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Journal of Physics A Mathematical and General

Critical dynamics in Heisenberg ferromagnets and antiferromagnets near the percolation threshold

Iveta R. Pimentel, Robin B. Stinchcombe

Published Sep 21, 1989• 4 citations
Publisher

Abstract

The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.

DOI
10.1088/0305-4470/22/18/029

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The damping of spin waves in bond-diluted Heisenberg antiferromagnets in d dimensions is considered using a method developed for the ferromagnetic case by the authors. They treat the hydrodynamic limit q xi identical to (k- pi ) xi <<1 and derive an expression for the damping Gamma c (q, xi ) varies as (p-p c ) - mu q d+1 , where mu = nu d- 1 / 2 (2t- beta -(d-2) nu ) which is consistent with the dynamic scaling principle. They also obtain a closed-form expression for the response function and consider finite-cluster response. The critical dynamic properties of the corresponding site-diluted system are then examined. Finally, they contrast d=1 and d>1 and then generalise the results to the ferrimagnet.
We investigate the quantum dynamics of site diluted S=1/2 Heisenberg antiferromagnetic clusters at the 2D percolation threshold. We use Lanczos diagonalization to calculate the lowest excitation gap Delta and, to reach larger sizes, use quantum Monte Carlo simulations to study an upper bound for Delta obtained from sum rules involving the staggered structure factor and susceptibility. Scaling the gap distribution with the cluster length L, Delta approximately L(-), we obtain a dynamic exponent z approximately 2D(f), where D(f)=91/48 is the fractal dimensionality of the percolating cluster. This is in contrast with previous expectations of z=D(f). We argue that the low-energy excitations are due to weakly coupled effective moments formed due to local imbalance in sublattice occupation.
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The decay of zero temperature spin wave excitations near the percolation threshold in bond-diluted d-dimensional (d>1) Heisenberg ferromagnets is investigated using a Green function equation-of-motion method. The diagrammatic perturbation theory is convergent in the hydrodynamic limit (k xi <<1) and the following expression is obtained there for the damping, or decay rate: Gamma c (k, xi ) varies as (p-p c ) -vd+(t- beta ) k d+2 ,(k xi <<1). This form satisfies the requirements of the dynamic scaling principle, by which Gamma c (k, xi )=k z g(k xi ) for general k xi , and z=2+(t- beta )/ nu . An explicit expression, applicable in the hydrodynamic regime and consistent with dynamic scaling, is also obtained for the infinite cluster transverse dynamic structure function. The approach is extended to incorporate the ensemble of finite clusters and to treat the case of non-zero temperature and also the experimentally more relevant situation of site dilution.
The critical dynamics of Heisenberg spin systems at the percolation threshold is obtained from the transformation of the equations of motion under a length scaling acheived by climination of sites. The method is applied exactly to ddd-dimensional nonrandom fractals representing the infinite-cluster backbone and approximatcly to a truly random twodimensional system to yield dynamic scaling forms for response function and characteristic frequency, and associated static and dynamic critical exponents.
The fracton dynamics of percolating Heisenberg antiferromagnets are studied in terms of a dynamic scaling argument and large-scale numerical analysis. It is shown that the spectral (or fracton) dimension for antiferromagnetic fractons, d~AF\tilde{d}_{\rm AF}d~AF​, is bounded by the relation d~AF≤1\tilde{d}_{\rm AF}\le 1d~AF​≤1. The densities of states (DOS) are calculated for very large percolating antiferromagnets at the critical concentration p c formed on d =2, 3, and 4 simple cubic lattices. It is concluded, from the calculated DOS's, that the spectral dimensions d~AF\tilde{d}_{\rm AF}d~AF​ are very close to unity for any Euclidean dimension d (≥2). We have also calculated the DOS for percolating antiferromagnets above p c in order to clarify the crossover behavior from extended magnons to fractons. In addition, the dynamical structure factor S ( q ,ω) is investigated both analytically and numerically. It is found, postulating a single-length scaling, that the asymptotic behaviors can be characterized by two exponents, the dynamical exponent z AF and a new exponent y . The numerical results for S ( q ,ω) support the validity of the single-length-scale postulate.