Table of contents

We consider a generalization of the contact process stochastic model, including an additional autocatalitic process. The phase diagram of this model in the proper 2-parameter space displays a line of transitions between an active and an absorbing phase which starts at the critical point of the contact process and ends at the transition point of the voter model. Thus, a crossover between the directed percolation and the compact percolation universality classes is observed at this latter point. We study this crossover by a variety of techniques. Using supercritical series expansions analysed with partial differential approximants, we obtain precise estimates of the crossover behaviour of the model. In particular, we find an estimate for the crossover exponent ϕ = 2.00 ± 0.02. We also show arguments that support the conjecture ϕ = 2.

The persistence exponents associated with the T = 0 quenching dynamics of the two-dimensional XY model and a two-dimensional uniaxial spin nematic model have been evaluated using a numerical simulation. The site persistence or the probability that the sign of a local spin component does not change starting from initial time t = 0 up to a certain time t, is found to decay as L(t)^−θ(L(t) is the linear domain length scale), with θ = 0.305(±0.020) for the two-dimensional XY model and 0.199(±0.009) for the two-dimensional uniaxial spin nematic model. We have also investigated the scaling (at the late time of phase ordering) associated with the correlated persistent sites in both models. The persistence correlation length was found to grow in the same way as L(t).

We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ_2nd/L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β_KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.

The two-dimensional RSOS model of adsorption on a chemically inhomogeneous and periodic substrate with point-like interaction between the interface and the substrate is discussed rigorously. We prove that for weakly inhomogeneous substrates critical wetting transitions exist. Their wetting temperatures are higher than in the case of a homogeneous substrate whose interaction parameter is equal to the spatial average of interaction parameters in the inhomogeneous cases.

We investigate a class of quantum symmetries of the perturbed cat map which exist only for a subset of possible values of Planck's constant. The effect of these symmetries is to change the spectral statistics along this positive-density subset. The symmetries are shown to be related to some simple classical symmetries of the map.

The asymptotic behaviour of a nonlinear oscillator subject to a multiplicative Ornstein–Uhlenbeck noise is investigated. When the dynamics is expressed in terms of energy–angle coordinates, it is observed that the angle is a fast variable as compared to the energy. Thus, an effective stochastic dynamics for the energy can be derived if the angular variable is averaged out. However, the standard elimination procedure, performed earlier for a Gaussian white noise, fails when the noise is coloured because of correlations between the noise and the fast angular variable. We develop here a specific averaging scheme that retains these correlations. This allows us to calculate the probability distribution function (PDF) of the system and to derive the behaviour of physical observables in the long time limit.

We consider a fractional generalization of Hamiltonian and gradient systems. We use differential forms and exterior derivatives of fractional orders. We derive fractional generalization of Helmholtz conditions for phase space. Examples of fractional gradient and Hamiltonian systems are considered. The stationary states for these systems are derived.

We investigate the low temperature behaviour of a two-parameter generalized bosonic quantum gas with -symmetry, where q₁ and q₂ are real independent deformation parameters. We calculate, in the thermodynamical limit, several statistical and thermodynamical functions of the system through an -invariant bosonic Hamiltonian. In the low and high temperature limits, the specific heat of the system is obtained in terms of some functions of the deformation parameters q₁ and q₂. At the critical temperature being higher than that of the free boson gas, the specific heat of the two-parameter generalized boson gas exhibits a λ-point transition behaviour. We also discuss the conditions under which the Bose–Einstein condensation would occur in the present two-parameter generalized boson model. However, the free boson gas results can be recovered in the limit q₁ = q₂ = 1.

In the first part we calculate the boundary susceptibility χ_B in the open XXZ-chain at zero temperature and arbitrary magnetic field h by the Bethe ansatz. We present analytical results for the leading terms when |h| ≪ α, where α is a known scale, and a numerical solution for the entire range of fields. In the second part we calculate susceptibility profiles near the boundary at finite temperature T numerically by using the density-matrix renormalization group for transfer matrices and analytically for T ≪ 1 by field theoretical methods. Finally, we compare χ_B at finite temperature with a low-temperature asymptotics which we obtain by combining our Bethe ansatz result with recent predictions from bosonization.

Similarly to quantum states, also quantum measurements can be 'mixed', corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable, i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points—the analogous of 'pure states' in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, 'informationally complete' measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: one class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like e.g. the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this paper we discuss and develop analytic methods for solving time-dependent tunnelling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunnelling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunnelling phenomena in Bose–Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions, which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pⁿ where p is a prime number. With this limitation, it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p² points. This is in contrast to the more usual choice of a two-dimensional phase space containing p²ⁿ points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p = 2) and our technique fails to establish the separability property for more than two qubits. Finally, we give a brief discussion of Hamiltonian dynamics in the language developed in the paper.

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory-based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well-defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.

We derive an upper bound on the action of a direct product of two quantum maps (channels) acting on bi-partite quantum states. We assume that the individual channels Λ_j affect single-particle states so that for an arbitrary input ρ_j, the distance D_j(Λ_j[ρ_j], ρ_j) between the input ρ_j and the output Λ_j[ρ_j] of the channel is less than . Given this assumption we show that for an arbitrary separable two-partite state ρ₁₂, the distance between the input ρ₁₂ and the output Λ₁ ⊗ Λ₂[ρ₁₂] fulfils the bound where d₁ and d₂ are the dimensions of the first and second quantum system respectively. In contrast, entangled states are transformed in such a way that the bound on the action of the local channels is , where d is the dimension of the smaller of the two quantum systems passing through the channels. Our results show that the fundamental distinction between the set of separable and the set of entangled states results in two different bounds which in turn can be exploited for discrimination between the two sets of states. We generalize our results to multi-partite channels.

Usually, in supersymmetric theories, it is assumed that the time evolution of states is determined by the Hamiltonian, through the Schrödinger equation. Here we explore the superevolution of states in superspace, in which the supercharges are the principal operators. The superevolution equation is consistent with the Schrödinger equation, but it avoids the usual degeneracy between bosonic and fermionic states. We discuss superevolution in supersymmetric quantum mechanics and in a simple supersymmetric field theory.

The maps of magnitudes of the axial crystal-field parameters, B_k0, for k = 2, 4, 6, which enter the parametrizations as functions of the z-axis spherical coordinates of the relevant reference frames are identical for all equivalent parametrizations with accuracy to the definite rotations of these frames. Therefore, it is possible to reduce all tested parametrizations to the one common reference frame providing that one can find for all these parametrizations the same distinguished space direction. This condition is fulfilled by the three z-axes of the frames for which the axial parameter, B_k0 (where k = 2, 4, 6 corresponds to the component multipoles), reaches its maximal value max B_k0 ⩽ [∑_m|B_km|²]^1/2. These maxima can serve as convenient discriminants of the entire classes of equivalent parametrizations. Based on the distinguished directions and transformational properties of parametrizations with respect to the reference frame rotations, the paper presents the method how to effectively verify the equivalence of these parametrizations and postulates the way of their standardization. This method can be applied to all point symmetries of the central ion, although it seems to be particularly useful and recommended for triclinic symmetry (C₁, C_i).