Site dilution of quantum spins in the honeycomb lattice

Abstract

We discuss the effect of site dilution on both the magnetization and the density of states of quantum spins in the honeycomb lattice, described by the antiferromagnetic Heisenberg spin-S model. Since the disorder introduced by the dilution process breaks translational invariance, the model has to be solved in real space. For this purpose a real-space Bogoliubov-Valatin transformation is used. In this work we show that for the S > 1/2 the system can be analyzed in terms of linear spin wave theory, in the sense that for all dilution concentrations the assumptions of validity for the theory hold. For spin S = 1/2, however, the linear spin wave approximation breaks down. In this case, we have studied the effect of dilution on the staggered magnetization using the Stochastic Series Expansion Monte Carlo method. Two main results are to be stressed from the Monte Carlo method: (i) a better value for the staggered magnetization of the undiluted system, mav(L → ∞) = 0.2677(6), relatively to the only result available to date in the literature, and based on Trotter error extrapolations; (ii) a finite value of the staggered magnetization of the percolating clustern at the classical percolation threshold, showing that there is no quantum critical transition driven by dilution in the Heisenberg model. In the solution of the problem using linear the spin wave method we pay special attention to the presence of zero energy modes. We show, for a finite-size system (in a bipartite lattice), that if the two sub-lattices are evenly diluted the system always has two zero energy modes, which play the role of Goldstone boson modes for a diluted lattice, having no translation symmetry but supporting long range magnetic order. We also discuss the case when the two sub-lattices are not evenly diluted. In this case, for finite size lattices, the Goldstone modes are not a well defined concept, and special care is needed in taking them into account in order for sensible physical results can be obtained. Using a combination of linear spin wave analysis and the recursion method we were able to obtain the thermodynamic limit behavior of the density of states for both the square and the honeycomb lattices. We have used both the staggered magnetization and the density of states to analyze neutron scattering experiments (determining the effect of dilution on the system’s magnetic moment) and Néel temperature measurements on quasi- two-dimensional honeycomb systems. Our results are in quantitative agreement with experimental results on MnpZn1−pPS3 (a diluted S = 5/2 system) and on the Ba(NipMg1−p)2V2O8 (a diluted S = 1 system). Our work should stimulate further experimental research in Heisenberg diluted honeycomb systems.