The canonical cell tiling is a geometrical framework that uses four kinds of basic polyhedra, called the canonical cells, to model the packing of atoms and clusters in icosahedral quasicrystals and related periodic approximants. Over the past three decades, it has become increasingly clear that this framework is the most sensible approach to describe related structures, albeit technically much less tractable than the Ammann-Kramer-Neri tiling, which is the simplest icosahedral tiling geometry based on the two Ammann rhombohedra. Geometrical arrangements of cells pose a number of combinatorial problems that cannot be handled using simple linear algebra, making it infeasible to determine structures using the standard six-dimensional scheme. This up-to-date review begins with the motivation, definition, and mathematical facts about the canonical cell tiling. Then the reader is taken through the zoo of concrete structures, from smaller periodic approximants to larger ones, along with an overview of the techniques and heuristics used to study them. The recent discovery of a quasiperiodic canonical cell tiling is also briefly illustrated. The latter half of this review surveys the atomistic modeling of real atomic structures in all the three existing structural families based on the decoration concept of the canonical cell tiling. image

We consider the potential for novel forms of magnetism arising from the subtle interplay between electrons and spins in the under-screened kagome Kondo lattice model. At weak coupling, we show that incommensurate non-coplanar multi-wave vector magnetic orders arise at nearly all fillings and that this results from Fermi surface effects that introduces competing interactions between the spins. At strong coupling, we find that such complex order survives near half filling despite the presence of ferromagnetism at all other fillings. We show this arises due to state selection among a massive degeneracy of states at infinite coupling. Finally, we show that at intermediate filling, only commensurate orders seem to survive. But these orders still include non-coplanar magnetism. So, the mere presence of both local moments and itinerant electrons enables complex orders to form unlike any currently observed in kagome materials.

We model the remarkable thin-film Ba-Ti-O structures formed by heat treatment of an initial perovskite BaTiO3 thin film on a Pt(111) surface. All structures contain a rumpled Ti-O network with all Ti threefold coordinated with O, and with Ba occupying the larger. mainly Ti7O7, pores. The quasicrystal structue is a simple decoration of three types of tiles: square, triangle and 30° rhombus, with edge lengths 6.85 Å, joined edge-to-edge in a quasicrystalline pattern; observed periodic crystals in ultrathin film Ba-Ti-O are built from these and other tiles. Simulated STM images reproduce the patterns seen experimentally, and identify the bright protrusions as Ba atoms. The models are consistent with all experimental observations.

Using an adaptation of Schwinger boson mean field theory (SBMFT) for nonuniform systems, we study the nature of low-energy spin excitations on the square and Bethe lattice at their percolation threshold. The optimal SBMFT parameters are interpreted as on-site potentials and pairing amplitudes, which enables an explanation of why emergent local moments develop in this system on dilution [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); H.J. Changlani et al., ibid 111, 157201 (2013)] and why the corresponding single particle frequencies are driven to anomalously low values. We discuss how our mean field calculations suggest the strong link between the presence of sublattice imbalance and long range antiferromagnetic order, and why linear spin wave theory is inadequate for capturing this relation. Within the SBMFT framework, we also extract an energy scale for the interaction between emergent moments, which show qualitative agreement with many-body calculations.

Experimental studies show that plant root morphologies can vary widely from straight gravity-aligned primary roots to fractal-like root architectures. However, the opaqueness of soil makes it difficult to observe how environmental factors modulate these patterns. Here, we combine a transparent hydrogel growth medium with a custom built 3D laser scanner to directly image the morphology of Medicago truncatula primary roots. In our experiments, root growth is obstructed by an inclined plane in the growth medium. As the tilt of this rigid barrier is varied, we find Medicago transitions between randomly directed root coiling, sinusoidal root waving, and normal gravity-aligned morphologies. Although these root phenotypes appear morphologically distinct, our analysis demonstrates the divisions are less well defined, and instead, can be viewed as a 2D biased random walk that seeks the path of steepest decent along the inclined plane. Features of this growth response are remarkably similar to the widely known run-and-tumble chemotactic behavior of Escherichia coli bacteria, where biased random walks are used as optimal strategies for nutrient uptake.

We discover the detailed atomic structure of d-MgZnY, a stable decagonal quasicrystal alloy of the layered Frank-Kasper type, and related phases, using the "tiling and decoration" approach. The atoms have invariable sites in the rectangle and triangle tiles of a 10-fold-symmetric planar tiling. To discover the lowest-energy structures, we combine the methods of density functional theory (DFT) total energy calculations, empirical oscillating pair potentials (fitted to DFT), fitting effective Hamiltonians for tilings, and discovering optimum tiling structures using a nonlocal tile-reshuffling algorithm. We find a family of practically stable compounds with varying composition, including the decagonal quasicrystal and the known Mg_4Zn_7 phase these are more stable than competing icosahedral structures by a small margi

We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the 'quasi-degenerate density matrix' (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit are manifested in finite systems as a set of low-energy 'quasi-degenerate' states (in addition to the ground state). This method, its original form due to Furukawa et al (2006 Phys. Rev. Lett. 96 047211), is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases: (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the 'correlation density matrix', which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.

We investigate the dynamical properties of the classical antiferromagnetic Heisenberg model on the kagome lattice using a combination of Monte Carlo and molecular dynamics simulations. We find that frustration induces a distribution of time scales in the cooperative paramagnetic regime (i.e., far above the onset of coplanarity), as recently reported experimentally in deuterium jarosite. At lower temperature, when the coplanar correlations are well established, we show that the weather-vane loop fluctuations control the system relaxation: the time distribution observed at higher temperatures splits into two distinct time scales associated with fluctuations in the plane and out of the plane of coplanarity. The temperature and wave-vector dependencies of these two components are qualitatively consistent with loops diffusing in the entropically dominated free energy landscape. Numerical results are discussed and compared with the O(N) model and recent experiments for both classical and quantum realizations of the kagome magnets.

We derive the shape of the high-energy features due to a weakly coupled boson in cuprate superconductors, as seen experimentally in Bi_2 Sr_2 Ca_1 Cu_2 O_8+x (BSCCO) by Lee et al. [Nature (London) 442, 546 (2006)]. A simplified model is used of d-wave Bogoliubov quasiparticles coupled to Einstein oscillators with a momentum-independent electron-boson coupling and an analytic fitting form is derived, which allows us (a) to extract the boson mode's frequency and (b) to estimate the electron-boson coupling strength. We further calculate the maximum possible superconducting gap due to an Einstein oscillator with the extracted electron-boson coupling strength, which is found to be less than 0.2 times of the observed gap indicating at the observed boson's non-dominant role in the superconductivity's mechanism. The extracted momentum-independent electron-boson coupling parameter (that we show a posteriori to indeed be in the weak-coupling regime) is then to be interpreted as an (band-structure detail dependent weighted) average over the Brillouin zone of the actual momentum-dependent electron-boson coupling in BSCCO.

We present Monte Carlo simulations on a new class of lattice models in which the degrees of freedom are elements of an Abelian or non-Abelian finite symmetry group G, placed on directed edges of a two-dimensional lattice. The plaquette group product is constrained to be the group identity. In contrast to discrete gauge models (but similar to past work on height models), only elements of symmetry-related subsets S∈G are allowed on edges. These models have topological sectors labeled by group products along topologically nontrivial loops. Measurement of relative sector probabilities and the distribution of distance between defect pairs are done to characterize the types of order (topological or quasi-long-range order) exhibited by these models. We present particular models in which fully local non-Abelian constraints lead to global topological liquid properties.

To understand the role of local sublattice imbalance in low-energy spectra of s=1/2 quantum antiferromagnets, we study the s=1/2 quantum nearest neighbor Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform many-body calculations using an implementation of the density matrix renormalization group (DMRG) technique for generic tree graphs. We discover that the bond-centered Cayley tree has a quasidegenerate set of a low-lying tower of states and an "anomalous" singlet-triplet finite-size gap scaling. For understanding the construction of the first excited state from the many-body ground state, we consider a wave function ansatz given by the single-mode approximation, which yields a high overlap with the DMRG wave function. Observing the ground-state entanglement spectrum leads us to a picture of the low-energy degrees of freedom being "giant spins" arising out of sublattice imbalance, which helps us analytically understand the scaling of the finite-size spin gap. The Schwinger-boson mean-field theory has been generalized to nonuniform lattices, and ground states have been found which are spatially inhomogeneous in the mean-field parameters.

We study the spin-1/2 quantum Heisenberg antiferromagnet on a Bethe lattice diluted to the percolation threshold. Dilution creates areas of even or odd sublattice imbalance resulting in "dangling spins" [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); Phys. Rev. B 81, 054417 (2010)]. These collectively act as "emergent" spin-1/2 degrees of freedom and are responsible for the creation of a set of low-lying "quasidegenerate states." Using density matrix renormalization group calculations, we detect the presence and location of these emergent spins. We find an effective Hamiltonian of these emergent spins, with Heisenberg interactions that decay exponentially with the distance between them.

Using pair potentials fitted to an ab--initio database, combined with replica--exchange simulated annealing, we show the complex, quasicrystal-related Al$_{11}$Ir$_4$ compound contains a new version of the "pseudo-Mackay" icosahedral cluster, with non-icosahedral inner Al$_{10}$Ir and/or Al$_9$Ir clusters that exist in various orientations and account for partial occupancies in the reported structure. Two different compositions show first-order transitions to orientationally ordered phases doubling the (cubic) unit cell, which are respectively metallic and insulating.

We consider methods to identify the classical ground state for an exchange-coupled Heisenberg antiferromagnet on a non-Bravais lattice with interactions $J_{ij}$ to several neighbor distances. Here we apply this to the unusual "octahedral" lattice in which spins sit on the edge midpoints of a simple cubic lattice. Our approach is informed by the eigenvectors of $J_{ij}$ with largest eigenvalues. We discovered two families of non-coplanar states: (i) two kinds of commensurate state with cubic symmetry, each having twelve sublattices with spins pointing in (1,1,0) directions in spin space (modulo a global rotation); (ii) varieties of incommensurate conic spiral. The latter family is addressed by projecting the three-dimensional lattice to a one-dimensional chain, with a basis of two (or more) sites per unit cell.

At low temperatures, spin dynamics in ideal spin ice is due mainly to dilute, thermally excited magnetic monopole excitations. I consider how these will affect the dynamics of a nuclear spin (the same theory applies to muon spin resonance if implanted muons do not diffuse). Up to the time scale for nearby monopoles to be rearranged, a stretched-exponential form of the relaxation functions is expected. I work out the expected exponent in that exponential and the formulas for the $T_1$ (longitudinal) and $T_2$ (dephasing) relaxations, as a function of the monopole density. Experimental NMR is incompatible with the predictions and I suggest is due to magnetic impurities.

I consider the microscopic mechanisms by which a particular left-right (L/R) asymmetry is generated at the organism level from the microscopic handedness of cytoskeletal molecules. In light of a fundamental symmetry principle, the typical pattern-formation mechanisms of diffusion plus regulation cannot implement the “right-hand rule”; at the microscopic level, the cell’s cytoskeleton of chiral filaments seems always to be involved, usually in collective states driven by polymerization forces or molecular motors. It seems particularly easy for handedness to emerge in a shear or rotation in the background of an effectively two-dimensional system, such as the cell membrane or a layer of cells, as this requires no pre-existing axis apart from the layer normal. I detail a scenario involving actin/myosin layers in snails and in C. elegans, and also one about the microtubule layer in plant cells. I also survey the other examples that I am aware of, such as the emergence of handedness in neurons, in eukaryote cell motility, and in non-flagellated bacteria.

We perform exact diagonalization studies in d=2 dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor excluding fermions that displays an exact non-relativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between 1/7 and ~1/5. At the lower end of that interval, at filling 1/7, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of ~0.178. At the higher end of the interval, we find zero energy ground states at fillings above 1/5, in contradiction to previous numerical studies and analytical suggestions; these display non-trivial amplitude degeneracies.