Local analysis of frustration based on Kagomé lattices

Abstract

Magnetic frustration in the framework of the Edwards–Anderson model is studied for the ground level ( T = 0 ) of the Kagomé lattice (KL). A sample consists of a realization of a random distribution of ferromagnetic (F) and antiferromagnetic (AF) interactions of the same strength along lines connecting nearest neighbors. Our goal is to compare two methods to calculate the following parameters: ground state energy, frustration and average frustration segment as functions of x , the concentration of F bonds. In doing so we make use of topological concepts such as plaquettes and frustration segments. The probability of a plaquette being unfrustrated (or flat) is ℘ p ( x ) , while the probability of a plaquette being frustrated (or curved) is ℘ c ( x ) . The analysis is done locally on a representative portion of the lattice which is called cell; cells of two different sizes are used in the present work. One method (which is simpler) is based on the probability function of any plaquette configuration ϕ ( ℘ p , ℘ c ) . The other method is more exact but also more complex and increasingly difficult to use for large cells; it is based on the probability of bond configurations ψ ( x ) . These methods are compared between themselves noting that the simpler method can be enough for most of the range for x . In addition, numerical simulations for many random samples at different concentrations x for a size given by 75 spins with periodic boundary conditions were done. This provides reference lines to compare with the properties under study. The local frustration analysis to obtain both ϕ ( ℘ p , ℘ c ) and ψ ( x ) is done over two cells of different size. Robustness of the criteria used in the local frustration analysis is also investigated.