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Journal of Physics C Solid State Physics

Spin wave damping in Heisenberg ferromagnets near the percolation threshold

A Christou, Robin B. Stinchcombe

Published Oct 20, 1986• 13 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/19/29/012

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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 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.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.
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.
The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta T =1. This function displays the crossover from spin wave (k- pi ) xi p >>1 to hydrodynamic (k- pi ) xi p <<1 response. The relevance of the results to the experimental situation is discussed.
A numerical study of the spin-wave density of states on the infinite percolating cluster of the site diluted Heisenberg ferromagnet is presented. The Gaussian elimination algorithm is employed for the accurate determination of the integrated density of states as a function of concentration in large two-dimensional strips. At the percolation threshold the energy dependence of the density of states is \ensuremath{\rho}(E)\ensuremath{\propto}${E}^{\mathrm{\ensuremath{-}}\ensuremath{\varphi}}$. We obtained the value of \ensuremath{\varphi}=0.32\ifmmode\pm\else\textpm\fi{}0.02 consistent with the presence of fracton excitations at all length scales, in agreement with previous estimates. For concentrations above pc{p}_{c}pc​ the density of states is constant in the magnon or long-wavelength regime while a crossover to the fracton regime occurs at shorter wavelengths. The magnon-fracton crossover in energy is smooth and no indication of a fracton-edge peak is found for this system.
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.