The journal article reviewed here is titled “Ising Model on Mixed two-
dimensional Lattices” by W. Lebrecht, E.E. Vogel, and J.F. Valdes. The paper was
published in volume 320 of Physica B in July 2002 pages 343 – 347. The purpose of the
paper is to find the ground level of Ising model for Kagome lattices (coordination number
of 4) and five-point star lattices (coordination number of 5). The basic principals applied
are to randomly assign ferromagnetic (-J) and anti-ferromagnetic (+J) nearest neighbor
interactions to each atom in each lattice and vary the concentration of this mixed
ferromagnetic and anti-ferromagnetic interactions between zero and one while having a
periodic boundary condition for N number of spins. With this setup, two methods are
utilized to find frustration segment, energy per interaction, fraction of non-frustrated
bonds in the entire ground manifold over total number of bonds, and frustration of
plaquettes. The methods used are: 1) numerical method related to multiple replicas and 2)
analytical method based on probabilistic analysis of flat or curved plaquettes. The results
from both these methods and results from honeycomb (HL), square (SL), and triangular
(TL) lattices (previously calculated) are used to form conclusions about the effects of
inhomogeneous geometry and coordination numbers.
Ising model is defined on a collection of discrete variables called spins. These
spins can take on the value 1 or -1. This model is especially helpful in dealing with short
range interactions which consequently gives rise to frustration in systems. In this study
ferromagnetic (F) interactions (-1 or –J) and anti-ferromagnetic (AF) interactions (+1 or
+J) are assigned randomly with x concentration for F and (1 – x) for AF. Two lattices are
studied here that have inhomogeneous geometries. The first is the Kagome Lattice (KL).
This lattice consists of a hexagonal honeycomb center with six triangular plaquettes
around it which has a coordination number of 4. The second lattice is the five-point star
lattice (FPSL) which consists of a square plaquette surrounded by four triangular
plaquettes with coordination number of 5. For both these lattices an array of L by M cells
is created which constitutes a total of N = (C – 1) L * M spins.
One method used to study these lattices is a numerical method. It uses the Ising
Hamiltonian equation and an algorithm (algorithm not discussed in the paper) to find all
lowest energy states.
With these lowest energy states known some physical properties can be found. Such
physical properties include frustration segment, energy per interaction, and fraction of
non-frustrated bonds in the entire ground manifold over total number of bonds. The
average energy per interaction is found to be slightly higher for FPSL than KL. This is
believed to be a result of higher coordination number of FPSL. These values are -0.66 for
FPSL and -0.63 for KL which fall within the range for TL and HL. The fraction of non-
frustrated bonds in the manifold is found to be 0.5 for KL and 0.7 for FPSL. The large
difference between the two values is due to the difference in coordination numbers. And
finally the average frustration segment is found to be 1.34 for KL and 1.22 for FPSL.
This difference is a result of the position and orientation of the triangles in each lattice.
The second method used in this study is a theoretical approach. Each lattice has a
specific number of interactions (KL = 10 and FPSL = 14). The process starts with zero
AF bond and weights the probability as a function of x. Then it goes to one AF bond and
does the same thing. It goes through all the different configurations of two, three, four,
etc. AF bonds and weights those probabilities. The sum of all configurations that results
in a frustrated plaquette gives a probabilistic frustration as a function of x. The general
equation from this approach is
The actual equations for the lattices studies here are
When plotted, the theoretical approach for each lattice matches almost exactly the
numerical simulation results previously done for each lattice. As further verification of
the validity of the equations, it is noted that at x = 0.5 the probability of a frustrated
plaquette is exactly 0.5 as is expected and at x = 0 the probability is 2/3 which
corresponds to only frustrated triangular plaquettes. Though the two lattices are very
different geometrically, they follow about the same tendencies. This is not what is
expected since each of the homogenous plaquettes (TL, HL, and SL) behave less similar.
Based on the analysis and calculations listed above several conclusions about
these types of lattices can be drawn. The two inhomogeneous lattices with their
incredibly different overall geometries have very similar distribution of curved plaquettes
which greatly differs from homogenous lattices (honeycomb, square, and triangular
lattices). They both also have very similar energy per interaction. However, these
exceptional similarities are not cause to create a specific class for these lattices.
Coordination number plays a big roll in the unfrustrated portion of each lattice. Odd
coordination numbers create a larger number of unfrustrated interactions than even
coordination numbers. Also, a smaller frustration segment is found in FPSL in
comparison to KL which is a direct result of the locations and orientation of the triangles
in each lattice. Therefore one can conclude that in spite their uniqueness in some ways,
these inhomogeneous lattices still have properties correlated to homogenous lattices.