of mean-field approximations are set up. The simple approximation is equivalent to an extended patch model, a simple metapopulation model with patches colonized by prey, patches colonized by predators and empty patches. This approximation is capable of describing the limited available space for species occupancy. The pair approximation is moreover able to describe two types of coexistence of prey and predators: one where population densities are constant in time and another displaying self-sustained time oscillations of the population densities. The oscillations are associated with limit cycles and arise through a Hopf bifurcation. They are stable against changes in the initial conditions and, in this sense, they differ from the Lotka–Volterra cycles which depend on initial conditions. In this respect, the present model is biologically more realistic than the Lotka–Volterra model.

We consider the continuum version of an elastic chain supporting topological and non-topological degrees of freedom; this generalizes a model for the dynamics of DNA recently proposed and investigated by ourselves. In a certain limit, the non-topological degrees of freedom are frozen, and the model reduces to the sine-Gordon equations and thus supports well-known topological soliton solutions. We consider a (singular) perturbative expansion around this limit and study in particular how the non-topological field assumes the role of an auxiliary field. This provides a more general framework for the slaving of this degree of freedom on the topological one, already observed elsewhere in the context of the mentioned DNA model; in this framework one expects such a phenomenon to arise in a quite large class of field-theoretical models.

We introduce a new technique—characterized by an arbitrary positive constant Ω, with which we associate the period T = 2π/Ω—to 'Ω-modify' a Hamiltonian so that the new Hamiltonian thereby obtained is entirely isochronous, namely it yields motions all of which (except possibly for a lower dimensional set of singular motions) are periodic with the same fixed period T in all their degrees of freedom. This technique transforms real autonomous Hamiltonians into Ω-modified Hamiltonians which are also real and autonomous, and it is widely applicable, for instance, to the most general many-body problem characterized by Newtonian equations of motion ('acceleration equal force') provided it is translation invariant. The Ω-modified Hamiltonians are of course not translation invariant, but for Ω = 0 they reduce (up to marginal changes) to the unmodified Hamiltonians they were obtained from. Hence, when this technique is applied to translation-invariant Hamiltonians yielding, in their center-of-mass systems, chaotic motions with a natural time scale much smaller than T, the corresponding Ω-modified Hamiltonians shall display a chaotic behavior for quite some time before the isochronous character of the motions takes over. We moreover show that the quantized versions of these Ω-modified Hamiltonians feature equispaced spectra.

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point—X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.

We construct the bi-Hamiltonian structure of the waterbag model of dKP and establish the third-order Hamiltonian operator associated with the waterbag model. Also, the symmetries and conserved densities of the rational type are discussed.

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda equation). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda equation. The formula for the fundamental cycle of the pBBS is obtained as a corollary.

A sufficiently general form of a two-component dark-bright (DB) vector optical soliton with seven free real parameters is obtained by solving the integrable coupled nonlinear Schrödinger equation (Manakov model) with the help of the Hirota method. We find that this solution unlike the other vector soliton solutions of the Manakov model (namely bright–bright, dark–dark and bright-guided dark vector optical solitons) possesses a singularity. This singularity is found to have a restricted movement in the parametric domain of the solution provided the dark component of the DB vector soliton is a gray dark soliton. However in the case of the DB soliton with fundamental dark component, this singularity is fixed to a particular value. In addition, two different physically interesting cases namely DB soliton with self-focusing nonlinearity and self-defocusing nonlinearity arise under two different parametric conditions. Finally, its collision dynamics is also investigated by constructing a more general DB multisoliton solution.

In this paper, we consider the (2+1)-dimensional discrete fourth-order nonisospectral problem. By using the Lax technique, three new (2+1)-dimensional nonisospectral four-field integrable lattice hierarchies are constructed. Their reductions yield three (1+1)-dimensional isospectral four-field integrable lattice hierarchies due to Mlaszak–Marciniak. We make a comparison between the (1+1)-dimensional discrete fourth-order nonisospectral problem and the third-order nonisospectral problem. We found that the integrable lattice hierarchies related to the discrete fourth-order nonisospectral problem have new characteristics.

In this paper, we study a set of functions, defined on an interval of finite width, which are orthogonal and which reduce to the sinc functions when the appropriate limit is taken. We show that these functions can be used within a variational approach to obtain accurate results for a variety of problems. We have applied them to the interpolation of functions on finite domains and to the solution of the Schrödinger equation, and we have compared the performance of the present approach with others.

Matrix quasi-exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite-dimensional invariant vector spaces. New examples of (2 × 2)-matrix quasi-exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.

We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.

The machinery of qubit portraits of qudit states, recently presented, is considered here in more details in order to characterize the presence of quantum correlations in bipartite qudit states. In the tomographic representation of quantum mechanics, Bell-like inequalities are interpreted as peculiar properties of a family of classical joint probability distributions which describe the quantum state of two qudits. By means of the qubit-portraits machinery a semigroup of stochastic matrices can be associated with a given quantum state. The violation of the CHSH inequalities is discussed in this framework with some examples; we found that quantum correlations in qutrit isotropic states can be detected by the suggested method while it cannot be in the case of qutrit Werner states.

We derive a multipartite generalized Bell inequality which involves the entire range of settings for each of the local observers. Especially, it is applied to show non-local behavior of a six-qubit mixture of Greenberger–Horne–Zeilinger correlations stronger than previous Bell inequalities. For certain noise admixture to the correlations an explicit local realistic model exists in the case of a standard Bell experiment. Bell experiments with many local settings reveal the non-locality of the state. It turns out that the new inequality is more stringent than many other Bell inequalities in the specific quantum state.

The interest of quadratic algebras for position-dependent mass Schrödinger equations is highlighted by constructing spectrum generating algebras for a class of d-dimensional radial harmonic oscillators with d ⩾ 2 and a specific mass choice depending on some positive parameter α. Via some minor changes, the one-dimensional oscillator on the line with the same kind of mass is included in this class. The existence of a single unitary irreducible representation belonging to the positive-discrete series type for d ⩾ 2 and of two of them for d = 1 is proved. The transition to the constant-mass limit α → 0 is studied and deformed su(1,1) generators are constructed. These operators are finally used to generate all the bound-state wavefunctions by an algebraic procedure.

We present a general scheme for multiparty-controlled teleportation of an arbitrary m-qudit (d-dimensional quantum system) state by using non-maximally entangled states as the quantum channel. The sender performs m generalized Bell-state measurements on her 2m particles, the controllers take some single-particle measurements with the measuring basis X_d and the receiver only needs to introduce one auxiliary two-level particle to extract quantum information probabilistically with the fidelity unit if he cooperates with all the controllers. All the parties can use some decoy photons to set up their quantum channel securely, which will forbid a dishonest party to eavesdrop freely. This scheme is optimal as the probability that the receiver obtains the originally unknown m-qudit state equals the entanglement of the quantum channel.

A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H^{(chain model)}, and all its up–down symmetrized special cases with J = [N/2] real couplings are considered, H^{(chain model)} → H^(N). Using symbolic manipulation and extrapolation techniques we find out that in the strongly non-Hermitian regime the secular equation gets partially factorized at all N. This enables us to reveal a fine-tuned alignment of the dominant couplings implying an asymptotically sharply spiked shape of the boundary of the J-dimensional quasi-Hermiticity domain in which all the spectrum of energies E^(N)_n remains real and observable.

We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure , these properties are related to the Ricci tensor of .

The radiation reaction problem for an electric charge moving in flat spacetime of three dimensions is discussed. The divergences stemming from the pointness of the particle are studied. A consistent regularization procedure is proposed, which exploits the Poincaré invariance of the theory. Effective equation of motion of radiating charge in an external electromagnetic field is obtained via the consideration of energy-momentum and angular momentum conservation. This equation includes the effect of the particle's own field. The radiation reaction is determined by the Lorentz force of point-like charge acting upon itself plus a non-local term which provides finiteness of the self-action.

Planar bilayer lipid membranes (BLMs) are currently employed to construct many bio-inspired material systems and structures. In order to characterize the pressure effects on the equilibrium configurations of these biological membranes, a novel continuum model is proposed. The BLM is assumed to be a two-layer smectic A liquid crystal. The mean orientation of the amphiphilic molecules comprising the membrane is postulated to be perpendicular to the layers and each layer is idealized as a two-dimensional liquid. Moreover, the BLM is modeled as a simply supported plate undergoing small deformations. It is subjected to a pressure load that acts perpendicularly to the layers. The equilibrium equations and boundary conditions are derived from the bulk elastic energy for smectic A liquid crystals as described by de Gennes and Prost (1993 The Physics of Liquid Crystals 2nd edn (Oxford Science Publications)) by using variational methods. The resulting fourth-order linear partial differential equation is solved by employing cylindrical functions and the series solution is proved to be convergent. The solution is numerically computed for values of the model parameters that are reported in the literature.