Infinite Hexagonal Lattice Research Articles

Directional ballistic transport in the two-dimensional metal PdCoO2

In an idealized infinite crystal, the material properties are constrained by the symmetries of the unit cell. The point-group symmetry is broken by the sample shape of any finite crystal, but this is commonly unobservable in macroscopic metals. To sense the shape-induced symmetry lowering in such metals, long-lived bulk states originating from an anisotropic Fermi surface are needed. Here we show how a strongly facetted Fermi surface and the long quasiparticle mean free path present in microstructures of PdCoO2 yield an in-plane resistivity anisotropy that is forbidden by symmetry on an infinite hexagonal lattice. We fabricate bar-shaped transport devices narrower than the mean free path from single crystals using focused ion beam milling, such that the ballistic charge carriers at low temperatures frequently collide with both of the side walls that define the channel. Two symmetry-forbidden transport signatures appear: the in-plane resistivity anisotropy exceeds a factor of 2, and a transverse voltage appears in zero magnetic field. Using ballistic Monte Carlo simulations and a numerical solution of the Boltzmann equation, we identify the orientation of the narrow channel as the source of symmetry breaking.

Zagreb Connection Indices for Some Benzenoid Systems

A benzenoid is a class of chemical compounds with at least one benzene ring(hexagon as a graph) and resonance bonds in the benzene ring give increased stability in benzenoids. A finite connected subgraph of the infinite hexagonal lattice without cut vertices or non-hexagonal interior faces is said to be a benzenoid system (or a hexagonal system) and these systems are geometric figures. Benzenoid systems are widely used because they are the representations of the skeletons (focussing on the structure induced by the carbon atoms) of molecules of benzenoid hydrocarbons. The first and second Zagreb indices are among the most studied topological indices. We now consider analogous graph invariants, based on the second degrees of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for six benzenoid systems(three catacondensed and three pericondensed systems). In the end, an application of the obtained results for these indices of some classes of benzenoid systems is also included. Mainly, a comparison among the Zagreb connection indices of some benzenoid systems is performed with the help of numerical tables and 3 D plots.

Tensor network simulation of the Kitaev-Heisenberg model at finite temperature

We investigate the Kitaev-Heisenberg (KH) model at finite temperature using the exact environment full update (eeFU), introduced in Phys. Rev. B 99, 035115 (2019), which represents the purification of a thermal density matrix on an infinite hexagonal lattice by an infinite projected entangled pair state (iPEPS). We show that thanks to a dynamical mapping from a hexagonal to a rhombic lattice, the eeFU on the hexagonal lattice is as efficient as the simple full update (FU) algorithm. Critical temperatures for coupling constants in the stripy and the antiferromagnetic phase are estimated. They are an order of magnitude less than the couplings in the Hamiltonian. By a duality transformation, these results can be mapped to, respectively, the ferromagnetic and zigzag phases. For the special case of the pure Kitaev model, which is tractable by quantum Monte-Carlo but the most challenging for tensor networks, the algorithm is benchmarked against the Monte-Carlo results. It recovers accurately the crossover to spin ordering and qualitatively the one to flux ordering.

Lattice spanners of low degree

Let [Formula: see text] denote the degree [Formula: see text] dilation of a point set [Formula: see text] in the domain of plane geometric spanners. If [Formula: see text] is the infinite square lattice, it is shown that [Formula: see text] and [Formula: see text]. If [Formula: see text] is the infinite hexagonal lattice, it is shown that [Formula: see text] and [Formula: see text]. All our constructions are planar lattice tilings constrained to degree [Formula: see text] or [Formula: see text].

The packing chromatic number of infinite product graphs

The packing chromatic number χ ρ ( G ) of a graph G is the smallest integer k such that the vertex set V ( G ) can be partitioned into disjoint classes X 1 , … , X k , where vertices in X i have pairwise distance greater than i . For the Cartesian product of a path and the two-dimensional square lattice it is proved that χ ρ ( P m □ Z 2 ) = ∞ for any m ≥ 2 , thus extending the result χ ρ ( Z 3 ) = ∞ of [A. Finbow, D.F. Rall, On the packing chromatic number of some lattices, Discrete Appl. Math. (submitted for publication) special issue LAGOS’07]. It is also proved that χ ρ ( Z 2 ) ≥ 10 which improves the bound χ ρ ( Z 2 ) ≥ 9 of [W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, J.M. Harris, D.F. Rall, Broadcast chromatic numbers of graphs, Ars Combin. 86 (2008) 33–49]. Moreover, it is shown that χ ρ ( G □ Z ) < ∞ for any finite graph G . The infinite hexagonal lattice H is also considered and it is proved that χ ρ ( H ) ≤ 7 and χ ρ ( P m □ H ) = ∞ for m ≥ 6 .

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into k packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ ρ ( G ) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.

Investigation of continuous-time quantum walk on root lattice [formula omitted] and honeycomb lattice

The continuous-time quantum walk (CTQW) on root lattice A n (known as hexagonal lattice for n = 2 ) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group Z m ⊗ n and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice A n and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice A n and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances, the scaling behavior of the probability distribution is deduced. Also, it is shown that, the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit (stratum of the graph) of the point group of the lattice. Apart from physical results, it is shown that the Bose–Mesner algebras of our constructed association schemes (called n-variable P-polynomial) can be generated by n commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of n-variable orthogonal polynomials which are special cases of generalized Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the supersymmetric structure of finite honeycomb lattices is revealed.

Benzenoids with branching graphs being trees

The branching graph of a benzenoid hydrocarbon molecule is the subgraph consisting of vertices of degree 3 and the bonds among them. In this paper, a practical method is established for recognizing trees embedded in the infinite hexagonal lattice which can be branching graphs of benzenoids. Moreover, a simple criterion is given to determine when a tree embedded in the infinite hexagonal lattice is the branching graph of a catacondensed benzenoid.

Hamiltonian circuits, Hamiltonian paths and branching graphs of benzenoid systems

A benzenoid systemH is a finite connected subgraph of the infinite hexagonal lattice with out cut bonds and non-hexagonal interior faces. The branching graphG ofH consists of all vertices ofH of degree 3 and bonds among them. In this paper, the following results are obtained: (1) A necessary condition for a benzenoid system to have a Hamiltonian circuit. (2) A necessary and sufficient condition for a benzenoid system to have a Hamiltonian path. (3) A characterization of connected subgraphs of the infinite hexagonal lattice which are branching graphs of benzenoid systems. (4) A proof that if a disconnected subgraph G of the infinite hexagonal lattice given along with the positions of its vertices is the branching graph of a benzenoid system H, then H is unique.

Equatorial X-ray diffraction by fibrous proteins: Short-range order in collagen, feather keratin and F-actin

It is pointed out that the observed spacings of the equatorial diffraction patterns from five fibrous proteins can be described by a simple general equation; the diffracting systems giving rise to these reflections are termed one-dimensional. Various two-dimensional packing arrangements of cylindrically symmetrical diffracting units that are restricted in lateral extent can give rise to one-dimensional diffraction patterns; the reflections from such systems are termed non-Bragg maxima. For a restricted hexagonal lattice two sets of non-Bragg maxima are possible. One set is at higher spacings than the 10ī0 spacing of the infinite hexagonal lattice, while the other set differentiates into the normal hexagonal set as the lattice size increases. The theory is applied to the equatorial diffraction of collagen and an alternative lattice is suggested to the cylindrical lattice of Ramachandran & Sasisekharan (1956) . New suggestions are made for the structures of both feather keratin and F-actin.