Rémy Mosseri

Entanglement in a second-order quantum phase transition

We consider a system of mutually interacting spins 1/2 embedded in a transverse magnetic field which undergoes a second-order quantum phase transition. We analyze the entanglement properties and the spin squeezing of the ground state and show that, contrarily to the one-dimensional case, a cusplike singularity appears at the critical point {lambda}{sub c} in the thermodynamical limit. We also show that there exists a value {lambda}{sub 0}{>=}{lambda}{sub c} above which the ground state is not spin squeezed despite a nonvanishing concurrence.

Geometry of entangled states, Bloch spheres and Hopf fibrations

We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of highdimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S 3 .T heS 2 base space of a suitably oriented S 3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S 7 , which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. The most striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.

Entanglement in a first-order quantum phase transition

We analyze the entanglement properties of the ground state for a system of spins half embedded in a magnetic field, mutually interacting antiferromagnetically. Contrary to the ferromagnetic case where a second-order quantum phase transition occurs, a first-order transition is obtained at zero field and the so-called concurrence which measures the two-spin entanglement displays a jump at the transition point.

CONFIGURATIONAL ENTROPY IN OCTAGONAL TILING MODELS

We calculate the configurational entropy of random tilings obtained by elementary flips from a perfect octagonal tiling with an octagonal boundary. We map the problem of generating all configurations onto a partition problem. We calculate numerically the number of configurations and the associated entropy. We give some exact expressions in restricted cases and upper bounds for the entropy in the asymptotic case.

Modelling of the structure of glasses

Abstract Order and disorder are the two main concepts discussed in this paper. We emphasize on the topological order and disorder. Defects appear by comparison to a non-frustrated order defined in a curved space.

The E8 lattice and quasicrystals

Abstract In this paper we study the quasiperiodic structures that can be derived from the 8-dimensional lattice E8. We develop a modified version of the cut-and-projection algorithm. The high-dimensional lattice is first foliated into successive shells surrounding a vertex. The shells are embedded in 7-dimensional S 7 sphers. We then use the so-called Hopf fibration of S 7 to gather families of E8 shell sites into S 3 fibres containing 24 sites (or multiples of 24) symmetrically disposed in S 3 . This selection process has two advantages: first, a whole fibre is either selected or rejected, which reduces the computation; more important, the fibre selection process is based on simple arithmetical criteria. The process leads to a shell-by-shell analysis of a 4D quasicrystal with {3, 3, 5} asymmetry, which can be brought onto a 2D-1D algorithm similar to the Fibonacci chain construction. We propose an arithmetic formula which gives the number of points on the successive shells. This 4D quasicrystal has the interesting feature that it can be sliced into lower-dimensional quasicrystals, for example with icosahedral and tetrahedral symmetry in 3 dimensions.

Orgons andbiolons in theoretical biology: Phenomenological analysis and quantum analogies

In this paper we define two types of formal biological entities corresponding to biological levels of organization, thebiolons and theorgons, the properties of which are phenomenologically analyzed and discussed.We examine then, in a rather speculative manner, how some characteristics of these entities may suggest analogies between properties of biological systems and some special features of quantum systems.These analogies are principally related to the specific roles played by these entities (relatively to matter-energy, for orgons, and to information, for biolons) in a biological system. They are funded on the formal equivalence between the temporal variations associated to the development of the orgons and the biolons, respectively, and the statistical distribution over the available energy levels of the two main types of quantum entities, the fermions and the bosons (the former being associated to the constitution of matter and the latter to the effects of interactions).This formal comparison leads us to put into correspondences the developmental duration in biological systems with the energetic structuration in quantum ones and the related characteristic times of the former with the temperature of the latter. We discuss briefly these correspondences.

Visible points in a lattice

The geometrical structure associated with visible points in a lattice is investigated. It appears to be the Mobius transform of the original lattice. The related symmetry group and structure factor are explicitly derived.

Configurational entropy in random tiling models

Abstract The configurational entropy of some random tiling models is calculated. We mainly focus on co-dimension one rhombohedral tilings for which a general approximate formula is given for the number of configurations as a function of boundary conditions. The formula gives numbers remarkably close to (numerically derived) exact values and is believed to be valid in the thermodynamic limit. The link with a related statistical mechanics problem is pointed out.

Generalized quasiperiodic Rauzy tilings

We present a geometrical description of new canonical d-dimensional codimension one quasiperiodic tilings based on generalized Fibonacci sequences. These tilings are made up of rhombi in 2d and rhombohedra in 3d as the usual Penrose and icosahedral tilings. Thanks to a natural indexing of the sites according to their local environment, we easily write down, for any approximant, the sites coordinates, the connectivity matrix and we compute the structure factor.