Table of contents

The phenomena of spontaneous symmetry breaking, associated with non-Hermitian Hamiltonians, are investigated. It is shown that spontaneous breakdown of symmetry is accompanied by the explicit breakdown of pseudo-supersymmetry. We also discuss in detail the resulting structure.

In a paper published in Am. Math. Mon. (1966 73 1–23), Kac asked his famous question 'Can one hear the shape of a drum?'. Gordon et al answered this question negatively by constructing planar isospectral pairs in their paper published in Invent. Math. (1992 110 1–22). Only a finite number of pairs have been constructed till now. Further in J. Phys. A: Math. Gen. (2005 38 L477–83), Giraud showed that most of the known examples can be generated from solutions of a certain equation which involves certain involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions when n = 2. In this letter we handle all dimensions, and show that no other examples arise.

Several results of black holes thermodynamics can be considered as firmly founded and formulated in a very general manner. From this starting point, we analyse in which way these results may improve our understanding in the thermodynamics of ordinary systems for which a pre-relativistic description is sufficient. First, we introduce a spacetime model and an entropy related to a local definition of the order in this spacetime. We show that such an approach leads to the traditional thermodynamics provided an equilibrium condition is assumed. From this condition a relation time/temperature is introduced. We show that such a relation extensively used in the black hole theory has a very general and physical meaning here. Our dynamical approach of thermodynamic equilibrium allows us to establish a relation between action and entropy identical to the one existing in the case of black holes. Since this relation exists for systems with very different underlying physics, we may expect that it corresponds to a general result in thermodynamics; it suggests that a definition of entropy in terms of order in spacetime might be more general than the Boltzmann definition related to a counting of microstates. All these results based on the fact that the paths introduced in the path-integral formalism have a physical meaning give a new approach of statistical mechanics. Finally, we compare our approach to other works based on a similar starting point.

The magnetic susceptibility of the one-dimensional Hubbard model with open boundary conditions at arbitrary filling is obtained from field theory at low temperatures and small magnetic fields, including leading and next-leading orders. Logarithmic contributions to the bulk part are identified as well as algebraic–logarithmic divergences in the boundary contribution. As a manifestation of spin–charge separation, the result for the boundary part at low energies turns out to be independent of filling and interaction strength and identical to the result for the Heisenberg model. For the bulk part at zero temperature, the scale in the logarithms is determined exactly from the Bethe ansatz. At finite temperature, the susceptibility profile as well as the Friedel oscillations in the magnetization are obtained numerically from the density-matrix renormalization group applied to transfer matrices. Agreement is found with an exact asymptotic expansion of the relevant correlation function.

We consider nonlinear Klein–Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbours) may preserve either standardly defined linear momentum or standardly defined total energy but not both. This has a variety of intriguing implications for the 'non-potential' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.

In planar cellular systems m_n denotes the average sidedness of a cell neighbouring an n-sided cell. Aboav's empirical law states that nm_n is linear in n. A downward curvature is nevertheless observed in the numerical nm_n data of the random Voronoi froth. The exact large-n expansion of m_n obtained in the present work, namely, , explains this curvature. Its inverse square root dependence on n sets a new theoretical paradigm. Similar curved behaviour may be expected, and must indeed be looked for, in experimental data of sufficiently high resolution. We argue that it occurs, in particular, in diffusion-limited colloidal aggregation on the basis of recent simulation data due to Fernández-Toledano et al (2005 Phys. Rev. E 71 041401) and earlier experimental results by Earnshaw and Robinson (1994 Phys. Rev. Lett.72 3682).

We study a non-Markovian random walk in dimension 1. It depends on two parameters _R and _L, the probabilities to go straight on when walking to the right, and respectively to the left. The position x of the walk after n steps and the number of reversals of direction k are used to estimate _R and _L. We calculate the joint probability distribution p_n(x, k) in a closed form and show that, approximately, it belongs to the exponential family.

The Baxter–Bazhanov–Stroganov model (also known as the τ⁽²⁾ model) has attracted much interest because it provides a tool for solving the integrable chiral -Potts model. It can be formulated as a face spin model or via cyclic L-operators. Using the latter formulation and the Sklyanin–Kharchev–Lebedev approach, we give the explicit derivation of the eigenvectors of the component B_n(λ) of the monodromy matrix for the fully inhomogeneous chain of finite length. For the periodic chain, we obtain the Baxter T-Q-equations via separation of variables. The functional relations for the transfer matrices of the τ⁽²⁾ model guarantee nontrivial solutions to the Baxter equations. For the N = 2 case, which is the free fermion point of a generalized Ising model, the Baxter equations are solved explicitly.

In this paper we construct a model for the simultaneous compaction by which clusters are restructured, and growth of clusters by pairwise coagulation. The model has the form of a multi-component aggregation problem in which the components are cluster mass and cluster diameter. Following suitable approximations, exact explicit solutions are derived which may be useful for the verification of simulations of such systems. Numerical simulations are presented to illustrate typical behaviour and to show the accuracy of approximations made in deriving the model. The solutions are then simplified using asymptotic techniques to show the relevant timescales of the kinetic processes and elucidate the shape of the cluster distribution functions at large times.

A Cauchy problem on the semiline for a nonlinear diffusion equation is considered, with a boundary condition corresponding to a prescribed thermal conductivity at the origin. The problem is mapped into a moving boundary problem for the linear heat equation with a Robin-type boundary condition. Such a problem is then reduced to a linear integral Volterra equation of II type which admits a unique solution.

We propose a new integrable Hamiltonian describing two interacting particles in a harmonic mean field in D = 1 dimensional space. This model is found to be both supersymmetric and shape invariant. We show that for a given domain of the coupling constant the irregular solution is acceptable and contributes to the spectrum. We also discuss two inequalities of the Bertlmann–Martin type, which link the ground state mean-square radius to the lowest excitation energy.

Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super Bäcklund transformation and is further used to obtain N super soliton solutions.

We generalize the construction of a class of type (1, 1) tensor fields R on a tangent bundle which was introduced in a preceding paper. The generalization comes from the fact that, apart from a given Lagrangian, the further data consist of a type (1, 1) tensor J along the tangent bundle projection τ: TQ →Q, rather than a tensor on Q. The main features under investigation are two kinds of recursion properties of R, namely its potential invariance under the flow of the given dynamics and the property of having vanishing Nijenhuis torsion. The theory is applied, in particular, to the case of second-order dynamics coming from a Finsler metric.

Apart from the rectangular barrier, other barriers having a single maximum generally display reflectivity, R(E), as a smoothly decreasing function of energy. We conjecture that symmetric potential barriers with a single maximum entail zeros or sharp minima in R(E) provided they have either their coefficient of kurtosis lying in the range (1.8, 3.0), or their top curvature as zero, or both.

From the early work of Hoffmann–Ostenhof, the electron density ρ(r) for He-like ions at sufficiently large r has the form where I is the ionization potential. A derivation of the relation between n and I is briefly presented. The non-relativistic ionization potential I is then calculated quantitatively from diffusion quantum Monte Carlo (DMC) calculations, as is the range of validity of the above asymptotic form for Z ⩽ 2. The above arguments are valid away from Z = Z_c, the critical atomic number 0.911 028 at which the ionization potential I tends to zero. Again analytic theory plus DMC is used at Z = Z_c to quantify ρ(r).

The embedding method (Inglesfield J E 1981 J. Phys. C: Solid State Phys.14 3795) is nowadays widely used for determining properties of bound and continuum eigenstates of the Schrödinger equation. Recently, Crampin (2004 J. Phys.: Condens. Matter16 8875) extended Inglesfield's formalism to systems described by the Dirac equation. In this paper, we reformulate the embedding method for bound states of Dirac particles in the language of the theory of surface integral operators, which are natural analogues of the Dirichlet-to-Neumann (DtN) and Neumann-to-Dirichlet (NtD) operators used in the non-relativistic theory. A method of constructing kernels of these Dirac counterparts of the DtN and NtD maps from solutions to an analogue of the Steklov (Stekloff) eigenproblem for an exterior Dirac problem is presented. A numerical example illustrating the utility of the DtN/NtD embedding method is provided.

Generalizing a method used previously for three-body, four-body and very recently for five-body systems, we derive a lower bound for the ground state energy of an N-body Hamiltonian, with arbitrary N. Because the expression of the lower bound obtained in this way depends on a number of parameters, we obtain in fact a family of lower bounds, a lower bound for each set of values of these parameters. The best of these is of course obtained by maximizing over these parameters and is correspondingly named optimized lower bound. The set of values of the parameters corresponding to the optimized lower bound satisfy a number of relations, named universal dynamical constraints, which result from the application of a dynamical principle and are independent of the particular form of the potential. For N = 3, 4, 5, they can be worked out in the most general case. For N = 6 up, they can be worked out only for particular mass configurations. Furthermore, the optimized lower bound proves to be saturated in the harmonic oscillator case.

We study the properties of 'semi-coherent' quantum states introduced by Mathews and Eswaran in 1973, establishing relations between them and other families of 'nonclassical' states, such as displaced number states and 'multiphoton' states. We also analyse the photon number distribution, photon statistics, the Wigner and marginal distribution functions, and show the ways for possible further generalizations.

In this paper we first derive a general method which enables one to create expressions for vortices and monopoles. By using this method we construct several order-parameters describing the vortices and monopoles of Bose–Einstein condensates with hyperfine spin F = 1 and F = 2. We concentrate on defects which are topologically stable in the absence of an external magnetic field. In particular we show that in a ferromagnetic condensate there can be a vortex which does produce any superfluid flow. We also point out that the order-parameter space of the cyclic phase of F = 2 condensate consists of two disconnected sets. Finally we examine the effect of an external magnetic field on the vortices of a ferromagnetic F = 1 condensate and discuss the experimental preparation of a vortex in this system.

The assumptions required for the derivation of Bell inequalities are not satisfied for random field models in which there are any thermal or quantum fluctuations, in contrast to the general satisfaction of the assumptions for classical two point particle models. Classical random field models that explicitly include the effects of quantum fluctuations on measurement are possible for experiments that violate Bell inequalities.

This paper presents a study of the free energy and particle density of the relativistic Landau problem, and their relevance to the quantum Hall effect. First we study the zero-temperature Casimir energy and fermion number for Dirac fields in a (2+1)-dimensional Minkowski spacetime, in the presence of a uniform magnetic field perpendicular to the spatial manifold. Then, we go to the finite-temperature problem, with a chemical potential, introduced as a uniform zero component of the gauge potential. By performing a Lorentz boost, we obtain Hall's conductivity in the case of crossed electric and magnetic fields.

The introduction of defects is discussed under the Lagrangian formalism and Backlund transformations for the N = 1 super sinh-Gordon model. Modified conserved momentum and energy are constructed for this case. Some explicit examples of different Backlund soliton solutions are discussed. The Lax formulation within the space split by the defect leads to the integrability of the model and henceforth to the existence of an infinite number of constants of motion.

By averaging over an ensemble of field configurations, a classical field theory can display many of the characteristics of quantum field theory, including Lorentz invariance, a loop expansion and renormalization effects. There is additional freedom in how the ensemble is chosen. When the field mode amplitudes have a Gaussian distribution, and the mode phases are randomly distributed, we review the known differences between the classical and quantum field theories. When the mode amplitudes are fixed, or have a non-Gaussian distribution, the quartic and higher correlations among the free fields are modified, seemingly in a nonlocal way. We show how this in turn affects the perturbative expansion. We focus on λϕ⁴ theory in 1 + 1 dimensions and use lattice simulations to augment our study. We give examples of how these nonlocal correlations induce the behaviour more similar to a quantum field theory, at both weak and strong couplings.

The pseudorelativistic no-pair Jansen–Hess operator is derived for the case where in addition to the Coulomb potential an external magnetic field B is permitted. With some restrictions on the vector potential, it is shown that this operator is positive provided the strength γ of the Coulomb potential is below a critical value (γ_c ⩽ 0.35, depending on the magnetic field energy E_f). Moreover, for γ < 0.32 and for B tending asymptotically to zero in a weak sense, the essential spectrum is given by [m, ∞) + E_f.

A Lorentz invariant formalism for the quasi-classical description of electromagnetic radiation from a neutral spin particle with an anomalous magnetic moment moving in an external electromagnetic field is developed. In the high symmetry fields, for which analytical solutions to the Bargmann–Michel–Telegdi equation are known, the so-called self-polarization axes, i.e. directions of preferred polarization of particles in the radiation process, are found. Expressions for the radiative transition probability and spectral-angular distribution of the radiation emitted by a polarized particle are obtained in the fields under consideration.

An exact analytic solution is obtained for a uniformly expanding, neutral, infinitely conducting plasma sphere in an external dipole magnetic field. The electrodynamical aspects related to the radiation and transformation of energy were considered as well. The results obtained can be used in analysing the recent experimental and simulation data.

An operator method, which was developed earlier for bianisotropic layered media with plane and cylindrical interfaces, is generalized to solve spherically symmetric problems: determination of eigenmodes of spherical waveguides, investigation of surface waves, calculation of light scattering cross-section by multi-layer spherical particles. Analytical results are applied to obtain the modes of the negative-refractive-index waveguide, to determine dispersion curves of the waves at the spherical interface between isotropic and bianisotropic media, to reveal the regularities of electromagnetic wave scattering by spherical dielectric particles.

We study anode-directed ionization fronts in curved geometries. An electric shielding factor determines the behaviour of the electric field and the charged particle densities. From a minimal streamer model, a Burgers type equation which governs the dynamics of the electric shielding factor is obtained when electron diffusion is neglected. A Lagrangian formulation is then derived to analyse the ionization fronts. Power laws for the velocity and the amplitude of streamer fronts are found numerically and calculated analytically by using the shielding factor formulation. The phenomenon of geometrical diffusion is explained and clarified, and a universal self-similar asymptotic behaviour is derived.

We give a description of the ideal plasma flow, which is governed by an exact partially invariant solution of the magnetohydrodynamics equations. The solution generalizes known one-dimensional flow with spherical waves. The generalization consists in addition of the special tangent vector components of the velocity and the magnetic field at any plasma particle. In the special case of zeroth tangential component the solution coincides with the classical one-dimensional one. This paper describes a three-dimensional picture of the plasma flow, governed by the obtained solution.

We study the influence of an equation-of-state (EOS) model on the interpretation of electrical conductivity measurements in strongly coupled plasma of tungsten by Korobenko et al (2002 Plasma Phys. Rep.28 1008–16). Three different semiempirical EOS models for tungsten are used. Discrepancies in obtaining thermodynamic parameters and specific resistivity values as compared with calculation results of Korobenko et al are analysed.

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