10 research outputs found
Mathematical diffraction of aperiodic structures
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details
Phase-Space Networks of Geometrical Frustrated Systems
Geometric frustration leads to complex phases of matter with exotic
properties. Antiferromagnets on triangular lattices and square ice are two
simple models of geometrical frustration. We map their highly degenerated
ground-state phase spaces as discrete networks such that network analysis tools
can be introduced to phase-space studies. The resulting phase spaces establish
a novel class of complex networks with Gaussian spectral densities. Although
phase-space networks are heterogeneously connected, the systems are still
ergodic except under periodic boundary conditions. We elucidate the boundary
effects by mapping the two models as stacks of cubes and spheres in higher
dimensions. Sphere stacking in various containers, i.e. square ice under
various boundary conditions, reveals challenging combinatorial questions. This
network approach can be generalized to phase spaces of some other complex
systems
Weighted Dirac combs with pure point diffraction
A class of translation bounded complex measures, which have the form of
weighted Dirac combs, on locally compact Abelian groups is investigated. Given
such a Dirac comb, we are interested in its diffraction spectrum which emerges
as the Fourier transform of the autocorrelation measure. We present a
sufficient set of conditions to ensure that the diffraction measure is a pure
point measure. Simultaneously, we establish a natural link to the theory of the
cut and project formalism and to the theory of almost periodic measures. Our
conditions are general enough to cover the known theory of model sets, but also
to include examples such as the visible lattice points.Comment: 44 pages; several corrections and improvement
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
Frustrated two dimensional quantum magnets
We overview physical effects of exchange frustration and quantum spin
fluctuations in (quasi-) two dimensional (2D) quantum magnets () with
square, rectangular and triangular structure. Our discussion is based on the
- type frustrated exchange model and its generalizations. These
models are closely related and allow to tune between different phases,
magnetically ordered as well as more exotic nonmagnetic quantum phases by
changing only one or two control parameters. We survey ground state properties
like magnetization, saturation fields, ordered moment and structure factor in
the full phase diagram as obtained from numerical exact diagonalization
computations and analytical linear spin wave theory. We also review finite
temperature properties like susceptibility, specific heat and magnetocaloric
effect using the finite temperature Lanczos method. This method is powerful to
determine the exchange parameters and g-factors from experimental results. We
focus mostly on the observable physical frustration effects in magnetic phases
where plenty of quasi-2D material examples exist to identify the influence of
quantum fluctuations on magnetism.Comment: 78 pages, 54 figure
Topology of Arrangements and Representation Stability
The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research