J.F. Valdés

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.

Thermodynamics of Small Magnetic Particles

In the present paper, we discuss the interpretation of some of the results of the thermodynamics in the case of very small systems. Most of the usual statistical physics is done for systems with a huge number of elements in what is called the thermodynamic limit, but not all of the approximations done for those conditions can be extended to all properties in the case of objects with less than a thousand elements. The starting point is the Ising model in two dimensions (2D) where an analytic solution exits, which allows validating the numerical techniques used in the present article. From there on, we introduce several variations bearing in mind the small systems such as the nanoscopic or even subnanoscopic particles, which are nowadays produced for several applications. Magnetization is the main property investigated aimed for two singular possible devices. The size of the systems (number of magnetic sites) is decreased so as to appreciate the departure from the results valid in the thermodynamic limit; periodic boundary conditions are eliminated to approach the reality of small particles; 1D, 2D and 3D systems are examined to appreciate the differences established by dimensionality is this small world; upon diluting the lattices, the effect of coordination number (bonding) is also explored; since the 2D Ising model is equivalent to the clock model with q = 2 degrees of freedom, we combine previous results with the supplementary degrees of freedom coming from the variation of q up to q = 20 . Most of the previous results are numeric; however, for the case of a very small system, we obtain the exact partition function to compare with the conclusions coming from our numerical results. Conclusions can be summarized in the following way: the laws of thermodynamics remain the same, but the interpretation of the results, averages and numerical treatments need special care for systems with less than about a thousand constituents, and this might need to be adapted for different properties or devices.

±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.

±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...