Planar Triangular Lattice Research Articles

Recent developments in the magnetic study of the deformed pyrochlore lattice M2(OH)3X (M = 3d magnetic ions, X = Cl, Br) - exotic magnetic order in Ni2(OH)3Cl and controlled spin-spin interactions in Co2(OH)3Cl1-xBrx and (Co1-xFex)2(OH)3Cl

Following the discovery of frustrated magnetism in deformed pyrochlore lattice Cu2(OH)3Cl and Co2(OH)3Cl we have extensively investigated the material series in the chemical formula of M2(OH)3X, with M = Cu, Co, Ni, Fe, Mn, and X = Cl, Br, or I. In atacamite-structure Ni2(OH)3Cl, strong geometric frustration and an exotic antiferromagnetic transition below 5 K was found. While neutron diffraction witnessed unambiguously an antiferromagnetic long-range order, the μSR method can't ‘see’ this order, instead, the detected local field behaved quite like a dynamically fluctuating one. For the system of Co2(OH)3Cl, the magnetic state is very sensitive to both the anion and cation substitution. While Co2(OH)3Cl behaves like a zero-field kagomé ice ferromagnet, a completely substituted version of Co2(OH)3Br becomes antiferromagnetic although there is little difference in the crystal structure. The antiferromagnetic Co2(OH)3Br showed complicated magnetic transitions. Meanwhile, partially substituted Co2(OH)3Cl1-xBrx transforms from ferromagnetic to antiferromagnetic with increasing the x ratio. The results suggest that the interaction on the kagome-lattice plane is antiferromagnetic while that on the triangular lattice plane is ferromagnetic. For the substituted series (Co1-xFex)2(OH)3Cl a spin glass state is observed.

Single Domain Magnetic Helicity and Triangular Chirality in Structurally EnantiopureBa3NbFe3Si2O14

A novel doubly chiral magnetic order is found in the structurally chiral langasite compound Ba3NbFe3Si2O14. The magnetic moments are distributed over planar frustrated triangular lattices of triangle units. On each of these they form the same triangular configuration. This ferrochiral arrangement is helically modulated from plane to plane. Unpolarized neutron scattering on a single crystal associated with spherical neutron polarimetry proved that a single triangular chirality together with a single helicity is stabilized in an enantiopure crystal. A mean-field analysis allows us to discern the relevance on this selection of a twist in the plane to plane super-superexchange paths.

Magnetoelectric control of spin-chiral ferroelectric domains in a triangular lattice antiferromagnet

We have grown single crystals of a triangular lattice antiferromagnet (TLA), ${\text{CuCrO}}_{2}$, and investigated the correlation between magnetic and dielectric properties. Two magnetic phase transitions are observed at ${T}_{\text{N}2}\ensuremath{\approx}24.2\text{ }\text{K}$ and ${T}_{\text{N}1}\ensuremath{\approx}23.6\text{ }\text{K}$. It was found that ferroelectric polarization along the triangular lattice plane develops at ${T}_{\text{N}1}$, suggesting that the system undergoes a transition into an out-of-plane $120\ifmmode^\circ\else\textdegree\fi{}$ spin-chiral phase at ${T}_{\text{N}1}$. The TLA provides an opportunity for unique magnetoelectric control of spin-chiral ferroelectric domain structures by means of electric and/or magnetic fields.

Magnetodielectric phenomena in a charge- and spin-frustrated system of layered iron oxide

Dielectric and magnetic phenomena in spin- and charge-frustrated system $R{\text{Fe}}_{2}{\text{O}}_{4}$ ($R$ is a rare-earth metal ion) are studied. An electronic model for charge, spin, and orbital degrees in a pair of triangular-lattice planes is derived. We analyze this model by utilizing the mean-field approximation and the Monte Carlo simulation in a finite size cluster. A threefold-type charge-ordered structure with charge imbalance between the planes is stabilized in finite temperatures. This polar charge order is reinforced by spin ordering of Fe ions. This magnetodielectric phenomenon is caused by spin frustration and charge-spin coupling in the exchange interaction. We show cross-correlation effect in magnetic- and electric-field responses. Oxygen-deficiency effect as an impurity effect in a frustrated charge-spin coupled system is also examined.

On the Packing Chromatic Number of Trees, Cartesian Products and Some Infinite Graphs

Abstract A subset A of vertices in a graph G is an r-packing if the distance between every pair of distinct vertices in A is more than r. The packing chromatic number, χ ρ ( G ) , is the smallest k for which there exists some partition V 1 , V 2 , … , V k of the vertex set of G such that V r is an r-packing. We obtain lower and upper bounds for the packing chromatic number of Cartesian products and subdivisions of finite graphs and study the existence of monotone colorings in trees. The infinite, planar triangular lattice and the three dimensional square lattice are shown to have unbounded packing chromatic number.

NMR Study of Spin State in Geometrically Frustrated Compound Co2Cl(OH)3

NMR studies of a geometrically frustrated system, Co 2 Cl(OH) 3 , are carried out at low temperature. The NMR results of 59 Co and 1 H nuclei show that long-range ordered ferromagnetic moments on the triangular lattice plane coexist microscopically with disordered moments on the kagome lattice plane. The moment direction is not ordered. Although the disordered moments freeze within the NMR observation time (10 -4 s) at low temperature, the frozen moments begin to fluctuate with increasing temperature. Up to 4.2 K (0.4 T C ), the magnitude of the ferromagnetically ordered moments is independent of temperature.

Permuting and Lifting Wavelet Coding for Structured Geometry Data of 3-D Polygonal Mesh

This paper presents a lifting wavelet coding technique with permutation and coefficient modification processes for coding the structured geometry data of 3-D polygonal mesh model. One promising method for coding 3-D geometry data is based on the structure processing of a 3-D model on a triangle lattice plane, while maintaining connectivity. In the structuring process, each vertex may be assigned to several nodes on the triangular lattice plane. One of the nodes to which a vertex is assigned is selected as a representative node and the others are called expanded nodes. Only the geometry data of the vertices at the representative nodes are required for reconstructing the 3-D model. In this paper we apply a lifting wavelet transform with a permutation process for an expanded node at an even location in each decomposition step and the neighboring representative node. This scheme arranges more representative nodes into the lower frequency band. Also many representative nodes separated from the connective expanded nodes are made to adjoin each other in lower frequency bands, and the correlation between the representative nodes will be reduced by the following decomposition process. A process is added to use the modified coefficients obtained from the coefficients of the adjacent representative nodes instead of the original coefficients in the permutation process. This has the effect of restraining increases in the decomposed coefficients with larger magnitude. Some experiments in which the proposed scheme was applied to structured geometry data of a 3-D model with complex connectivity show that the proposed scheme gives better coding performance and the reconstructed models are more faithful to the original in comparison with the usual schemes.

Frustrated magnetism in superconducting hexagonal Fe: Calculation of inter-atomic pair exchange interactions

We calculate magnetic exchange interaction parameters in hcp Fe using magnetic force theorem in density functional theory in a range of volumes, which covers the region of stability of the superconducting state. The calculations are done for a paramagnetic state using disordered local moment model within a local fixed spin-moment constraint. It is found that exchange interaction parameters strongly favor antiferromagnetic inter-plane stacking of the basal hcp planes. At the same time, the nearest neighbor in-plane exchange interaction parameters are also strongly antiferromagnetic, which should lead to magnetic frustrations in the planar triangular lattice. Eventual consequences of the magnetic frustration effects on the interplay between magnetism and superconductivity in hcp Fe under pressure are discussed.

Antiferromagnetism and geometric frustration in tetrahedral lattice hydroxyhalidesM2(OH)3X

A systematic study of magnetic susceptibility measurementsis carried out on distorted tetrahedral lattice hydroxyhalidesFe2(OH)3Cl,Mn2(OH)3Cl,Mn2(OH)3Br,and Co2(OH)3Br, following our recent finding of coexisting antiferromagnetic order and disorder inCu2(OH)3Cl (clinoatacamite). These transition metal hydroxyhalides are found to undergo antiferromagnetic transitions,at TN = 14 K forFe2(OH)3Cl, at two successivetransitions of TN1 = 3.4 K and TN2 = 2.7 K for Mn2(OH)3Cl, at TN1 = 3.3 Kand TN2 = 2.4 Kfor Mn2(OH)3Br,and at TN = 5 K for Co2(OH)3Br, respectively. All hydroxychlorides show coexisting glassiness below the magnetictransitions; meanwhile, the glassiness is not found in the hydroxybromides. The distortedtetrahedral lattice for the magnetic ions consists of stacked layers of triangularlattice planes and Kagome lattice planes, with the magnetic ions on the Kagomelattice planes being bonded by the halogen ions. The contrasting behaviours inMn2(OH)3Cl and Mn2(OH)3Br suggest that the glassiness arises from competing magnetic interactions in the tetrahedron,specifically on the Kagome lattice planes.

Successive Magnetic Phase Transitions in α-Tb2S3 Studied by Neutron Diffraction Technique

Neutron diffraction measurements on a powder sample of α-Tb 2 S 3 were performed. This material consists of a stacking of square lattice Tb1 and triangular lattice Tb2 planes and exhibits two magnetic phase transitions at ∼3.5 and ∼12 K. It was found that the Tb1 and Tb2 moments behave quite independently. The Tb1 moments show a collinear antiferromagnetic ordering at ∼12 K with the magnetic unit cell same as the chemical one. On the other hand, the Tb2 moments start to order at ∼3.5 K without affecting the magnetic order of the Tb1 moments. The magnetic structure is noncollinear with a magnetic unit cell double of the chemical one along the a - and c -axes. The ordered moments of the Tb1 and Tb2 moments are ∼8µ B , similar to the full moment for the free Tb 3+ ion. These results indicate that the Tb1 and Tb2 moments are completely decoupled due to an orbital contribution, which probably gives rise to a noncollinear structure in the triangular lattice plane. The results are discussed in comparison with th...

Exact finite-size scaling functions for the interfacial tensions of the Ising model on planar lattices

Exact finite-size scaling functions of the interfacial tensions are obtained for the Ising model with isotropic coupling on a set of M x N planar lattices, including square (sq), plane triangular (pt), and honeycomb (hc) lattices. The analyses of transitive behaviors at criticality revise the knowledge of the interfacial tensions as a function of the aspect ratio defined by R = M/N for R approaching to zero gradually. The amplitudes of the interfacial tensions for the sq, pt, and hc lattices are further shown to have relative proportions 1:square root of 3 / 2:square root of 3 which are related to the aspect ratios for the three lattices to have similar domains.

Structure of spherical Yukawa clusters

Structure of clusters of Yukawa particles is analysed by simulations and theoretical approaches in an isotropic environment which can be realized under microgravity or by active cancellation of the effect of gravity. Yukawa particles model dust particles in dusty plasmas and such an environment is suitable to observe their inherent properties. At low temperatures, clusters are composed of spherical shells and, when scaled by the mean distance, the structure seems to depend almost only on the system size or the number of particles. The positions and populations of shells are given by simple interpolation formulae. It is shown that shells have an approximately equal spacing close to that of triangular lattice planes in the bulk close-packed structures. When the cohesive energy in each shell is properly taken into account, the shell model reproduces these structures of spherical Yukawa clusters to a good accuracy.

Structure of spherical Yukawa clusters: A model for dust particles in dusty plasmas in an isotropic environment

The structure of spherical clusters composed of Yukawa particles is analyzed by molecular dynamics simulations and theoretical approaches as a model for dust particles in dusty plasmas in the isotropic environment. The latter condition is expected to be realized under microgravity or by active cancellation of the effect of gravity on the ground. It is found that, at low temperatures, Yukawa particles form spherical shells and, when scaled by the mean distance, the structure is almost independent of the strength of screening including the case of the Coulomb interaction. The positions and populations of shells and the conditions for the change of the number of shells are expressed by simple interpolation formulas. Shells have an approximately equal spacing close to that of triangular lattice planes in the bulk close-packed structures. It is shown that, when the cohesive energy in each shell is properly taken into account, the shell model reproduces the structure of spherical Yukawa clusters to a good accuracy.

Field-Induced Phase Transitions Driven by Quantum Fluctuation inS=1/2 Anisotropic Triangular Antiferromagnet Cs2CuBr4

The field induced magnetic phase transitions of Cs 2 CuBr 4 were investigated by means of magnetization process and neutron scattering experiments. Cs 2 CuBr 4 should be characterized as S = 1/2 two-dimensional triangular antiferromagnet. Below the ordering temperature T N = 1.4 K, the Spin structure is the helical incommensurate structure almost within the triangular lattice plane. In the field direction within the triangular lattice plane, Cs 2 CuBr 4 exhibits the magnetization plateaux at one-third and two-thirds of the saturation magnetization. The spin structure in the one-third plateau phase is found to be almost collinear up-up-down structure which should be stabilized by quantum fluctuation as predicted by the theoretical studies.

Cut‐and‐Unfold Approach to Fullerene Enumeration.

A simple and effective approach for the enumeration of fullerene structures is proposed. The method combines the formalism of a fullerene graph cut-and-unfold onto a planar triangular lattice and a topological description of closed quasi-2D clusters. A tabulation of possible fullerene graphs Cn is given for the number of atoms 20 < or = n < or = 150.

Cut-and-unfold approach to fullerene enumeration.

A simple and effective approach for the enumeration of fullerene structures is proposed. The method combines the formalism of a fullerene graph cut-and-unfold onto a planar triangular lattice and a topological description of closed quasi-2D clusters. A tabulation of possible fullerene graphs Cn is given for the number of atoms 20 < or = n < or = 150.

Universal finite-size scaling functions with exact nonuniversal metric factors

Using exact partition functions and finite-size corrections for the Ising model on finite square, plane triangular, and honeycomb lattices and extending a method [J. Phys. 19, L1215 (1986)] to subtract leading singular terms from the free energy, we obtain universal finite-size scaling functions for the specific heat, internal energy, and free energy of the Ising model on these lattices with exact nonuniversal metric factors.

Ground states of adsorbates on single-walled carbon nanotubes

We have investigated the ground states of single-walled carbon nanotubes with atoms adsorbed in the center of the hexagonal unit using the Monte Carlo method. We have determined the ground state structures of adsorbate on carbon nanotubes with different diameters. We have shown that the confinement has strong effects on the ground state structures. For the case of the zigzag $(n,0)$ nanotubes, when $n+1$ can be divided exactly by 6, the ground state structures is the same as those of planar triangular lattice. Otherwise, novel ground state structures are found as a result of the size confinement of the system. We have also compared the results of zigzag type structures with those of armchair type. It is shown that the rolling way influences the ground state structures.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Exact partition functions of the Ising model on M $times$ N planar lattices with periodic$ndash$aperiodic boundary conditions

The Grassmann path integral approach is used to calculate exact partition functions of the Ising model on MxN square (sq), plane triangular (pt) and honeycomb (hc) lattices with periodic-periodic (pp), periodic-antiperiodic (pa), antiperiodic-periodic (ap) and antiperiodic-antiperiodic (aa) boundary conditions. The partition functions are used to calculate and plot the specific heat, $C/k_B$, as a function of the temperature, $\theta =k_BT/J$. We find that for the NxN sq lattice, $C/k_B$ for pa and ap boundary conditions are different from those for aa boundary conditions, but for the NxN pt and hc lattices, $C/k_B$ for ap, pa, and aa boundary conditions have the same values. Our exact partition functions might also be useful for understanding the effects of lattice structures and boundary conditions on critical finite-size corrections of the Ising model.