Table of contents

We argue that frustrated Josephson junction arrays may support a topologically ordered superconducting ground state, characterized by a non-trivial ground-state degeneracy on the torus. This superconducting quantum fluid provides an explicit example of a system in which superconductivity arises from a topological mechanism rather than from the usual Landau–Ginzburg mechanism.

We study the φ⁴ lattice model with anisotropic short-range interactions. We show that the nonuniversal finite-size scaling functions of the anisotropic φ⁴ model in a d-dimensional rectangular geometry are determined by the universal finite-size scaling functions of the isotropic φ⁴ theory in a d-dimensional parallelepiped. Predictions are made for the Binder cumulant. In the bulk limit two-scale factor universality is absent in those universal critical-point amplitude relations that involve the correlation length.

In a recent paper (Aldaya V and Guerrero J 2005 J. Phys. A: Math. Gen.38 6939) it is shown that the bound states of the Modified Pöschl–Teller potential should be described by the non-compact dynamical group SU(1, 1) instead of the usual compact group SU(2). Here we prove that SU(2) cannot be the dynamical group for a potential with a finite number of bound states on the basis of the Modified Pöschl–Teller potential and the Morse potential. This suggests that a quantum system with a continuum spectrum and a finite number of bound states should be described, both in the continuum and the discrete spectrum, by a non-compact dynamical group instead of a compact one.

We consider quantum global vortex string correlation functions, within the Kalb–Ramond framework, in the presence of a background field-strength tensor and investigate the conditions under which this yields a nontrivial contribution to those correlation functions. We show that a background field must be supplemented to the Kalb–Ramond theory in order to correctly describe the quantum properties of the vortex strings. The explicit form of this background field and the associated quantum vortex string correlation function are derived. The complete expression for the quantum vortex creation operator is explicitly obtained. We discuss the potential applicability of our results in the physics of superfluids and rotating Bose–Einstein condensates.

We study the semiclassical Wigner–Kirkwood (WK) expansion of the partition function Z(t) for arbitrary even homogeneous potentials, starting from the Bloch equation. As is well known, the phase-space kernel of Z satisfies the so-called Uhlenbeck–Beth equation, which depends on the gradients of the potential. We perform a chain of transformations to obtain novel forms of this equation that invite analogies with various physical phenomena and formalisms, such as diffusion processes, the Fokker–Planck equation and supersymmetric quantum mechanics.

When treating problems of vector diffraction in electromagnetic theory, the evaluation of the integral involving Bessel and associated Legendre functions is necessary. Here we present the analytical result for this integral that will make numerical quadrature techniques or localized approximations unnecessary. The solution is presented using the properties of the Bessel and associated Legendre functions.

I investigate the thermodynamic equivalence between finite systems of constant density of states below the Bose–Einstein condensation temperature. I show that in order to preserve the equivalence, a phenomenon called Fermi condensation has to be taken into account. This phenomenon leads to a correction of the intensive parameters of the Fermi gas and of the Fermi distribution in canonical Fermi systems at low temperatures.

We analyse the statistics of electrostatic energies (and their differences) for a quantum dot system composed of a finite number K of electron islands (metallic grains) with random capacitance–inductance matrix C, for which the total charge is discrete, Q = Ne (where e is the charge of an electron and N is an integer). The analysis is based on a generalized charging model, where the electrons are distributed among the grains such that the electrostatic energy E(N) is minimal. Its second difference (inverse compressibility) χ_N = E(N + 1) − 2E(N) + E(N − 1) represents the spacing between adjacent Coulomb-blockade peaks appearing when the conductance of the quantum dot is plotted against gate voltage. The statistics of this quantity for single grain quantum dots has been the focus of experimental and theoretical investigations during the last two decades. In the more general case of quantum dots composed of several grains, we provide an algorithm for calculating the distribution function corresponding to χ_N and show that this function is piecewise polynomial.

We study the structure of chains which have anisotropic short-range contact interactions that depend on the alignment of the binormal vectors of chain segments. This represents a crude model of hydrogen bonding or 'stacking' interactions out of the plane of curvature. The polymers are treated as ribbon-like semi-flexible chains, where the plane of the ribbon is determined by the local binormal. We show that with dipole–dipole interactions between the binormals of contacting chain segments, mean-field theory predicts a first-order transition to a binormally aligned state. We describe the onset of this transition as a function of the temperature-dependent parameters that govern the chain stiffness and the strength of the binormal interaction, as well as the binormal alignment's coupling to chain collapse. We also examine a metastable state governing the folding kinetics. Finally, we discuss the possible mesoscopic structure of the aligned phase, and application of our model to secondary structure motifs like β-sheets and α-helices, as well as composite structures like β-(amyloid) fibrils.

Numerical computations explored for this paper show that an upper bound on the ropelength of large knots with crossing number n grows as fast as nln²n. The algorithms to randomly generate samples of such large knots and to determine an upper bound on the ropelength for each knot are described. The numeric results are presented and compared to the smallest known theoretical upper bounds on ropelength.

We analyse the thermodynamic Bethe Ansatz equations of the one-dimensional half-filled Hubbard model in the 'spin-disordered regime', which is characterized by the temperature being much larger than the magnetic energy scale, but small compared to the Mott–Hubbard gap. In this regime the thermodynamics of the Hubbard model can be thought of in terms of gapped charged excitations with an effective dispersion and spin degrees of freedom that only contribute entropically. In particular, the internal energy and the effective dispersion become essentially independent of temperature. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal–insulator transition at a finite interaction strength above which the gap opens linearly. We relate these observations to studies of the Mott–Hubbard transition in the limit of infinite dimensions.

We study a general mass transport model on an arbitrary graph consisting of L nodes each carrying a continuous mass. The graph also has a set of directed links between pairs of nodes through which a stochastic portion of mass, chosen from a site-dependent distribution, is transported between the nodes at each time step. The dynamics conserves the total mass and the system eventually reaches a steady state. This general model includes as special cases various previously studied models such as the zero-range process and the asymmetric random average process. We derive a general condition on the stochastic mass transport rules, valid for arbitrary graph and for both parallel and random sequential dynamics, that is sufficient to guarantee that the steady state is factorizable. We demonstrate how this condition can be achieved in several examples. We show that our generalized result contains as a special case the recent results derived by Greenblatt and Lebowitz for d-dimensional hypercubic lattices with random sequential dynamics.

The Langevin dynamics of a d-dimensional mean spherical model with competing interactions along m ⩽ d directions of a hypercubic lattice is analysed. After a quench at high temperatures, the dynamical behaviour is characterized by two distinct time scales associated with stationary and ageing regimes. The asymptotic expressions for the autocorrelation and response functions, in supercritical, critical and subcritical cases, were calculated. Ageing effects, which are known to be present in the ferromagnetic version of this model system, are not affected by the introduction of competing interactions.

Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning trees in the associated factor graph. We introduce a modified survey propagation algorithm which returns null edges of the factor graph and helps us to find satisfiable spanning trees. This allows us to study organization of satisfiable spanning trees in the space spanned by spanning trees.

In a positive answer to the open problem proposed by R Metzler and J Klafter, it is proved that the asymptotic shape of the solution for a wide class of fractional Fokker–Planck equations is a stretched Gaussian for the initial condition being a pulse function in the homogeneous and heterogeneous fractal structures, whose mean square displacement behaves like ⟨(Δx)²(t)⟩ ∼ t^γ and ⟨(Δx)²(t)⟩ ∼ x^−θt^γ(0 < γ < 1, − ∞ < θ < +∞), respectively.

We investigate the critical behaviour of the two-dimensional Ising model defined on a curved surface with a constant negative curvature. Finite-size scaling analysis reveals that the critical exponents for the zero-field magnetic susceptibility and the correlation length deviate from those for the Ising lattice model on a flat plane. Furthermore, when reducing the effects of boundary spins, the values of the critical exponents tend to those derived from the mean field theory. These findings evidence that the underlying geometric character is responsible for the critical properties of the Ising model when the lattice is embedded on negatively curved surfaces.

Bose–Einstein condensation of an ideal relativistic Bose gas trapped in a generic power-law potential is investigated. The analytical expressions for some important parameters such as the critical temperature, ground-state fraction and heat capacity are derived. The general criteria on the occurrence of Bose–Einstein condensation and the discontinuity of heat capacity at the critical temperature are obtained. The results obtained here present a unified description for the Bose–Einstein condensation of a class of ideal Bose systems so that many important conclusions in the literature are included in this paper.

We study the spectra of a finite-dimensional Frobenius–Perron operator (matrix) of the Bernoulli map derived from phase space discretization. The eigenvalues and (right and left) eigenvectors are analytically calculated, which are closely related to periodic orbits on the partition points. In the degenerate case, Jordan decomposition of the matrix is explicitly constructed. Except for the isolated eigenvalue 1, there is no definite limit with respect to eigenvalues when n → ∞. The behaviour of the eigenvectors is discussed in the limit of large n.

Using asymptotic methods, we investigate whether discrete breathers are supported by a two-dimensional Fermi–Pasta–Ulam lattice. A scalar (one-component) two-dimensional Fermi–Pasta–Ulam lattice is shown to model the charge stored within an electrical transmission lattice. A third-order multiple-scale analysis in the semi-discrete limit fails, since at this order, the lattice equations reduce to the (2 + 1)-dimensional cubic nonlinear Schrödinger (NLS) equation which does not support stable soliton solutions for the breather envelope. We therefore extend the analysis to higher order and find a generalized (2 + 1)-dimensional NLS equation which incorporates higher order dispersive and nonlinear terms as perturbations. We find an ellipticity criterion for the wave numbers of the carrier wave. Numerical simulations suggest that both stationary and moving breathers are supported by the system. Calculations of the energy show the expected threshold behaviour whereby the energy of breathers does not go to zero with the amplitude; we find that the energy threshold is maximized by stationary breathers, and becomes arbitrarily small as the boundary of the domain of ellipticity is approached.

We investigate the fundamental cycle of a periodic box–ball system (PBBS) from a relation between the PBBS and a solvable lattice model. We show that the fundamental cycle of the PBBS is obtained from eigenvalues of the transfer matrix of the solvable lattice model.

We introduce a generalization of the Dobiński relation through which we define a family of Bell-type numbers and polynomials. For all these sequences, we find the weight function of the moment problem and give their generating functions. We provide a physical motivation of this extension in the context of the boson normal ordering problem and its relation to an extension of the Kerr Hamiltonian.

The Hamiltonian formalism of the derivative nonlinear Schrödinger equation with a nonvanishing boundary value is developed by the standard procedure. The action-angle variables are given explicitly. At the end of this work, a Galileo transformation is introduced to ensure that the conservative quantities obtained satisfy the Hamiltonian equation.

The (2^w)! reversible logic circuits of width w, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group G, isomorphic to the symmetric group . We define two conjugate subgroups G₁ and G₂. Together they partition the group G into 2^w−1 + 1 double cosets. These allow us to decompose an arbitrary member of G into a cascade of three simpler members. This decomposition is a far relative of the well-known LU decomposition of a square matrix.

The solution of the problem of one-dimensional, unsteady, isentropic gas flow with the use of Riemann's invariants gives rise to a linear hyperbolic partial differential equation with the ratio of specific heats as an essential parameter. The partial differential equation has 3 + 1 + ∞ Lie point symmetries. The classical solutions are recovered with the use of the nongeneric symmetries to construct similarity solutions. Further solutions, both polynomial and other, are constructed using the invariants of the Lie point symmetries as seed solutions and the property of mapping solutions into solutions. These solutions are analogous to the well-known heat polynomials.

A generalization of the Hirota–Miwa equation to an abstract non-Abelian associative algebra is considered. This system is integrable in the sense that it arises as compatibility condition for a linear system and has solutions constructed by means of the application of an arbitrary number of Darboux transformations. These solutions are in general expressed in terms of quasideterminants.

Following an earlier derivation by Miller, we give a concise derivation of the Herman–Kluk propagator that fully exploits the symplectic structure of the problem. We also show directly that the version of the thawed Gaussian semiclassical propagator obtained by Baranger and co-workers is equivalent to the linearized Herman–Kluk propagator.

Decoherence of two atoms initially prepared in Bell states in free space is discussed theoretically through calculation of the probabilities for the atoms to remain respectively in these states. It is found that, due to quantum interference, the atoms never follow a purely exponential law to decohere from their initial states, even though spontaneous emission is illustrated to cause the decoherence.

The elastic atomic photon scattering form factor amplitude A^f0, obtained from the high energy and low momentum transfer (K) limit of the exact (within the independent particle approximation (IPA)) Rayleigh relativistic second order S-matrix scattering amplitude A^sm is decomposed into an infinite sum of its electric and magnetic field multipoles. The energy-dependent part of A^f0 (the scattering angle-dependent part of A^f0 is incorrect due to the low K approximation), combined with the angle-dependent part of A^sm, yields a form factor amplitude A^f (different versions of A^f and A^f0 will be described in the text) that can be used to decompose the anomalous scattering factor A^asf into its multipoles. A^f0 which initially contains coherent and incoherent factors is reduced to a single sum over the multipoles of only coherent terms by assuming that the charge density is spherically symmetric. At photon energies up to about ten times the K-shell binding energy it takes up to quadrupole A^asf for light atoms of Z ⩽ 20 and up to the fourth multipole for heavy atoms such as uranium for an accurate A^asf (within IPA). The multipole contributions were found to become somewhat larger as the scattering angle becomes larger (greater than 120°); however there were partial cancellations of such effects.

For mathematical models of quantum waveguides, we show that in some situations two interacting particles can be trapped more easily than a single particle. In particular, we give an example of a waveguide that cannot bind a single particle, but does have a geometrically induced bound state for two bosons that attract each other via a harmonic potential. We also show that Neumann boundary conditions are 'stickier' for two interacting bosons than for a single one.

For a nonseparable bipartite quantum state violating the Clauser–Horne–Shimony–Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both 'sides'.

Using the momentum space representation, we determine the energy eigenvalues, eigenfunctions and the high-temperature thermodynamic properties of the Dirac oscillator in one dimension in the presence of a minimal length given by , where β is the deformation parameter of the modified commutation relation [X, P] = iℏ(1 + βP²). The obtained results suggest that the effect of the minimal length could be detected in ultrarelativistic heavy-ion collisions.

Probabilistic cloning was first proposed by Duan and Guo. Then Pati established a novel cloning machine (NCM) for copying superposition of multiple clones simultaneously. In this paper, we deal with the novel cloning machine with supplementary information (NCMSI). For the case of cloning two states, we demonstrate that the optimal efficiency of the NCMSI in which the original party and the supplementary party can perform quantum communication equals that achieved by a two-step cloning protocol wherein classical communication is only allowed between the original and the supplementary parties. From this equivalence, it follows that NCMSI may increase the success probabilities for copying. Also, an upper bound on the unambiguous discrimination of two nonorthogonal pure product states is derived. Our investigation generalizes and completes the results in the literature.

A Hilbert space in M dimensions is shown explicitly to accommodate representations that reflect the decomposition of M into prime numbers. Representations that exhibit the factorization of M into two relatively prime numbers: the kq representation (Zak J 1970 Phys. Today23 51), and related representations termed q₁q₂ representations (together with their conjugates) are analysed, as well as a representation that exhibits the complete factorization of M. In this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up M.

The characteristic stretching and squeezing of chaotic motion is linearized within the finite number of phase space domains which subdivide a classical baker map. Tensor products of such maps are also chaotic, but a more interesting generalized baker map arises if the stacking orders for the factor maps are allowed to interact. These maps are readily quantized, in such a way that the stacking interaction is entirely attributed to primary qubits in each map, if each jth subsystem has Hilbert space dimension . We here study the particular example of two baker maps that interact via a controlled-not interaction, which is a universal gate for quantum computation. Numerical evidence indicates that the control subspace becomes an ideal Markovian environment for the target map in the limit of large Hilbert space dimension.

By attaching five (complex) anticommuting property coordinates to the four (real) commuting spacetime ones, it is possible to accommodate all the known fundamental particles in their three generations. A general relativistic extension to spacetime–property can be carried out such that the gauge fields find their place in the space–property sector and the Higgs scalars in the property–property sector. The full curvature is the sum of the gravitational curvature, the gauge field Lagrangian and the Higgs field contribution; property curvature may be linked to the cosmological term.

We describe an algebraic approach to the time-dependent noncommutative geometry of a six-dimensional Cahen–Wallach pp-wave string background supported by a constant Neveu–Schwarz flux and develop a general formalism to construct and analyse quantum field theories defined thereon. Various star products are derived in closed explicit form and the Hopf algebra of twisted isometries of the plane wave is constructed. Scalar field theories are defined using explicit forms of derivative operators, traces and noncommutative frame fields for the geometry, and various physical features are described. Noncommutative worldvolume field theories of D-branes in the pp-wave background are also constructed.

In this paper stationary nondiffracting vector fields are discussed. Using the definition of propagating and evanescent nondiffracting waves based on the invariance of the intensity distribution during propagation, all the types of such fields in stratified inhomogeneous media are found.

Non-local boundary conditions have been considered in theoretical high-energy physics with emphasis on one-loop quantum cosmology, one-loop conformal anomalies, Bose–Einstein condensation models and spectral branes. In the present paper, for the first time in the literature, the Wightman function, the vacuum expectation values of the field square and the energy–momentum tensor are investigated for a massive scalar field satisfying non-local boundary conditions on a single and two parallel plates. The vacuum forces acting on the plates are evaluated. Interestingly, suitable choices of the kernel in the non-local boundary conditions lead to forces acting on the plates that can be repulsive for intermediate distances. It is then possible to obtain a locally stable equilibrium value of the interplate distance stabilized by the vacuum forces.

The authors have discovered some errors in their paper. For corrections and additional results, please refer to the PDF of this corrigendum.