W. Lebrecht

Frustration in mixed two-dimensional ±J Ising lattices

Method of the sublattice previously introduced for homogeneous lattices is adapted here to characterize ground state properties of two inhomogeneous lattices: Kagome lattice with coordination 4 and Five-points-star lattice with coordination 5. A representative cell must be chosen in each case in such way that main geometrical and topological properties of the lattice are well represented. Ferromagnetic interactions (in concentration x) and antiferromagnetic interactions (in concentration 1−x) define different possible configurations in the cell. By means of combinatorial and probability analysis weight functions are obtained allowing to calculate properties such as frustration length, energy per bond, and fractional content of unfrustrated interactions for the entire ground manifold. We report analytic expressions as functions of x which are later compared to average results coming from numerically solving 40000 randomly prepared samples in an exact way. The good agreement between numerical and theoretical analysis validates the use of the method for inhomogeneous lattices and helps to interpret results. Values for these topological and physical parameters are also compared to those available for homogeneous two-dimensional lattices, looking for general trends. Roles of coordination number, plaquette shape and topology in the previously mentioned properties are discussed and established.

Ising model on mixed two-dimensional lattices

We inform results on physical and topological magnitudes related to the ground level of Ising model on mixed two-dimensional lattices of coordination numbers 4 (Kagome lattices) and 5 (five-point star lattices). We consider little clusters of size N, where N represents the total number of spins, subject to periodic boundary conditions. On these systems we randomly distribute ±J nearest-neighbor interactions (+J: antiferromagnetic, −J: ferromagnetic (F)). Concentration x of F interactions is varied in the interval (0,1). Two different methods are used to obtain results reported here. First, a numerical method related to multiple replicas. Second, an analytical method based on probabilistic analysis of flat and curved plaquettes. Both methods are complementary to each other. Initially, this study is restricted to calculate frustration of plaquettes and bonds, energy and bond order parameter at T=0. The results of magnitudes informed here are compared with the similar ones obtained for honeycomb, square and triangular lattices.

Algoritmo Matemático para Cuantificar Directamente el Ángulo de Convergencia en Preparaciones de Dientes Artificiales

El objetivo de este trabajo fue determinar un algoritmo matematico para cuantificar directamente el angulo de convergencia (AC) en troqueles de preparaciones dentarias. El modelo experimental consistio en preparaciones coronarias simuladas sobre troqueles de yeso, en el cual el AC fue calculado por tres formulas trigonometricas. Las formulas fueron obtenidas de un modelo matematico en el cual la preparacion coronaria representa una forma de piramide truncada, la cual permite una proyeccion triangular en un plano. Fueron realizadas 60 mediciones in situ sobre las paredes de 15 troqueles. Se obtuvo una imagen de cada troquel usando una camara digital (Schick® CDR). El AC fue medido usando la herramientas del software (Control). Los datos fueron analizados estadisticamente y se aplicaron test de propagacion de errores. Los angulos calculados con las tres formulas matematicas ([F1], [F2] y [F3]) mostraron un alto nivel de correlacion con el grupo control excepto para dos muestras. Dentro de las limitaciones de este estudio podemos concluir que a traves de este algoritmo matematico, es posible cuantificar directamente el AC de las preparaciones coronarias en troqueles. Actualmente la evaluacion de los AC tanto en preparaciones realizadas por alumnos de pregrado de odontologia como por dentistas, se hacen de manera subjetiva. Las tres formulas presentadas en el algoritmo tiene una correlacion alta para cuantificar el AC en troqueles. La [F3], es la que mas correlacion logra en todas las muestras (0,89).

±J Ising model on Archimedean (4,82) lattices

We report results on ground state properties for a ±J±J Ising model defined on the Archimedean (4,82)(4,82) lattice. The sublattice method is adapted to this system. By means of combinatorics and probability analysis, weight functions are obtained allowing to calculate properties such as frustrated plaquette distribution, frustration length, energy per bond, and fractional content of unfrustrated bonds; these analytic expressions are presented as functions of x (concentration of ferromagnetic bonds). On the other hand, these parameters are also calculated by an exact numerical algorithm applied to a large number of samples for increasing size N (number of spin sites) and values of x in the range [0.0,1.0]. Analytical and numerical results tend to agree, which makes these two techniques complementary to each other. Finally, comparison is made to results previously reported for other Archimedean lattices.

Site-bond percolation on simple cubic lattices: numerical simulation and analytical approach

The site-percolation problem on simple cubic lattices has been studied by means of numerical simulation and analytical calculations based on exact counting of configurations on finite cells. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as and ) have been considered. In (), two points are said to be connected if a sequence of occupied sites and (or) bonds joins them. Theoretical and numerical results, supplemented by analysis using finite-size scaling theory, were used to calculate the complete phase diagram of the system in the () space. Our study allowed us also to determine the critical exponents (and universality) characterizing the phase transition occurring in the system.

±J±J Ising model on Archimedean (4,82)(4,82) lattices

We report results on ground state properties for a ±J±J Ising model defined on the Archimedean (4,82)(4,82) lattice. The sublattice method is adapted to this system. By means of combinatorics and probability analysis, weight functions are obtained allowing to calculate properties such as frustrated plaquette distribution, frustration length, energy per bond, and fractional content of unfrustrated bonds; these analytic expressions are presented as functions of x (concentration of ferromagnetic bonds). On the other hand, these parameters are also calculated by an exact numerical algorithm applied to a large number of samples for increasing size N (number of spin sites) and values of x in the range [0.0,1.0]. Analytical and numerical results tend to agree, which makes these two techniques complementary to each other. Finally, comparison is made to results previously reported for other Archimedean lattices.

Ising model on Archimedean lattices

We report results on ground state properties for a ±J±J Ising model defined on the Archimedean (4,82)(4,82) lattice. The sublattice method is adapted to this system. By means of combinatorics and probability analysis, weight functions are obtained allowing to calculate properties such as frustrated plaquette distribution, frustration length, energy per bond, and fractional content of unfrustrated bonds; these analytic expressions are presented as functions of x (concentration of ferromagnetic bonds). On the other hand, these parameters are also calculated by an exact numerical algorithm applied to a large number of samples for increasing size N (number of spin sites) and values of x in the range [0.0,1.0]. Analytical and numerical results tend to agree, which makes these two techniques complementary to each other. Finally, comparison is made to results previously reported for other Archimedean lattices.

Analytic and Computational Analysis for Bond Percolation on Two‐Dimensional Lattices

Beginning in the 60t’s it has been a growing number of studies referred to disordered systems. Percolation theory has been one of the most popular techniques applied to solving these problems. We report here an analytical method to deal with percolation in two‐dimensional lattices. In particular, we inform results on bond percolation related to square, triangular and honeycomb lattices, selecting different types of cells associated to each geometry. Thus, we choose two cells related to square lattices whose sizes are 5 [1] and 13 [2] and two cells associated to triangular lattices with sizes 7 and 19 respectively. For the case of honeycomb lattices we have considered one cell of size 15. For each cell, we calculate probabilities of generating bond percolation along one direction. We must include all possible configurations associated to each cell, where p represents an occupied bond and 1 − p represents an empty bond, according to usual treatments [1]. Percolation function, p′, is expressed as a polynomia...

Size and Shape Dependence for Triangular and Honeycomb Finite Lattices with Mixed Exchange Interactions

By means of exact numerical calculations, the properties of small lattices with mixed exchange interactions (or bonds) are studied. A particular bond can be either ferromagnetic (F) or antiferromagnetic (AF). We assume equal magnitudes and equal concentrations for each kind of bonds. An important independent variable is the number of spins (N) or size of the lattice. We consider here two different two-dimensional geometries: triangular lattices (TL) and honeycomb lattices (HL).1 The maximum size is 42 for TL and 64 for HL. The distribution of these spins considers all possible rectangular and square arrays. The shape is an interesting independent variable. Each distribution of bonds for a given array is a sample. Once the sample is generated no mutation or migration of bonds is allowed. Thus for each sample the distribution of bonds is fixed. Periodic boundary conditions are used to keep the coordination number constant through the lattice. To get a statistical representation of these systems, 500 samples are considered for each array. Our main interest is to study the properties of the ground level of these lattices and their dependence with size and shape. In the present article we report the following properties for TL and HL: average energy per bond, remnant entropy, and order parameters p and h.2 The results agree well with the expected behavior toward the thermodynamic limit. A comparison with similar results reported for square lattices (SL) will be also performed. A deeper discussion is carried out for the recently defined order parameters p and h.2,3 Open image in new window Figure 1 Distribution of sites for a triangular lattice (filled circles) and for a honeycomb lattice (open circles).