The Local Structure of Tilings and Their Integer Group of Coinvariants

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the K ₀-group of the groupoid C ^*-algebra for tilings which reduce to decorations of . The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and v-primitive. This yields in particular the set of possible gap labels predicted by K-theory for Schr?dinger operators describing the particle motion in such a tiling. Received: 22 September 1995 / Accepted: 2 December 1996     

PV Cohomology of the Pinwheel Tilings, their Integer Group of Coinvariants and Gap-Labeling

In this paper, we first remind how we can see the “hull” of the pinwheel tiling as an inverse limit of simplicial complexes (Anderson and Putnam in Ergod Th Dynam Sys 18:509–537, 1998) and we then adapt the PV cohomology introduced in Savinien and Bellissard (Ergod Th Dynam Sys 29:997–1031, 2009) to define it for pinwheel tilings. We then prove that this cohomology is isomorphic to the integer Čech cohomology of the quotient of the hull by S ¹ which let us prove that the top integer Čech cohomology of the hull is in fact the integer group of coinvariants of the canonical transversal Ξ of the hull. The gap-labeling for pinwheel tilings is then proved and we end this article by an explicit computation of this gap-labeling, showing that m^t ( C(X,\mathbb Z) ) = \frac1264\mathbb Z [ \frac15]{\mu^t \left( C(\Xi,\mathbb {Z}) \right) = \frac{1}{264}\mathbb {Z} \left [ \frac{1}{5}\right ]}.     

Ground states of two-dimensional quasicrystals

Hamiltonians for nonperiodic tilings are considered. It is shown that the quasicrystalline tiling obtained by the cut-and-strip method from aD-dimensional cubic lattice may bs a ground state only if the tiling possesses a high orientational symmetry: the (2,D)-quasicrystal should haveD-fold symmetry ifD is even and 2D-fold symmetry ifD is odd. For interactions of a finite range the restrictions are stronger: only a (2, 5)-quasicrystal (Penrose tiling) may be a stable ground state.     

Spaces of Tilings, Finite Telescopic Approximations and Gap-Labeling

The continuous Hull of a repetitive tiling T in ℝ^d with the Finite Pattern Condition (FPC) inherits a minimal ℝ^d-lamination structure with flat leaves and a transversal which is a Cantor set. This class of tiling includes the Penrose & the Amman Benkker ones in 2D, as well as the icosahedral tilings in 3D. We show that the continuous Hull, with its canonical ℝ^d-action, can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. As a consequence, the longitudinal cohomology and the K-theory of the corresponding C*-algebra are obtained as direct limits of cohomology and K-theory of ordinary manifolds. Moreover, the space of invariant finite positive measures can be identified with a cone in the d^th homology group canonically associated with the orientation of ℝ^d. At last, the gap labeling theorem holds: given an invariant ergodic probability measure μ on the Hull the corresponding Integrated Density of States (IDS) of any selfadjoint operators affiliated to takes on values on spectral gaps in the ℤ-module generated by the occurrence probabilities of finite patches in the tiling. Accepted in Revised Form: 7 May 2005     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Archimedean-like colloidal tilings on substrates with decagonal and tetradecagonal symmetry

Two-dimensional colloidal suspensions subjected to laser interference patterns with decagonal symmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodically along one direction (J. Mikhael et al., Nature 454, 501 (2008)). In experiments as well as in Monte Carlo and Brownian dynamics simulations, we identify a similar phase when the laser field possesses tetradecagonal symmetry. We characterize the structure of both Archimedean-like tilings in detail and point out how the tilings differ from each other. Furthermore, we also estimate specific particle densities where the Archimedean-like tiling phases occur. Finally, using Brownian dynamics simulations we demonstrate how phasonic distortions of the decagonal laser field influence the Archimedean-like tiling. In particular, the domain size of the tiling can be enlarged by phasonic drifts and constant gradients in the phasonic displacement. We demonstrate that the latter occurs when the interfering laser beams are not ideally adjusted.     

The structure of an Al–Rh–Cu decagonal quasicrystal studied by spherical aberration (Cs)-corrected scanning transmission electron microscopy

The structure of an Al–Rh–Cu decagonal quasicrystal formed with two quasiperiodic planes along the periodic axis in an Al₆₃Rh_18.5Cu_18.5 alloy has been studied by spherical aberration (Cs)-corrected high-angle annular detector dark-field (HAADF)- and annular bright-field (ABF)-scanning transmission electron microscopy (STEM). Heavy atoms of Rh and mixed sites (MSs) of Al and Cu atoms projected along the periodic axis can be clearly represented as separate bright dots in observed HAADF-STEM images, and consequently arrangements of Rh atoms and MSs on the two quasiperiodic planes can be directly determined from those of bright dots in the observed HAADF-STEM image. The Rh atoms are arranged in pentagonal tiling formed with pentagonal and star-shaped pentagonal tiles with an edge-length of 0.76 nm, and also MSs with a pentagonal arrangement are located in the pentagonal tiles with definite orientations. The star-shaped pentagonal tiles in the pentagonal tiling are arranged in τ²(τ: golden ratio)-inflated pentagonal tiling with a bond-length of 2 nm. From arrangements of Rh atoms placed in pentagonal tilings with a bond-length of 2 nm, which are generated by the projection of a five-dimensional hyper-cubic lattice, occupation domains in the perpendicular space are derived. Al atoms as well as Rh atoms and MSs are represented as dark dots in an observed ABF-STEM image, and arrangements of Al atoms in well-symmetric regions are discussed.     

A remark on Schrödinger operators on aperiodic tilings

We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl ²( ^d ) to operators on thel ²-spaces of sets of vertices of strictly ergodic tilings.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.     

Studies on the cell structure of quasicrystals

The geometric and crystallographic ideas for cell models and tilings in non-periodic ordered structures are outlined. The basic concepts, among them the cell geometry and duality in a lattice are explained for the example of a one dimensional section through the root lattice A ₂. The corresponding constructions for 3D icosahedral sections through the 6D face-centered hypercubic lattice, equivalent to the root lattice D ₆, are described. Two different tilings, one with equivalent, one with three inequivalent vertex positions, are derived and discussed.     

On the stability of Archimedean tilings formed by patchy particles

We have investigated the possibility of decorating, using a bottom-up strategy, patchy particles in such a way that they self-assemble in (two-dimensional) Archimedean tilings. Except for the trihexagonal tiling, we have identified conditions under which this is indeed possible. The more compact tilings, i.e., the elongated triangular and the snub square tilings (which are built up by triangles and squares only) are found to be stable up to intermediate pressure values in the vertex representation, i.e., where the tiling is decorated with particles at its vertices. The other tilings, which are built up by rather large hexagons, octagons and dodecagons, are stable over a relatively large pressure range in the centre representation where the particles occupy the centres of the polygonal units.     

Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A⁵. We also show that this A⁵ root lattice can be related to the F₄⁽¹⁾ root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F₄⁽¹⁾ root lattice.     

Deceptions in quasicrystal growth

We discuss a new general phenomenon pertaining to tiling models of quasicrystal growth. It is known that with Penrose tiles no (deterministic) local matching rules exist which guarantee defect-free tiling for regions of arbitrary large size. We prove that this property holds quite generally: namely, that the emergence of defects in quasicrystal growth is unavoidable for all aperiodic tiling models in the plane with local matching rules, and for many models inR ³ satisfying certain conditions.Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grants 003658113 and 003658007.     

Local rules for quasiperiodic tilings of quadratic 2-planes inR 4

We prove that quasiperiodic tilings of the plane, appearing in the strip projection method always admit local rules, when the linear embedding ofR ² inR ⁴ has quadratic coefficients. These local rules are constructed and studied. The connection between Novikov quasicrystallographic groups and the quasiperiodic tilings of Euclidean space is explained. All the point groups in Novikov's sense, compatible with these local rules, are enlisted. The two-dimensional quasicrystals with infinite-fold rotational symmetry are constructed and studied.Address after September 1, 1992: Dept. of Physics, Harvard University, Cambridge, MA 02138, USA     

A Finite Analog of the AGT Relation I: Finite <Emphasis Type="Italic">W</Emphasis>-Algebras and Quasimaps’ Spaces

Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on \mathbbP²{\mathbb{P}^2} . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.     

Tensor Fields of Mixed Young Symmetry Type¶and N-Complexes

We construct N-complexes of non-completely antisymmetric irreducible tensor fields on ℝ ^D which generalize the usual complex (N=2) of differential forms. Although, for N≥ 3, the generalized cohomology of these N-complexes is nontrivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10]. Received: 25 October 2001 / Accepted: 13 November 2001     

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φ ^t defined on an attracting basic set Λ. A map from the first (Čech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φ ^t . Received: 17 June 1998 / Accepted: 24 February 1999     

String center of mass operator and its effect on BRST cohomology

We consider the theory of bosonic closed strings on the flat background ℝ^25,1. We show how the BRST complex can be extended to a complex where the string center of mass operator,x ₀ ^μ is well defined. We investigate the cohomology of the extended complex. We demonstrate that this cohomology has a number of interesting features. Unlike in the standard BRST cohomology, there is no doubling of physical states in the extended complex. The cohomology of the extended complex is more physical in a number of aspects related to the zero-momentum states. In particular, we show that the ghost number one zero-momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the backgroundi.e., the Poincaré algebra. Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818     

Q-Systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the Q-system for A _r as a tool for relating Q-system solutions to all possible sets of initial data. Considered as a discrete integrable dynamical system, we show that the conserved quantities are partition functions of hard particles on certain weighted graphs determined by the choice of initial data. This allows us to interpret the solutions of the system as partition functions of Viennot’s heaps on these graphs, or as partition functions of weighted paths on dual graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the Q-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the A _r Q-system. We also give the relation to domino tilings of deformed Aztec diamonds with defects.     

Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings

We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to generate three-dimensional Euclidean (E³) networks. The technique involves projection of edges of tilings of the hyperbolic plane (H²) onto three-periodic minimal surfaces, embedded in E³. Given the extraordinary wealth of symmetries commensurate with H², we can generate networks in E³ that are difficult to construct otherwise. In particular, we form four-, five- and seven-connected (E³) nets containing three- and five-rings, viz. (3, 7), (5, 4) and (5, 5) tilings in H². Received 14 January 2002 / Received in final form 12 August 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: stephen.hyde@anu.edu.au