Table of contents

The off-diagonal profile ϕ_od^b(v) associated with a local operator (v) (order parameter or energy density) close to the boundary of a semi-infinite strip with width L is obtained at criticality using conformal methods. It involves the surface exponent x_ϕ^s and displays a simple universal behaviour which crosses over from surface finite-size scaling when v/L is held constant to corner finite-size scaling when v/L→0.

The role of linearity in the definition of entropy is examined. While discussions of entropy often treat extensivity as one of its fundamental properties, the extensivity of entropy is not axiomatic in thermodynamics. It is shown that systems in which entropy is an extensive quantity are systems in which a entropy obeys a generalized principle of linear superposition.

The spherical limit of strongly commensurate dirty bosons is studied perturbatively at weak disorder and numerically at strong disorder in two dimensions (2D). We argue that disorder is not perfectly screened by interactions and consequently that the ground state in the effective Anderson localization problem always remains localized. As a result there is only a gapped Mott insulator phase in the theory. Comparisons with other studies and the parallel with disordered fermions in 2D are discussed. We conjecture that while for the physical cases N = 2 (XY) and N = 1 (Ising) the theory should have the ordered phase, it may not for N = 3 (Heisenberg).

A regular lattice in which the sites can have long-range connections at a distance l with a probabilty P(l) ∼ l ^−δ, in addition to the short-range nearest neighbour connections, shows small-world behaviour for 0 ≤ δ < δ_c. In the most appropriate physical example of such a system, namely, the linear polymer network, the exponent δ is related to the exponents of the corresponding n-vector model in the n → 0 limit, and its value is less than δ_c. Still, the polymer networks do not show small-world behaviour. Here, we show that this is due to a (small value) constraint on the number, q, of long-range connections per monomer in the network. In the general δ-q space, we obtain a phase boundary separating regions with and without small-world behaviour, and show that the polymer network falls marginally in the regular lattice region.

Considering a charged three-dimensional harmonic oscillator coupled to the photon field by the usual coupling constant, we show that a qualitative change in the possible states of the system occurs when a length and an energy, characteristic quantities of the oscillator, satisfy a simple relation. The frequency being fixed, oscillator–photon resonances change to oscillator–photon bound states when the length increases.

In applications of the density matrix renormalization group to nonhermitean problems, the choice of the density matrix is not uniquely prescribed by the algorithm. We demonstrate that for the recently introduced stochastic transfer matrix DMRG (stochastic TMRG) the necessity to use open boundary conditions makes asymmetrical reduced density matrices, as used for renormalization in quantum TMRG, an inappropriate choice. An explicit construction of the largest left and right eigenvectors of the full transfer matrix allows us to show why symmetrical density matrices are the correct physical choice.

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight `polymer', but in addition there are isolated eigenvalues unless N = 2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically.

Generalized principal models on non-semisimple groups are defined. An ansatz for the Lax form of the equations of motion is chosen and models on two- and three-dimensional non-semisimple groups that admit this Lax formulation are classified. Only one of these models has truly nonlinear equations of motion, and the Lax pair is explicitly given. The equations of motion of all the other models can be brought to linear partial differential equations.

Following a conjecture of Berry and Howls (1994) concerning the geometric information contained within the high orders of Weyl series, we examine such series for the average spectral properties of two- and three-dimensional quantum ball billiards threaded by a single flux line at the centre. We adapt a Mellin-based scheme of Bordag et al (1996) to generate the Weyl series. It is shown that for a circular billiard, only a single Weyl series term is changed and thus the flux line only induces a simple constant shift in the average properties of the spectrum, although the fluctuations about this average will still be flux dependent. This implies that the late terms in the expansion are dominated by the diametrical periodic orbit of the unfluxed circle, rather than the shorter diffractive orbits encountering both the billiard boundary and the flux line. For a spherical billiard with flux the late terms suffer modifications which can be linked to diffractive orbits. The origins of the differences between the structure of the series are traced to the interaction of the geometry and symmetry breaking.

The amplitude–phase formulation of the Schrödinger equation is investigated within the context of uncoupled Ermakov systems, whereby the amplitude function is given by the auxiliary non-linear equation. The classical limit of the amplitude and phase functions is analysed by setting up a semi-classical Ermakov system. In this limit, it is shown that classical quantities, such as the classical probability amplitude and the reduced action, are obtained only when the semi-classical amplitude and the accumulated phase are non-oscillating functions respectively of the space and energy variables. Conversely, among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction, only the non-oscillating ones yield classical quantities in the limit ℏ → 0.

The exact equation describing the shape of a fluid drop under the action of local surface stresses induced by colloidal interactions is derived without resorting to any of the approximations inherent in the profile equation currently employed in the literature. The exact equation implies, and numerical examples confirm, that repulsive external (i.e. positive) surface energies assist in stabilizing the drop against deformation, while attractive (i.e. negative) energies destabilize the drop, promoting or enhancing deformation. An inherent singularity in the governing differential equation (absent from the approximate equations currently used) when the surface energy (surface tension) is identically matched by an external attractive energy represents an instability limit. Explicit bounds are established for a further instability criterion and for the hydrostatic pressure difference across the interface. An exact equation for the radial extent of the sessile drop and some numerical examples are also presented.

We discuss an exactly solvable model for the creation of entanglement between two subsystems by the observation of decay products. The system consists of two identical decaying boson modes, and the decay channels are observed through a beam splitter. For a reasonable class of initial states the decay process is completely decoupled from the buildup of the relative phase. Exact expressions are derived for the distribution over the two output channels, and for the conditional density matrix after a given detection history.

The two-parametric quantum superalgebra U_p,q[gl(2/2)] and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and non-typical ones according to a proposition proved in the present paper. This proposition is a non-trivial deformation from the one for the classical superalgebra gl(2/2), unlike the case of one-parametric deformations.

It is known that magnetic fields in ideal chaotic plasmas tend to become extremely irregular and to concentrate in a fractal set, and it is assumed that the presence of a positive resistivity will have a smoothing effect. Here we try to quantify this effect by proving new inequalities which, on the one hand, relate the local and global size of velocity and magnetic field with the gradient of this field, and on the other provide a bound of the area of generalized level surfaces.

We represent quantum phase observables as phase shift covariant normalized positive operator measures. The phase operators are the first moment operators of the phase observables. A phase operator determines the associated phase observable uniquely. We show that the Cahill–Glauber s-ordered phase operators are determined by phase shift covariant generalized operator measures, which are ordinary operator measures whenever Re s < 0 and phase observables when s ≤ −1. The Wigner–Weyl quantized phase operator is not determined by any phase observable. We investigate the classical limit of covariant (generalized) operator measures in coherent states.