Table of contents

We investigate the separability of arbitrary n-dimensional multipartite identical bosonic systems. An explicit relation between the dimension and the separability is presented. In particular, for n = 3, it is shown that the property of PPT (positive partial transpose) and the separability are equivalent for tripartite systems.

We review here some recent work in the field of quantum annealing, alias adiabatic quantum computation. The idea of quantum annealing is to perform optimization by a quantum adiabatic evolution which tracks the ground state of a suitable time-dependent Hamiltonian, where 'ℏ' is slowly switched off. We illustrate several applications of quantum annealing strategies, starting from textbook toy-models—double-well potentials and other one-dimensional examples, with and without disorder. These examples display in a clear way the crucial differences between classical and quantum annealing. We then discuss applications of quantum annealing to challenging hard optimization problems, such as the random Ising model, the travelling salesman problem and Boolean satisfiability problems. The techniques used to implement quantum annealing are either deterministic Schrödinger's evolutions, for the toy models, or path-integral Monte Carlo and Green's function Monte Carlo approaches, for the hard optimization problems. The crucial role played by disorder and the associated non-trivial Landau–Zener tunnelling phenomena is discussed and emphasized.

The subcritical stationary properties of the contact process in one dimension are studied by perturbative series expansions in powers of the creation rate. To avoid entering the absorbing state and thus obtaining a nonzero stationary distribution of particles three small modifications on the rules of the contact process are considered. The series expansions for the average number of particles and the spatial correlation length are analysed by Padé approximants from which the values of the critical creation rate and the critical exponents are obtained.

We consider the model of a strongly correlated system of electrons on a two-dimensional square lattice described by an extended Falicov–Kimball Hamiltonian where the stability of some axial and diagonal striped phases was proved. Introducing a next-nearest-neighbour hopping, small enough not to destroy the striped structure, we examine rigorously how the presence of the next-nearest-neighbour hopping anisotropy reduces the π/2-rotation degeneracy of the diagonal-striped phase. The effect appears to be similar to that in the case of anisotropy of the nearest-neighbour hopping: diagonal stripes are oriented in the direction of the weaker next-nearest-neighbour hopping.

In this work, we numerically study the dynamical evolution and heat transport properties of a system that consists of two time-reversible thermostats connected either by a one-dimensional Fermi–Pasta–Ulam or a Frenkel–Kontorova oscillator lattice, which are representative models of momentum-conserving and nonconserving systems, respectively. The thermostats were described by a chain of variables, Nosé–Hoover chains, which enhances the ergodicity of the thermostats in comparison to the Nosé–Hoover method. The time evolution of both lattices is not significantly altered by the dynamics of the thermostats. The temperature profile and heat flux of the Fermi–Pasta–Ulam model are more sensitive to the dynamics of the extended variables than those corresponding to the Frenkel–Kontorova model. Nevertheless we reproduce the scaling properties of the thermal conductivity with system size obtained by other authors.

We examine partitions and their natural three-dimensional generalizations, plane partitions, as models of vesicles undergoing an inflation–deflation transition. The phase diagrams of these models include a critical point corresponding to an inflation–deflation transition, and exhibits multicritical scaling in the vicinity of a multicritical point located elsewhere on the critical curve. We determine the locations of the multicritical points by analysing the generating functions using analytic and numerical means. In addition, we determine the numerical values of the multicritical scaling exponents associated with the multicritical scaling regimes in these models.

The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su(2) algebra (BCS theory).

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving many states of the quantum baker's map. These transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry.

We investigate the bi-Hamiltonian structure of the waterbag model of dispersionless K (dKP) for the two-component case. One can establish the third- and first-order Hamiltonian operators associated with the waterbag model. Also, the dispersive corrections are discussed.

A previously introduced scheme for describing integrable deformations of algebraic curves is completed. Lenard relations are used to characterize and classify these deformations in terms of hydrodynamic-type systems. A general solution of the compatibility conditions for consistent deformations is given and expressions for the solutions of the corresponding Lenard relations are provided.

A classical problem of free-space Green's function G_0Λ representations of the Helmholtz equation is studied in various quasi-periodic cases, i.e., when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space. Exponentially convergent series for the free-space quasi-periodic G_0Λ and for the expansion coefficients D_L of G_0Λ in the basis of regular (cylindrical in two dimensions and spherical in three dimension (3D)) waves, or lattice sums, are reviewed and new results for the case of a one-dimensional (1D) periodicity in 3D are derived. From a mathematical point of view, a derivation of exponentially convergent representations for Schlömilch series of cylindrical and spherical Hankel functions of any integer order is accomplished. Exponentially convergent series for G_0Λ and lattice sums D_L hold for any value of the Bloch momentum and allow G_0Λ to be efficiently evaluated also in the periodicity plane. The quasi-periodic Green's functions of the Laplace equation are obtained from the corresponding representations of G_0Λ of the Helmholtz equation by taking the limit of the wave vector magnitude going to zero. The derivation of relevant results in the case of a 1D periodicity in 3D highlights the common part which is universally applicable to any of remaining quasi-periodic cases. The results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics, remote sensing of periodic surfaces, periodic gratings, and infinite arrays of resonators coupled to a waveguide, in many contexts of simulating systems of charged particles, in molecular dynamics, for the description of quasi-periodic arrays of point interactions in quantum mechanics, and in various ab initio first-principle multiple-scattering theories for the analysis of diffraction of classical and quantum waves.

In this paper we explore further the connections between convex bodies related to quantum correlation experiments with dichotomic variables and related bodies studied in combinatorial optimization, especially cut polyhedra. Such a relationship was established in Avis et al (2005 J. Phys. A: Math. Gen.38 10971–87) with respect to Bell inequalities. We show that several well-known bodies related to cut polyhedra are equivalent to bodies such as those defined by Tsirelson (1993 Hadronic J. Suppl.8 329–45) to represent hidden deterministic behaviours, quantum behaviours and no-signalling behaviours. Among other things, our results allow a unique representation of these bodies, give a necessary condition for vertices of the no-signalling polytope, and give a method for bounding the quantum violation of Bell inequalities by means of a body that contains the set of quantum behaviours. Optimization over this latter body may be performed efficiently by semidefinite programming. In the second part of the paper we apply these results to the study of classical correlation functions. We provide a complete list of tight inequalities for the two party case with (m, n) dichotomic observables when m = 4, n = 4 and when min{m, n} ⩽ 3, and give a new general family of correlation inequalities.

A generalization of the matrix Jaynes–Cummings model in the rotating wave approximation is proposed by means of the shape-invariant hierarchies of scalar factorized Hamiltonians. A class of Darboux transformations (sometimes called SUSY transformations in this context) suitable for these generalized Jaynes–Cummings models is constructed. Finally one example is worked out using the methods developed.

The properties and structure of the eight-dimensional parameter space of three-state density matrices are examined using the SU(3) generator expansion. Three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method and an appropriate classification of 56 different three-parameter density matrices into ten types, according to the shape of the corresponding region of allowed values of three relevant parameters, is introduced. Positions of the representative points corresponding to the pure states are determined for all ten types.

We consider a massless scalar field in 1+1 dimensions satisfying a Robin boundary condition (BC) at a non-relativistic moving boundary. We derive a Bogoliubov transformation between the input and output bosonic field operators, which allows us to calculate the spectral distribution of created particles. The cases of Dirichlet and Neumann BC may be obtained from our result as limiting cases. These two limits yield the same spectrum, which turns out to be an upper bound for the spectra derived for Robin BC. We show that the particle emission effect can be considerably reduced (with respect to the Dirichlet/Neumann case) by selecting a particular value for the oscillation frequency of the boundary position.

We present a general formulation of the matrix product ansatz for exactly integrable chains on periodic lattices. This new formulation extends the matrix product ansatz present in our previous articles (F C Alcaraz and M J Lazo 2004 J. Phys. A: Math. Gen.37 L1–L7 and F C Alcaraz and M J Lazo 2004 J. Phys. A: Math. Gen.37 4149–82).