Quasiperiodic Heisenberg antiferromagnets in two dimensions

This is a review of the properties of 2d quantum quasiperiodic antiferromagnets as reported in studies that have been carried out in the last decade. Many results have been obtained for perfectly ordered as well as for disordered two dimensional bipartite quasiperiodic tilings. The theoretical methods used include spin wave theory, and renormalization group along with Quantum Monte Carlo simulations. These methods all show that the ground state of these unfrustrated antiferromagnets have N\'eel type order but with a highly complex spatial distribution of local staggered magnetization. The ground state properties, excitation energies and spatial dependence, structure factor, and local susceptibilities are presented. The effects of introducing geometrical disorder on the magnetic properties are discussed.Comment: 21 pages, 29 figure

Open theoretical problems in the physics of aperiodic systems

Quasicrystals have intrigued and stimulated research in a large number of disciplines. Mathematicians, physicists, chemists, metallurgists and materials scientists have found in them a fertile ground for new insights and discoveries. In the quarter century that has ensued since the publication of the experimental observation of a quasiperiodic Al-Mn alloy \cite{shecht}, many different kinds of quasiperiodic alloys have been manufactured and studied. The physical properties of quasicrystals are no less interesting than the unusual structural properties that led to their discovery in 1984. In this review, I present some of the properties that characterize quasicrystals, briefly discuss several types of theories that have been put forward, and describe some new behaviors that might be investigated by experiment.Comment: 6 pages, 5 figures, plenary lecture for CMAC (Complex metallic alloys) workshop (Zagreb, 2009

The eight-fold way for optical quasicrystals

In a recent Letter we proposed a means to realize a quasicrystal with eight-fold symmetry by trapping particles in an optical potential created by four lasers. The quasicrystals obtained in this way, which are closely related to the well-known octagonal tiling, offer unique possibilities to study the effects of quasiperiodicity on physical properties. This method allows to transform the structures, to inflate or deflate them, include interactions or disorder and thus realize a large variety of theoretical models, both classical and quantum. In this paper we derive a number of interesting geometrical properties of the optical quasicrystals, and present some results obtained by numerical calculations.Comment: Paper following short report published in Europhys.Lett. vol.104 p. 66003 (2013

Geometry fluctuations and Casimir effect in a quantum antiferromagnet

We show the presence of a Casimir type force between domain walls in a two dimensional Heisenberg antiferromagnet subject to geometrical fluctuations. The type of fluctuations that we consider, called phason flips, are well known in quasicrystals, but less so in periodic structures. As the classical ground state energy of the antiferromagnet is unaffected by this type of fluctuation, energy changes are purely of quantum origin. We calculate the effective interaction between two parallel domain walls, defining a slab of thickness d, in such an antiferromagnet within linear spin wave theory. The interaction is anisotropic, and for a particular orientation of the slab we find that it decays as 1/d, thus, more slowly than the electromagnetic Casimir effect in the same geometry.Comment: 5 pages, 5 figures, minor modifications, accepted for publication in EPJ

Tight-binding models in a quasiperiodic optical lattice

This paper describes how one can use four standing wave laser fields to realize a two dimensional optical quasicrystal with eight-fold symmetry, closely related to the well-known octagonal or Ammann-Beenker tiling quasicrystal. We describe the structure and its properties, and the effective tight-binding model for atoms in this optical quasicrystal. Such a system, if realized experimentally, should provide valuable insights into the quantum properties of quasicrystals.Comment: revised version (to appear in the Proceedings of the International Conference on Quasicrystals ICQ12 Kracow, Poland

Closing of gaps and gap labeling and passage from molecular states to critical states in a 2D quasicrystal

The single electron spectrum and wavefunctions in quasicrystals continue to be a fascinating problem, with few known solutions, especially in two and higher dimensions. This paper investigates the energy spectra and gap structures in tight-binding models on a quasiperiodic tiling in two dimensions. By varying a continuous parameter, we follow the evolution of the band structure from the discrete molecular or atomic states, to the multifractal states well-known from previous studies. We propose a scheme for labeling gaps in finite approximants. It is equivalent in the limit of infinite systems to the one presented by Kellendonk and Putnam based on the algebraic structure of this quasiperiodic system